Research Article The Spreading of Information in Online Social Networks through Cellular Automata Yuda Wang and Gang Li School of Economics and Management, Beijing University of Posts and Telecommunications, Beijing 100876, China Correspondence should be addressed to Yuda Wang; [email protected] Received 10 September 2018; Accepted 15 October 2018; Published 1 November 2018 Academic Editor: Dimitri Volchenkov Copyright © 2018 Yuda Wang and Gang Li. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Epidemic dynamics in complex networks have been extensively studied. Due to the similarity between information and disease spreading, most studies on information dynamics use epidemic models and merely consider the characteristics of online social networks and individual’s cognitive. In this paper, we propose an online social networks information spreading (OSIS) model combining epidemic models and individual’s cognitive psychology. en we design a cellular automata (CA) method to provide a computational method for OSIS. Finally, we use OSIS and CA to simulate the spreading and evolution of information in online social networks. e experimental results indicate that OSIS is effective. Firstly, individual’s cognition affects online information spreading. When infection rate is low, it prevents the spreading, whereas when infection rate is sufficiently high, it promotes transmission. Secondly, the explosion of online social network scale and the convenience of we-media greatly increase the ability of information dissemination. Lastly, the demise of information is affected by both time and heat decay rather than probability. We believe that these findings are in the right direction for perceiving information spreading in online social networks and useful for public management policymakers seeking to design efficient programs. 1. Introduction Networks such as Internet, communication networks, and social networks can be used to describe the interconnections among individuals. Since most of these networks have non- trivial structural properties, they are called complex networks [1–4]. According to the degree of each network node, complex networks are divided into regular networks, Erdos-Renyi (ER) networks [5], small-world networks [6], and scale-free (SF) networks [7]. Among those networks, the small-world model proposed by Watts and Strogatz describes the features of high clustering and small average path length, which is most suitable for real social networks. ese small-world features [8] have been found to have a significant impact on the dynamics in networks [9]. Spreading [10] is one of the dynamical processes on com- plex networks, especially the spreading of epidemic, which has been investigated for decades and achieves a lot of the- ories. Most of these studies investigate susceptible-infected- susceptible (SIS), susceptible-infected-recovered (SIR), or susceptible-exposed-infected-removed (SEIR) model [11–15]. With the growth of the Internet scale and the convenience of we-media, information such as rumors’ spreading and evolution in online social networks has received special atten- tion [16, 17]. Due to the similarity between information and disease, the models mentioned above are widely considered. Qian [18] investigated the spreading and evolution of public opinion on Weibo [19] with SIR model. Xiong et al. [20] proposed a new model called SCIR to describe the spreading of information. Centola et al. [21] have illustrated the effects of network structure on diffusion experimentally by studying the spread of health behavior through artificially structured online communities. However, all these studies are based on a probabilistic approach, regardless of individual’s cognitive psychology and the characteristics of online social networks. Recently, researchers start to seriously take into consideration the specific features of online social networks information spreading, such as individual’s characteristics and the impact of friends around [22–31]. Nevertheless, there are more factors affecting online information spreading. Hindawi Complexity Volume 2018, Article ID 1890643, 9 pages https://doi.org/10.1155/2018/1890643
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Research ArticleThe Spreading of Information in Online Social Networksthrough Cellular Automata
Yuda Wang and Gang Li
School of Economics and Management Beijing University of Posts and Telecommunications Beijing 100876 China
Correspondence should be addressed to Yuda Wang wangydbupteducn
Received 10 September 2018 Accepted 15 October 2018 Published 1 November 2018
Academic Editor Dimitri Volchenkov
Copyright copy 2018 Yuda Wang and Gang Li This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Epidemic dynamics in complex networks have been extensively studied Due to the similarity between information and diseasespreading most studies on information dynamics use epidemic models and merely consider the characteristics of online socialnetworks and individualrsquos cognitive In this paper we propose an online social networks information spreading (OSIS) modelcombining epidemic models and individualrsquos cognitive psychology Then we design a cellular automata (CA) method to provide acomputationalmethod forOSIS Finally we useOSIS andCA to simulate the spreading and evolution of information in online socialnetworksThe experimental results indicate thatOSIS is effective Firstly individualrsquos cognition affects online information spreadingWhen infection rate is low it prevents the spreading whereas when infection rate is sufficiently high it promotes transmissionSecondly the explosion of online social network scale and the convenience of we-media greatly increase the ability of informationdissemination Lastly the demise of information is affected by both time andheat decay rather than probabilityWe believe that thesefindings are in the right direction for perceiving information spreading in online social networks and useful for public managementpolicymakers seeking to design efficient programs
1 Introduction
Networks such as Internet communication networks andsocial networks can be used to describe the interconnectionsamong individuals Since most of these networks have non-trivial structural properties they are called complex networks[1ndash4] According to the degree of eachnetwork node complexnetworks are divided into regular networks Erdos-Renyi(ER) networks [5] small-world networks [6] and scale-free(SF) networks [7] Among those networks the small-worldmodel proposed byWatts and Strogatz describes the featuresof high clustering and small average path length which ismost suitable for real social networks These small-worldfeatures [8] have been found to have a significant impact onthe dynamics in networks [9]
Spreading [10] is one of the dynamical processes on com-plex networks especially the spreading of epidemic whichhas been investigated for decades and achieves a lot of the-ories Most of these studies investigate susceptible-infected-susceptible (SIS) susceptible-infected-recovered (SIR) or
susceptible-exposed-infected-removed (SEIR)model [11ndash15]With the growth of the Internet scale and the convenienceof we-media information such as rumorsrsquo spreading andevolution in online social networks has received special atten-tion [16 17] Due to the similarity between information anddisease the models mentioned above are widely considered
Qian [18] investigated the spreading and evolution ofpublic opinion on Weibo [19] with SIR model Xiong etal [20] proposed a new model called SCIR to describe thespreading of information Centola et al [21] have illustratedthe effects of network structure on diffusion experimentallyby studying the spread of health behavior through artificiallystructured online communities However all these studies arebased on a probabilistic approach regardless of individualrsquoscognitive psychology and the characteristics of online socialnetworks Recently researchers start to seriously take intoconsideration the specific features of online social networksinformation spreading such as individualrsquos characteristicsand the impact of friends around [22ndash31] Nevertheless thereare more factors affecting online information spreading
HindawiComplexityVolume 2018 Article ID 1890643 9 pageshttpsdoiorg10115520181890643
2 Complexity
Furthermore most previous studies of dynamical pro-cesses on complex networks are based on mean field method[32] or analytical equation [33] However these approacheshas some serious drawbacks The method based on meanfield can only reflect an approximate trend frommacroscopicview which cannot accurately describe the spreading stateof each individual Meanwhile analytical equation neglectsvariable susceptibility of individuals and cannot handle thedifferent boundary and initial condition in addition it willbe very complicated for large-scale online social networksCellular automata (CA) [34 35] proposed by Von Neumanncan overcome the above drawbacks and have been used byseveral researchers as an alternative method of modelingepidemics In this model every cell of the grid representsan individual updating state of each cell by state transitioncriterion [36 37]
In summary complex networks provide theoretical basisfor social network topology and the extensive investigationof epidemic models promotes the quantitative study ofinformation spreading By referring to complex networkdynamics as well as combining the features of public opiniona breakthroughhas beenmade in the research on informationtransmission in social networks With the rapid developmentof mobile Internet and Internet of things the online socialnetwork is fundamentally different from the traditional socialnetwork The spreading and control of information espe-cially public opinion has become a difficulty and key pointconcerned by public management policymakers Meanwhiledue to the complexity of network topology and the explosionof network access nodes traditional analytical methods havebeen unable to accurately calculate the results By introducingthe cellular automata theory the bottleneck can be solvedthrough the iterative calculation
In this paper we propose an online social networksinformation spreading (OSIS) model combining individualrsquoscognitive psychology with traditional SEIRmodelMoreoverwe provide a computational method to prove the accuracyand authenticity of OSIS by cellular automata We obtainedexperimental results with the help of computer graph theoryThe results reveal that individualrsquos cognition affects onlineinformation spreading When infection rate is low cognitivepsychology prevents the spreading By contrast when infec-tion rate is sufficiently high it promotes transmission Wealso demonstrate that the explosion of online social networkscale and the convenience of we-media can greatly increasethe ability of information dissemination especially rumorsFinally we show that the demise of information is affected bythe dual effects of the decay of time and heat In particularthe major contributions of our work can be summarized asfollows
(i) a new information spreading model that takes intoaccount individualrsquos cognitive psychology and thecharacteristics of online social networks herd men-tality factor memory superposition factor and atten-tion attenuation factor
(ii) a two-dimensional CA method that can accuratelyanalyze the process of online information spreading
(iii) revealing the characteristic of online social networksinformation dissemination
The sections in this paper are structured as follows Section 1presents the background while Section 2 describes themodelSection 3 shows the design of cellular automata followed bythe experimental results in Section 4 finally we conclude ourpaper in Section 5 with the future work discussed
2 Online Social Network InformationSpreading Model
21 Analysis of Information in Online Social Network Com-paredwith traditional social networks online social networksuse Internet as the medium of communication rather thanface-to-face which makes information transmission moreeasy to forward Meanwhile in social networks such as Twit-ter information is transmitted synchronously to all friendswhich is different from the peer-to-peer communicationof offline networks The communication of information ismore highly timely Meanwhile because of the intelligence ofterminals the carrier of online information varies from textto sound picture video etcThe content becomes interestingand easy to understand Based on the features describedabove we consider the information spreading model fromthree aspects
(i) Propagation factor Due to the characteristics ofonline social networks and individualrsquos fast-foodreading habits the cognition of information dependsnot only on the content itself but also on the heatThe more the friends who are concerned around atopic the higher the probability that an individual willspread it
(ii) Memory superposition factor Online social net-works shorten the distance among people whichmakes people access the same information over acontinuous period of time we call this phenomenonmemory superposition The memory superpositionenhances peoplersquos curiosity thus prompting thespreading
(iii) Information attenuation factor In traditional epi-demic models a disease is cured via a certain prob-ability whereas the demise of information is affectedby both time and heat decay
22 Description of OSIS Model Since information spreadingis similar to epidemic we propose our online social networkinformation spreading (OSIS) model by combining the char-acteristics of online social networks mentioned above withepidemic SEIR model In OSIS each individual adopts one offour states unaware concerned interested and abandonedUnaware represents the individual who has not heard theinformation and has the potential to accept it Concernedstands for the person who is aware of the news but notwilling to transmit it currently Interested means the onebelieves in the news and has the ability to spread it Andabandoned denotes the individual who loses interests andnever transmits the information again We use complex
Complexity 3
unaware interested abandoned
concerned
Figure 1 The state transfer diagram
networks as the model of real social networks in whicheach node and edge present individual and online socialrelationship
As shown in Figure 1 the state of each node can betransformed over time Initially we randomly pick up onenode as an interested node and the rest are in unaware stateAt each time step an unaware state individual receives a topicfrom her neighbor and turns to interested or concerned stateaccording to the propagation factor Furthermore whether ornot a concerned state node turns to interested state is basedon memory superposition factor Additionally informationattenuation factor determines interested to abandoned state
23 The Algorithm Based on OSIS model described abovean algorithm for simulation of information spreading oncomplex networks has been developed The core idea inAlgorithm 1 is presented with pseudocode
(a) If a node is in unaware state and there are interestedstate nodes around propagation factor determinesstate transition In our model the disseminationof information is due to not only the informationitself but also to herd mentality For that reason thepropagation factor consists of two parts One partis due to the information itself here we adopt thealgorithm in traditional SIR modelThe infection rateis 120572 and the number of interested neighbors is m(t)at the tth time The node will have a probability of1minus(1minus120572)119898(119905) being infected The other part is causedby herd mentality which includes both positive andnegative effect We assume that for a given individualat the tth time she has k adjacent m(t) interestedand n(t) abandoned neighbors The proportion ofinterested individuals in the neighbors determinesthe positive influence of the herd effect When noone around propagates the information (ie 119898(119905)119896= 0) the influence probability of herd effect is 0While all friends are in interested state (ie 119898(119905)119896= 0) the influence of herd effect is the greatest theprobability of which is 1 Our observations show thatthe effect of this ratio is a positively exponential effectrather than a linear effect we define the effect as(1minus119890minus119898(119905)119896)(1minus1119890) By contrast if most friends arein abandoned state this may reduce the transmissi-bilityThemore the neighbors in abandoned state thelower the probability that the truth of the information
Define(1) Statelarr997888 current state of a node(2) 120591 larr997888 time step(3) Durationlarr997888 the period in current stateProcedure(4) for each t isin 120591 do(5) Traverse the network and fill in State of each node(6) for each node in small-world networks do(7) if State = unaware then(8) if Propagation factor then(9) Statelarr997888 interested(10) else(11) Statelarr997888 concerned(12) end if(13) else if State = concerned then(14) if Memory superposition factor then(15) Statelarr997888 interested(16) else(17) Durationlarr997888 Duration + 1(18) end if(19) else if State = interested then(20) if Information attenuation factor then(21) Statelarr997888 abandoned(22) else(23) Durationlarr997888 Duration + 1(24) end if(25) end if(26) end for(27) end for
Algorithm 1 The information spreading model pseudocode
is believed which reduces the probability of spread-ing Similarly we use the proportion of abandonedindividuals in the neighbors to calculate the impact ofherd effect and the expression is (1minus119890minus119899(119905)119896)(1minus1119890)Above all the herd mentality effect can be describedas ((1minus119890minus119898(119905)119896)minus(1minus119890minus119899(119905)119896))(1minus1119890)In this paper the propagation factor is composed ofthese two parts by the assigned 12 weight for theexpressiveness of the model and the computability ofsimulations which can be expressed in (1)Thegreaterp(t) the more possibility of transferring state fromunaware to interested
119901 (119905) = 12lowast (1 minus (1 minus 120572)119898(119905)) + 1
2
lowast(1 minus 119890minus119898(119905)119896) + (119890minus119899(119905)119896 minus 1)
1 minus 1119890
(1)
(b) If a node is in concerned state memory superpositionfactor determines state transition In our designedmodel we believe that human memory has an impacton the spread of information It is easy for anindividual in concerned state to change their stateby constantly having neighbors to spread or rejectthe message within a short period of time When
4 Complexity
different friends forward the same topic multipletimes it will promote the concerned state to becomethe interested state By contrast if not heard for along time the topic will be abandoned We definethe one who receives the information over the timeas s(t) which quantitatively describes the cumulativeeffect of the memory Initially set s(t) to 0 ie thevariable is 0 when a node changes from an unawarestate to a concerned state for the first time at tth timestep If there are new turned interested nodes aroundthe next time step then s(t+1) = s(t) +1 Otherwises(t+1) = s(t) - 1 We set threshold 120576 When memorysuperposition factor accumulates above 120576 the stateturns to interested Contrarily if s(t) lt -120576 it becomesabandoned state
(c) If the node is in interested state information atten-uation determines state transition Attenuation is acompound process here we consider both the interestdecay and time decayWe use exponential function torepresent time decay and the proportion of interestedindividuals in the neighbors to represent interestdecay Thus information attenuation factor can beexpressed as (2) We set threshold 120575 when a(t) decaysbelow the threshold the interested state node willturn to abandoned state
119886 (119905) = 119890minus119905 lowast 119898 (119905)119896
(2)
3 Design of Cellular AutomataCellular automata (CA) constitute the model of physicalsystems where space and time are discrete and interactionsare local It is an effective theoretical tool for studyingcomplex systems and at the same time can be simulatedexactly by computer program because of their intrinsicdiscreteness For those reasons we adopt CA to realize OSISmodel In CA the value of each cell presents the local statewhich can only interact with its neighbors and is incapable ofglobal communicationThe state is updated simultaneously atdiscrete time steps according to the states of their neighborsat the preceding time step Here are the elements of ourproposed cellular automata
(a) Cellular space Our model is set up in complexnetwork each node in the network consists of thecellular space
(b) Cellular neighborhood For the reason that our cellu-lar space is not the regular two-dimensional networkwe cannot adopt traditional Von Neumann or Moorestructure The neighborhood in Figure 2 illustratesthat the nodes associated with edges can be regardedas adjacent relationship which we call neighbors
(c) Cellular state According to OSIS model the stateof each cell is unaware concerned interested orabandoned
(d) State transfer rule As the methodology describedabove unaware state turns to whether interested orconcerned state is decided by propagation factor
Figure 2 The neighborhood of each cell is formed by the cellsassociated with edges
Memory superposition factor is able to determineconcerned nodersquos next state Furthermore informa-tion attenuation factor can turn interested to aban-doned state The detailed transfer relationship can befound in Figure 1
4 Experiments and Evaluation
41 Implementation of Online Social Network We use small-world network to simulate online social network in ourexperiments On the one hand small-world theory is alsocalled six degrees of separation which means the distancebetween two strangers is at most six people This feature isthe closest to the online social network On the other handon the basis of sampling and analysis on millions of leveldata excavated from the social groups in [38] online socialnetwork has small-world effect
The implementation of a small-world network comesfrom a regular lattice with a linear size L and with N =LlowastL nodes The initial node degree is k which means eachnode is connected to its k2 nearest neighbors clockwise andcounterclockwise deleting one link randomly adding a linkbetween two randomly chosen nodes with probability p andavoiding duplicate links and self-loops The link probabilityp turns the nature of the network between that of a regularnetwork (p = 0) and that of a random network (p = 1)
42 Experimental Results In our experiments we use C++programming language to build the small-world networkswith 2500 nodes and an average degree of 6The shortest dis-tance between two nodes is calculated by Dijkstrarsquos algorithmin computer graph theory The results are averaged over 100realizations
421 The Comparison with Traditional SEIR Model Todemonstrate the effectiveness of OSIS we compare ourmodel with traditional epidemics SEIR model We firstlyobserve the information spreading over time and spaceconsidering various infection rates Afterwards we illustratethe evolution Particularly unaware concerned interestedand abandoned state correspond to susceptible exposedinfected and recovered respectively in SEIR
Complexity 5
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
unawareconcerned
interestedabandoned
(a)
0 5 10 15 20 25 30
t
00
02
04
06
08
10
perc
entage
unawareconcerned
interestedabandoned
(b)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
unawareconcerned
interestedabandoned
(c)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
susceptibleexposed
infectedrecovered
(d)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
susceptibleexposed
infectedrecovered
(e)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
susceptibleexposed
infectedrecovered
(f)
Figure 3 The percentage of each state in a continuous period of time The spreading model is OSIS ((a) (b) (c)) and SEIR ((d) (e) (f))separatelyThe infection rate is 035 in (a) and (d) 055 in (b) and (f) 075 in (c) and (e)
Firstly we set up three sets of contrastive experimentsWithout generality we set the rewiring probability of thesmall-world networks 1 ie a ldquocompleterdquo small-world net-work And the threshold of the memory superposition factoris 1 and the threshold of information attenuation is 05 The
infection rate is 035 and 055 in the first and second set whichis 075 in the third one
Initially one node in the centre is infected After 30 timeperiods the percentage of each state is recorded and the resultis shown in Figure 3
6 Complexity
t
(a)
(b)
(c)
(d)
Figure 4 The information spreading evolution in small-world The spreading model is OSIS ((a) (b)) and SEIR ((c) (d)) separately (a) and(c) present the evolution of interested nodes (b) and (d) stand for abandoned nodes The infection rate is 035 The time period is 0 5 10 15
It can be seen from Figures 3(a) and 3(d) that wheninfection rate is low the percentage of nodes in final aban-doned state is 94 in SEIR which decreases as much as38 to 58 in OSIS This is because when the numberof initial infected nodes as well as information spreadingcapacity is low herd mentality inhibits the transmission ofinformation The phenomenon explains herdmentality effecton online social networks from the reverse Figures 3(b) and3(e) illustrate the peak of interested nodes is significantlydelayed which is from the 7th time step in SEIR to the 13thtime step in OSIS The result reveals that both propagationand disappearance of a topic is gradual over time or with theattention of people around which is different from epidemicFigures 3(c) and 3(f) demonstrate thatwhen the infection rateis high the number of infected nodes is increased by 72
to 057 from 033 This is due to the fact that herd mentalityand memory superposition effect promote the spreading ofinformation
Furthermore to discover the evolution of OSIS weinvestigate the information spreading over space In thisexperiment rewiring probability of the small-world networkis that 01 and 1 nodes in the centre of the network are infectedinitially It can be seen from Figure 4 that the evolution ofonline social network information is a gradual process whilethe disease transmission has the characteristics of suddenoutbreak and the one exposed to the disease eventuallydevelops antibodies Further comparing Figures 4(a) and4(c) it is evident that the disease with SEIR model spreadsvery fast and suddenly dies out By contrast the informationspreads throughout the network smoothly Meanwhile the
Complexity 7
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60 70t
num
ber o
f aba
ndon
ed
initially 1 interested initially 10 interested initially 100 interested
(a)
0
500
1000
1500
2000
2500
num
ber o
f aba
ndon
ed
0 40 50 60 7010 20 30t
initially 1 interested initially 10 interested initially 100 interested
(b)
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60 70t
num
ber o
f aba
ndon
ed
initially 1 interested initially 10 interested initially 100 interested
(c)
Figure 5 The influence of different initial infection ratio in small-word The rewiring probability is 0 01 1 in (a) (b) (c) respectively Eachexperiment randomly selects 1 10 100 initial interested nodes
contrast between Figures 4(b) and 4(d) shows that theabandoned nodes in SEIR are in the large area while thenumber of immunized nodes in OSIS is flat The results areconsistent with the above conclusions
422 The Influence of Different Initial Infection Ratio Inorder to prove the influence of initial infection ratio oninformation spreading we randomly select 1 10 and 100initial interested nodes The rewiring probability in each setof experiments is 0 01 and 1 respectively and the infectionrate is 035The threshold of thememory superposition factoris 1 and the threshold of information attenuation is 05 Theabandoned state nodes are recorded
It is observable fromFigure 5(a) that the number of initialinfected nodes plays a decisive role in spreading when small-world effect is not obvious (ie rewiring probability is 0)When the number of initial interested nodes is 10 the numberof the eventually abandoned nodes is 8 times more than thatof 1 initial interested node while the multiple is 60 timeswhen the number of initial interested nodes is 100 and 1 Asshown in Figures 5(b) and 5(c) a higher rewiring probability
leads to a faster spreading and a higher steady-state infectiondensity When the rewiring probability is 01 the peak isreached at the 15th time period while it reduces to 10th timeperiodwhen the rewiring probability is 1 All the results abovereveal that the more initial infected nodes the easier it is tospread information However this effect tends to convergeas the rewiring probability increases Additionally with thestrengthening of the small-world features (ie the averagenumber of friends per person in social networks is greater)the information can spread farther and inhibited harder
423 The Effect of Memory Superposition To explore thememory superposition effect on information spreading Weperform simulations to obtain the infection density in thesteady state as a function of the effective infection rate 120572for various superposition thresholds In this experiment therewiring probability of the network is 01 and the threshold ofinformation attenuation is 05
We set up three sets of experiments The infection ratein each set is 01 05 and 07 meanwhile the superpositionthreshold is 1 3 and 5 Initially one node in network is
8 Complexity
0501 07infection rate
superposition threshold=1superposition threshold=3superposition threshold=5
0
500
1000
1500
2000
2500nu
mber o
f aba
ndon
ed
Figure 6The number of abandoned node under different superpo-sition threshold
randomly chosen to be infected After 100 time periods thenumber of nodes in abandoned state is recorded
As shown in Figure 6 the results of the experimentsall indicate that the effect of superposition is obvious andthe higher the threshold the fewer the nodes affected Forexample when the infection rate is 05 the number ofabandoned nodes with the threshold 1 is almost 4 timesmore than that with threshold 3 which even goes to 19 timeswhen compared with the result of the threshold 5The resultsaccurately explain why information spreads widely in onlinesocial networks With the explosion of Internet and we-media the spreading of information costs little investigationand individuals spread information with less filters Such lessfrequent contact to information leads to the rapid spreadingMemory superposition factor in our model stands for thenumber of contacts
5 Conclusions
In summary we introduce an OSIS model to describeinformation spreading in online social networks Firstly weproposeOSIS by combining epidemics SEIRmodelwith indi-vidualrsquos cognitive psychology Secondly a cellular automatamodel of information transmission is designed which pro-vides a computational method for information spreadingfrom a microscopic perspective Additionally we use small-world network to simulate online social network whichverifies the spreading and evolution of information Theexperimental results reveal that our model is consistent withthe propagation of information in online social networksWe find that individualrsquos cognition affects the informationspreading behavior When the infection rate is low cognitivepsychology prevents the spread of information By contrastwhen the infection rate is sufficiently high it promotes trans-mission We demonstrated that the explosion of online socialnetwork scale and the convenience of we-media can greatlyincrease the ability of information disseminationMeanwhile
our research shows that the demise of information is affectedby the dual effects of the decay of time and heat
Our current work mainly concentrates on the spreadingmodel itself As for the future work we believe that the realonline social network is not a single small-world networkDue to the vigorous development ofWeb 20 online networkssuch as Weibo Twitter and Facebook are intertwined Wedeem the information spreading in interconnected small-world networks as the promising and challenging futurework
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A L Barabasi and R E Crandall ldquoLinked the new science ofnetworksrdquo American Journal of Physics vol 71 no 4 pp 409ndash411 2003
[2] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003
[3] A Barrat M Barthlemy and A Vespignani Dynamical pro-cesses on complex networks 2008
[4] R Cohen and S Havlin ldquoComplex networks structure robust-ness and function London UKrdquo Cambridge University Press2010
[5] P Erdos and A Renyi ldquoOn the evolution of random graphsrdquoTransactions of the AmericanMathematical Society vol 286 no1 pp 257ndash274 2011
[6] D J Watts and S H Strogatz Collective Dynamics of rsquoSmall-Worldrsquo Networks 2011
[7] A Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquo Science vol 286 no 5439 pp 509ndash512 1999
[8] M Li Y Fan D Wang D Li J Wu and Z Di ldquoSmall-worldeffect induced by weight randomization on regular networksrdquoPhysics Letters A vol 364 p 488 2007
[9] Q Jin L Wang C Xia and Z Wang ldquoSpontaneous symmetrybreaking in interdependent networked gamerdquo Scientic Reportsvol 4 p 4095 2015
[10] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001
[11] N Bailey ldquoThe mathematical theory of infectious diseases andits applicationsrdquo The Journal of Immunology vol 34 no 5 pp955-956 1978
[12] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press 1991
[13] N C Grassly and C Fraser ldquoMathematicalmodels of infectiousdisease transmissionrdquo Nature Reviews Microbiology vol 6 no6 pp 477ndash487 2008
[14] R M May and A L Lloyd ldquoInfection dynamics on scale-freenetworksrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 64 no 6 Article ID 066112 2001
Complexity 9
[15] M H A Biswas L T Paiva and M D R D Pinho ldquoASEIR model for control of infectious diseases with constraintsrdquoMathematical Biosciences amp Engineering vol 11 no 4 p 7612014
[16] X Fu A Passarella D Quercia A Sala and T Strufe ldquoOnlineSocial Networksrdquo Computer Communications vol 73 pp 163ndash166 2016
[17] X Ke Z Sai C Hao et al ldquoMeasurement and analysis of onlinesocial networksrdquoChinese Journal of Computers pp 29ndash42 2014
[18] Q Ying Z Nan and Z Laijun ldquoResearch on the law of weibopublicity transmissionrdquo Journal of the China Society for Scienticand Technical Information vol 31 no 12 pp 1299ndash1304 2012(Chinese)
[19] weibo httpsweibocom[20] F Xiong Y Liu Z J Zhang J Zhu and Y Zhang ldquoAn
information diffusion model based on retweeting mechanismfor online social mediardquo Physics Letters A vol 376 p 2103 2012
[21] D Centola ldquoThe spread of behavior in an online social networkexperimentrdquo Science vol 329 no 5996 pp 1194ndash1197 2010
[22] H Wu A Arenas and S Gmez ldquoInfluence of trust in thespreading of informationrdquo Physical Review vol 95 2016
[23] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 no 11 Article ID e100096815 pages 2010
[24] L Lu D-B Chen and T Zhou ldquoSmall world yields the mosteffective information spreadingrdquo New Journal of Physics p825834 2011
[25] J X Zhang D B Chen Q Dong and Z D Zhao ldquoIdentifyinga set of influential spreaders in complex networksrdquo ScienticReports vol 6 2016
[26] W Wang M Cai and M Zheng ldquoSocial contagions oncorrelated multiplex networksrdquo Physica A Statistical Mechanicsamp Its Applications vol 499 2017
[27] W Wang X L Chen and L F Zhong ldquoSocial contagions withheterogeneous credibilityrdquo Physica A Statistical Mechanics amp ItsApplications 2018
[28] W Wang M Tang S H Eugene and L A BraunsteinldquoUnification of theoretical approaches for epidemic spreadingon complex networksrdquoReports on Progress in Physics p 0366032017
[29] W Wang M Tang H F Zhang H Gao Y Do and Z HLiu ldquoEpidemic spreading on complex networks with generaldegree and weight distributionsrdquo Physical Review E StatisticalNonlinear ampa Soft Matter Physics vol 90 p 042803 2014
[30] X Zhu W Wang S Cai and H E Stanley ldquoStanley Dynamicsof social contagions with local trend imitationrdquo Scientic Reportsvol 8 2018 Scientic Reports
[31] X Zhu W Wang S Cai and H E Stanley ldquoOptimal imitationcapacity and crossover phenomenon in the dynamics of socialcontagionsrdquo Journal of Statistical Mechanics Theory amp Experi-ment Article ID 063405 p 063405 2018
[32] A L Barabsi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 p 173 1999
[33] S A Pandit and R E Amritkar ldquoCharacterization and controlof small-world networksrdquo Physical Review E Statistical Nonlin-ear and SoftMatter Physics vol 60 no 2 pp R1119ndashR1122 1999
[34] J VonNeumann Papers of John Von Neumann on Computing ampComputer Theory vol 1 1951
[35] B Chopard and M Droz Cellular Automata for ModelingPhysics Cambridge University Press Cambridge UK 1998
[36] G J Martnez A Adamatzky F Chen and L Chua ldquoOn SolitonCollisions between Localizations in Complex Elementary Cel-lular Automata Rules 54 and 110 andBeyondrdquoComplex Systemsvol 117 2013
[37] H F Gagliardi and D Alves ldquoSmall-world effect in epidemicsusing cellular automatardquo Mathematical Population Studies AnInternational Journal of Mathematical Demography vol 17 no2 pp 79ndash90 2010
[38] X Jin J Li and L Zhang Online Social Networks Based onComplex Network Theory and Simulation Analysis SpringerInternational Publishing 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
2 Complexity
Furthermore most previous studies of dynamical pro-cesses on complex networks are based on mean field method[32] or analytical equation [33] However these approacheshas some serious drawbacks The method based on meanfield can only reflect an approximate trend frommacroscopicview which cannot accurately describe the spreading stateof each individual Meanwhile analytical equation neglectsvariable susceptibility of individuals and cannot handle thedifferent boundary and initial condition in addition it willbe very complicated for large-scale online social networksCellular automata (CA) [34 35] proposed by Von Neumanncan overcome the above drawbacks and have been used byseveral researchers as an alternative method of modelingepidemics In this model every cell of the grid representsan individual updating state of each cell by state transitioncriterion [36 37]
In summary complex networks provide theoretical basisfor social network topology and the extensive investigationof epidemic models promotes the quantitative study ofinformation spreading By referring to complex networkdynamics as well as combining the features of public opiniona breakthroughhas beenmade in the research on informationtransmission in social networks With the rapid developmentof mobile Internet and Internet of things the online socialnetwork is fundamentally different from the traditional socialnetwork The spreading and control of information espe-cially public opinion has become a difficulty and key pointconcerned by public management policymakers Meanwhiledue to the complexity of network topology and the explosionof network access nodes traditional analytical methods havebeen unable to accurately calculate the results By introducingthe cellular automata theory the bottleneck can be solvedthrough the iterative calculation
In this paper we propose an online social networksinformation spreading (OSIS) model combining individualrsquoscognitive psychology with traditional SEIRmodelMoreoverwe provide a computational method to prove the accuracyand authenticity of OSIS by cellular automata We obtainedexperimental results with the help of computer graph theoryThe results reveal that individualrsquos cognition affects onlineinformation spreading When infection rate is low cognitivepsychology prevents the spreading By contrast when infec-tion rate is sufficiently high it promotes transmission Wealso demonstrate that the explosion of online social networkscale and the convenience of we-media can greatly increasethe ability of information dissemination especially rumorsFinally we show that the demise of information is affected bythe dual effects of the decay of time and heat In particularthe major contributions of our work can be summarized asfollows
(i) a new information spreading model that takes intoaccount individualrsquos cognitive psychology and thecharacteristics of online social networks herd men-tality factor memory superposition factor and atten-tion attenuation factor
(ii) a two-dimensional CA method that can accuratelyanalyze the process of online information spreading
(iii) revealing the characteristic of online social networksinformation dissemination
The sections in this paper are structured as follows Section 1presents the background while Section 2 describes themodelSection 3 shows the design of cellular automata followed bythe experimental results in Section 4 finally we conclude ourpaper in Section 5 with the future work discussed
2 Online Social Network InformationSpreading Model
21 Analysis of Information in Online Social Network Com-paredwith traditional social networks online social networksuse Internet as the medium of communication rather thanface-to-face which makes information transmission moreeasy to forward Meanwhile in social networks such as Twit-ter information is transmitted synchronously to all friendswhich is different from the peer-to-peer communicationof offline networks The communication of information ismore highly timely Meanwhile because of the intelligence ofterminals the carrier of online information varies from textto sound picture video etcThe content becomes interestingand easy to understand Based on the features describedabove we consider the information spreading model fromthree aspects
(i) Propagation factor Due to the characteristics ofonline social networks and individualrsquos fast-foodreading habits the cognition of information dependsnot only on the content itself but also on the heatThe more the friends who are concerned around atopic the higher the probability that an individual willspread it
(ii) Memory superposition factor Online social net-works shorten the distance among people whichmakes people access the same information over acontinuous period of time we call this phenomenonmemory superposition The memory superpositionenhances peoplersquos curiosity thus prompting thespreading
(iii) Information attenuation factor In traditional epi-demic models a disease is cured via a certain prob-ability whereas the demise of information is affectedby both time and heat decay
22 Description of OSIS Model Since information spreadingis similar to epidemic we propose our online social networkinformation spreading (OSIS) model by combining the char-acteristics of online social networks mentioned above withepidemic SEIR model In OSIS each individual adopts one offour states unaware concerned interested and abandonedUnaware represents the individual who has not heard theinformation and has the potential to accept it Concernedstands for the person who is aware of the news but notwilling to transmit it currently Interested means the onebelieves in the news and has the ability to spread it Andabandoned denotes the individual who loses interests andnever transmits the information again We use complex
Complexity 3
unaware interested abandoned
concerned
Figure 1 The state transfer diagram
networks as the model of real social networks in whicheach node and edge present individual and online socialrelationship
As shown in Figure 1 the state of each node can betransformed over time Initially we randomly pick up onenode as an interested node and the rest are in unaware stateAt each time step an unaware state individual receives a topicfrom her neighbor and turns to interested or concerned stateaccording to the propagation factor Furthermore whether ornot a concerned state node turns to interested state is basedon memory superposition factor Additionally informationattenuation factor determines interested to abandoned state
23 The Algorithm Based on OSIS model described abovean algorithm for simulation of information spreading oncomplex networks has been developed The core idea inAlgorithm 1 is presented with pseudocode
(a) If a node is in unaware state and there are interestedstate nodes around propagation factor determinesstate transition In our model the disseminationof information is due to not only the informationitself but also to herd mentality For that reason thepropagation factor consists of two parts One partis due to the information itself here we adopt thealgorithm in traditional SIR modelThe infection rateis 120572 and the number of interested neighbors is m(t)at the tth time The node will have a probability of1minus(1minus120572)119898(119905) being infected The other part is causedby herd mentality which includes both positive andnegative effect We assume that for a given individualat the tth time she has k adjacent m(t) interestedand n(t) abandoned neighbors The proportion ofinterested individuals in the neighbors determinesthe positive influence of the herd effect When noone around propagates the information (ie 119898(119905)119896= 0) the influence probability of herd effect is 0While all friends are in interested state (ie 119898(119905)119896= 0) the influence of herd effect is the greatest theprobability of which is 1 Our observations show thatthe effect of this ratio is a positively exponential effectrather than a linear effect we define the effect as(1minus119890minus119898(119905)119896)(1minus1119890) By contrast if most friends arein abandoned state this may reduce the transmissi-bilityThemore the neighbors in abandoned state thelower the probability that the truth of the information
Define(1) Statelarr997888 current state of a node(2) 120591 larr997888 time step(3) Durationlarr997888 the period in current stateProcedure(4) for each t isin 120591 do(5) Traverse the network and fill in State of each node(6) for each node in small-world networks do(7) if State = unaware then(8) if Propagation factor then(9) Statelarr997888 interested(10) else(11) Statelarr997888 concerned(12) end if(13) else if State = concerned then(14) if Memory superposition factor then(15) Statelarr997888 interested(16) else(17) Durationlarr997888 Duration + 1(18) end if(19) else if State = interested then(20) if Information attenuation factor then(21) Statelarr997888 abandoned(22) else(23) Durationlarr997888 Duration + 1(24) end if(25) end if(26) end for(27) end for
Algorithm 1 The information spreading model pseudocode
is believed which reduces the probability of spread-ing Similarly we use the proportion of abandonedindividuals in the neighbors to calculate the impact ofherd effect and the expression is (1minus119890minus119899(119905)119896)(1minus1119890)Above all the herd mentality effect can be describedas ((1minus119890minus119898(119905)119896)minus(1minus119890minus119899(119905)119896))(1minus1119890)In this paper the propagation factor is composed ofthese two parts by the assigned 12 weight for theexpressiveness of the model and the computability ofsimulations which can be expressed in (1)Thegreaterp(t) the more possibility of transferring state fromunaware to interested
119901 (119905) = 12lowast (1 minus (1 minus 120572)119898(119905)) + 1
2
lowast(1 minus 119890minus119898(119905)119896) + (119890minus119899(119905)119896 minus 1)
1 minus 1119890
(1)
(b) If a node is in concerned state memory superpositionfactor determines state transition In our designedmodel we believe that human memory has an impacton the spread of information It is easy for anindividual in concerned state to change their stateby constantly having neighbors to spread or rejectthe message within a short period of time When
4 Complexity
different friends forward the same topic multipletimes it will promote the concerned state to becomethe interested state By contrast if not heard for along time the topic will be abandoned We definethe one who receives the information over the timeas s(t) which quantitatively describes the cumulativeeffect of the memory Initially set s(t) to 0 ie thevariable is 0 when a node changes from an unawarestate to a concerned state for the first time at tth timestep If there are new turned interested nodes aroundthe next time step then s(t+1) = s(t) +1 Otherwises(t+1) = s(t) - 1 We set threshold 120576 When memorysuperposition factor accumulates above 120576 the stateturns to interested Contrarily if s(t) lt -120576 it becomesabandoned state
(c) If the node is in interested state information atten-uation determines state transition Attenuation is acompound process here we consider both the interestdecay and time decayWe use exponential function torepresent time decay and the proportion of interestedindividuals in the neighbors to represent interestdecay Thus information attenuation factor can beexpressed as (2) We set threshold 120575 when a(t) decaysbelow the threshold the interested state node willturn to abandoned state
119886 (119905) = 119890minus119905 lowast 119898 (119905)119896
(2)
3 Design of Cellular AutomataCellular automata (CA) constitute the model of physicalsystems where space and time are discrete and interactionsare local It is an effective theoretical tool for studyingcomplex systems and at the same time can be simulatedexactly by computer program because of their intrinsicdiscreteness For those reasons we adopt CA to realize OSISmodel In CA the value of each cell presents the local statewhich can only interact with its neighbors and is incapable ofglobal communicationThe state is updated simultaneously atdiscrete time steps according to the states of their neighborsat the preceding time step Here are the elements of ourproposed cellular automata
(a) Cellular space Our model is set up in complexnetwork each node in the network consists of thecellular space
(b) Cellular neighborhood For the reason that our cellu-lar space is not the regular two-dimensional networkwe cannot adopt traditional Von Neumann or Moorestructure The neighborhood in Figure 2 illustratesthat the nodes associated with edges can be regardedas adjacent relationship which we call neighbors
(c) Cellular state According to OSIS model the stateof each cell is unaware concerned interested orabandoned
(d) State transfer rule As the methodology describedabove unaware state turns to whether interested orconcerned state is decided by propagation factor
Figure 2 The neighborhood of each cell is formed by the cellsassociated with edges
Memory superposition factor is able to determineconcerned nodersquos next state Furthermore informa-tion attenuation factor can turn interested to aban-doned state The detailed transfer relationship can befound in Figure 1
4 Experiments and Evaluation
41 Implementation of Online Social Network We use small-world network to simulate online social network in ourexperiments On the one hand small-world theory is alsocalled six degrees of separation which means the distancebetween two strangers is at most six people This feature isthe closest to the online social network On the other handon the basis of sampling and analysis on millions of leveldata excavated from the social groups in [38] online socialnetwork has small-world effect
The implementation of a small-world network comesfrom a regular lattice with a linear size L and with N =LlowastL nodes The initial node degree is k which means eachnode is connected to its k2 nearest neighbors clockwise andcounterclockwise deleting one link randomly adding a linkbetween two randomly chosen nodes with probability p andavoiding duplicate links and self-loops The link probabilityp turns the nature of the network between that of a regularnetwork (p = 0) and that of a random network (p = 1)
42 Experimental Results In our experiments we use C++programming language to build the small-world networkswith 2500 nodes and an average degree of 6The shortest dis-tance between two nodes is calculated by Dijkstrarsquos algorithmin computer graph theory The results are averaged over 100realizations
421 The Comparison with Traditional SEIR Model Todemonstrate the effectiveness of OSIS we compare ourmodel with traditional epidemics SEIR model We firstlyobserve the information spreading over time and spaceconsidering various infection rates Afterwards we illustratethe evolution Particularly unaware concerned interestedand abandoned state correspond to susceptible exposedinfected and recovered respectively in SEIR
Complexity 5
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
unawareconcerned
interestedabandoned
(a)
0 5 10 15 20 25 30
t
00
02
04
06
08
10
perc
entage
unawareconcerned
interestedabandoned
(b)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
unawareconcerned
interestedabandoned
(c)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
susceptibleexposed
infectedrecovered
(d)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
susceptibleexposed
infectedrecovered
(e)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
susceptibleexposed
infectedrecovered
(f)
Figure 3 The percentage of each state in a continuous period of time The spreading model is OSIS ((a) (b) (c)) and SEIR ((d) (e) (f))separatelyThe infection rate is 035 in (a) and (d) 055 in (b) and (f) 075 in (c) and (e)
Firstly we set up three sets of contrastive experimentsWithout generality we set the rewiring probability of thesmall-world networks 1 ie a ldquocompleterdquo small-world net-work And the threshold of the memory superposition factoris 1 and the threshold of information attenuation is 05 The
infection rate is 035 and 055 in the first and second set whichis 075 in the third one
Initially one node in the centre is infected After 30 timeperiods the percentage of each state is recorded and the resultis shown in Figure 3
6 Complexity
t
(a)
(b)
(c)
(d)
Figure 4 The information spreading evolution in small-world The spreading model is OSIS ((a) (b)) and SEIR ((c) (d)) separately (a) and(c) present the evolution of interested nodes (b) and (d) stand for abandoned nodes The infection rate is 035 The time period is 0 5 10 15
It can be seen from Figures 3(a) and 3(d) that wheninfection rate is low the percentage of nodes in final aban-doned state is 94 in SEIR which decreases as much as38 to 58 in OSIS This is because when the numberof initial infected nodes as well as information spreadingcapacity is low herd mentality inhibits the transmission ofinformation The phenomenon explains herdmentality effecton online social networks from the reverse Figures 3(b) and3(e) illustrate the peak of interested nodes is significantlydelayed which is from the 7th time step in SEIR to the 13thtime step in OSIS The result reveals that both propagationand disappearance of a topic is gradual over time or with theattention of people around which is different from epidemicFigures 3(c) and 3(f) demonstrate thatwhen the infection rateis high the number of infected nodes is increased by 72
to 057 from 033 This is due to the fact that herd mentalityand memory superposition effect promote the spreading ofinformation
Furthermore to discover the evolution of OSIS weinvestigate the information spreading over space In thisexperiment rewiring probability of the small-world networkis that 01 and 1 nodes in the centre of the network are infectedinitially It can be seen from Figure 4 that the evolution ofonline social network information is a gradual process whilethe disease transmission has the characteristics of suddenoutbreak and the one exposed to the disease eventuallydevelops antibodies Further comparing Figures 4(a) and4(c) it is evident that the disease with SEIR model spreadsvery fast and suddenly dies out By contrast the informationspreads throughout the network smoothly Meanwhile the
Complexity 7
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60 70t
num
ber o
f aba
ndon
ed
initially 1 interested initially 10 interested initially 100 interested
(a)
0
500
1000
1500
2000
2500
num
ber o
f aba
ndon
ed
0 40 50 60 7010 20 30t
initially 1 interested initially 10 interested initially 100 interested
(b)
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60 70t
num
ber o
f aba
ndon
ed
initially 1 interested initially 10 interested initially 100 interested
(c)
Figure 5 The influence of different initial infection ratio in small-word The rewiring probability is 0 01 1 in (a) (b) (c) respectively Eachexperiment randomly selects 1 10 100 initial interested nodes
contrast between Figures 4(b) and 4(d) shows that theabandoned nodes in SEIR are in the large area while thenumber of immunized nodes in OSIS is flat The results areconsistent with the above conclusions
422 The Influence of Different Initial Infection Ratio Inorder to prove the influence of initial infection ratio oninformation spreading we randomly select 1 10 and 100initial interested nodes The rewiring probability in each setof experiments is 0 01 and 1 respectively and the infectionrate is 035The threshold of thememory superposition factoris 1 and the threshold of information attenuation is 05 Theabandoned state nodes are recorded
It is observable fromFigure 5(a) that the number of initialinfected nodes plays a decisive role in spreading when small-world effect is not obvious (ie rewiring probability is 0)When the number of initial interested nodes is 10 the numberof the eventually abandoned nodes is 8 times more than thatof 1 initial interested node while the multiple is 60 timeswhen the number of initial interested nodes is 100 and 1 Asshown in Figures 5(b) and 5(c) a higher rewiring probability
leads to a faster spreading and a higher steady-state infectiondensity When the rewiring probability is 01 the peak isreached at the 15th time period while it reduces to 10th timeperiodwhen the rewiring probability is 1 All the results abovereveal that the more initial infected nodes the easier it is tospread information However this effect tends to convergeas the rewiring probability increases Additionally with thestrengthening of the small-world features (ie the averagenumber of friends per person in social networks is greater)the information can spread farther and inhibited harder
423 The Effect of Memory Superposition To explore thememory superposition effect on information spreading Weperform simulations to obtain the infection density in thesteady state as a function of the effective infection rate 120572for various superposition thresholds In this experiment therewiring probability of the network is 01 and the threshold ofinformation attenuation is 05
We set up three sets of experiments The infection ratein each set is 01 05 and 07 meanwhile the superpositionthreshold is 1 3 and 5 Initially one node in network is
8 Complexity
0501 07infection rate
superposition threshold=1superposition threshold=3superposition threshold=5
0
500
1000
1500
2000
2500nu
mber o
f aba
ndon
ed
Figure 6The number of abandoned node under different superpo-sition threshold
randomly chosen to be infected After 100 time periods thenumber of nodes in abandoned state is recorded
As shown in Figure 6 the results of the experimentsall indicate that the effect of superposition is obvious andthe higher the threshold the fewer the nodes affected Forexample when the infection rate is 05 the number ofabandoned nodes with the threshold 1 is almost 4 timesmore than that with threshold 3 which even goes to 19 timeswhen compared with the result of the threshold 5The resultsaccurately explain why information spreads widely in onlinesocial networks With the explosion of Internet and we-media the spreading of information costs little investigationand individuals spread information with less filters Such lessfrequent contact to information leads to the rapid spreadingMemory superposition factor in our model stands for thenumber of contacts
5 Conclusions
In summary we introduce an OSIS model to describeinformation spreading in online social networks Firstly weproposeOSIS by combining epidemics SEIRmodelwith indi-vidualrsquos cognitive psychology Secondly a cellular automatamodel of information transmission is designed which pro-vides a computational method for information spreadingfrom a microscopic perspective Additionally we use small-world network to simulate online social network whichverifies the spreading and evolution of information Theexperimental results reveal that our model is consistent withthe propagation of information in online social networksWe find that individualrsquos cognition affects the informationspreading behavior When the infection rate is low cognitivepsychology prevents the spread of information By contrastwhen the infection rate is sufficiently high it promotes trans-mission We demonstrated that the explosion of online socialnetwork scale and the convenience of we-media can greatlyincrease the ability of information disseminationMeanwhile
our research shows that the demise of information is affectedby the dual effects of the decay of time and heat
Our current work mainly concentrates on the spreadingmodel itself As for the future work we believe that the realonline social network is not a single small-world networkDue to the vigorous development ofWeb 20 online networkssuch as Weibo Twitter and Facebook are intertwined Wedeem the information spreading in interconnected small-world networks as the promising and challenging futurework
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A L Barabasi and R E Crandall ldquoLinked the new science ofnetworksrdquo American Journal of Physics vol 71 no 4 pp 409ndash411 2003
[2] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003
[3] A Barrat M Barthlemy and A Vespignani Dynamical pro-cesses on complex networks 2008
[4] R Cohen and S Havlin ldquoComplex networks structure robust-ness and function London UKrdquo Cambridge University Press2010
[5] P Erdos and A Renyi ldquoOn the evolution of random graphsrdquoTransactions of the AmericanMathematical Society vol 286 no1 pp 257ndash274 2011
[6] D J Watts and S H Strogatz Collective Dynamics of rsquoSmall-Worldrsquo Networks 2011
[7] A Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquo Science vol 286 no 5439 pp 509ndash512 1999
[8] M Li Y Fan D Wang D Li J Wu and Z Di ldquoSmall-worldeffect induced by weight randomization on regular networksrdquoPhysics Letters A vol 364 p 488 2007
[9] Q Jin L Wang C Xia and Z Wang ldquoSpontaneous symmetrybreaking in interdependent networked gamerdquo Scientic Reportsvol 4 p 4095 2015
[10] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001
[11] N Bailey ldquoThe mathematical theory of infectious diseases andits applicationsrdquo The Journal of Immunology vol 34 no 5 pp955-956 1978
[12] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press 1991
[13] N C Grassly and C Fraser ldquoMathematicalmodels of infectiousdisease transmissionrdquo Nature Reviews Microbiology vol 6 no6 pp 477ndash487 2008
[14] R M May and A L Lloyd ldquoInfection dynamics on scale-freenetworksrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 64 no 6 Article ID 066112 2001
Complexity 9
[15] M H A Biswas L T Paiva and M D R D Pinho ldquoASEIR model for control of infectious diseases with constraintsrdquoMathematical Biosciences amp Engineering vol 11 no 4 p 7612014
[16] X Fu A Passarella D Quercia A Sala and T Strufe ldquoOnlineSocial Networksrdquo Computer Communications vol 73 pp 163ndash166 2016
[17] X Ke Z Sai C Hao et al ldquoMeasurement and analysis of onlinesocial networksrdquoChinese Journal of Computers pp 29ndash42 2014
[18] Q Ying Z Nan and Z Laijun ldquoResearch on the law of weibopublicity transmissionrdquo Journal of the China Society for Scienticand Technical Information vol 31 no 12 pp 1299ndash1304 2012(Chinese)
[19] weibo httpsweibocom[20] F Xiong Y Liu Z J Zhang J Zhu and Y Zhang ldquoAn
information diffusion model based on retweeting mechanismfor online social mediardquo Physics Letters A vol 376 p 2103 2012
[21] D Centola ldquoThe spread of behavior in an online social networkexperimentrdquo Science vol 329 no 5996 pp 1194ndash1197 2010
[22] H Wu A Arenas and S Gmez ldquoInfluence of trust in thespreading of informationrdquo Physical Review vol 95 2016
[23] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 no 11 Article ID e100096815 pages 2010
[24] L Lu D-B Chen and T Zhou ldquoSmall world yields the mosteffective information spreadingrdquo New Journal of Physics p825834 2011
[25] J X Zhang D B Chen Q Dong and Z D Zhao ldquoIdentifyinga set of influential spreaders in complex networksrdquo ScienticReports vol 6 2016
[26] W Wang M Cai and M Zheng ldquoSocial contagions oncorrelated multiplex networksrdquo Physica A Statistical Mechanicsamp Its Applications vol 499 2017
[27] W Wang X L Chen and L F Zhong ldquoSocial contagions withheterogeneous credibilityrdquo Physica A Statistical Mechanics amp ItsApplications 2018
[28] W Wang M Tang S H Eugene and L A BraunsteinldquoUnification of theoretical approaches for epidemic spreadingon complex networksrdquoReports on Progress in Physics p 0366032017
[29] W Wang M Tang H F Zhang H Gao Y Do and Z HLiu ldquoEpidemic spreading on complex networks with generaldegree and weight distributionsrdquo Physical Review E StatisticalNonlinear ampa Soft Matter Physics vol 90 p 042803 2014
[30] X Zhu W Wang S Cai and H E Stanley ldquoStanley Dynamicsof social contagions with local trend imitationrdquo Scientic Reportsvol 8 2018 Scientic Reports
[31] X Zhu W Wang S Cai and H E Stanley ldquoOptimal imitationcapacity and crossover phenomenon in the dynamics of socialcontagionsrdquo Journal of Statistical Mechanics Theory amp Experi-ment Article ID 063405 p 063405 2018
[32] A L Barabsi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 p 173 1999
[33] S A Pandit and R E Amritkar ldquoCharacterization and controlof small-world networksrdquo Physical Review E Statistical Nonlin-ear and SoftMatter Physics vol 60 no 2 pp R1119ndashR1122 1999
[34] J VonNeumann Papers of John Von Neumann on Computing ampComputer Theory vol 1 1951
[35] B Chopard and M Droz Cellular Automata for ModelingPhysics Cambridge University Press Cambridge UK 1998
[36] G J Martnez A Adamatzky F Chen and L Chua ldquoOn SolitonCollisions between Localizations in Complex Elementary Cel-lular Automata Rules 54 and 110 andBeyondrdquoComplex Systemsvol 117 2013
[37] H F Gagliardi and D Alves ldquoSmall-world effect in epidemicsusing cellular automatardquo Mathematical Population Studies AnInternational Journal of Mathematical Demography vol 17 no2 pp 79ndash90 2010
[38] X Jin J Li and L Zhang Online Social Networks Based onComplex Network Theory and Simulation Analysis SpringerInternational Publishing 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 3
unaware interested abandoned
concerned
Figure 1 The state transfer diagram
networks as the model of real social networks in whicheach node and edge present individual and online socialrelationship
As shown in Figure 1 the state of each node can betransformed over time Initially we randomly pick up onenode as an interested node and the rest are in unaware stateAt each time step an unaware state individual receives a topicfrom her neighbor and turns to interested or concerned stateaccording to the propagation factor Furthermore whether ornot a concerned state node turns to interested state is basedon memory superposition factor Additionally informationattenuation factor determines interested to abandoned state
23 The Algorithm Based on OSIS model described abovean algorithm for simulation of information spreading oncomplex networks has been developed The core idea inAlgorithm 1 is presented with pseudocode
(a) If a node is in unaware state and there are interestedstate nodes around propagation factor determinesstate transition In our model the disseminationof information is due to not only the informationitself but also to herd mentality For that reason thepropagation factor consists of two parts One partis due to the information itself here we adopt thealgorithm in traditional SIR modelThe infection rateis 120572 and the number of interested neighbors is m(t)at the tth time The node will have a probability of1minus(1minus120572)119898(119905) being infected The other part is causedby herd mentality which includes both positive andnegative effect We assume that for a given individualat the tth time she has k adjacent m(t) interestedand n(t) abandoned neighbors The proportion ofinterested individuals in the neighbors determinesthe positive influence of the herd effect When noone around propagates the information (ie 119898(119905)119896= 0) the influence probability of herd effect is 0While all friends are in interested state (ie 119898(119905)119896= 0) the influence of herd effect is the greatest theprobability of which is 1 Our observations show thatthe effect of this ratio is a positively exponential effectrather than a linear effect we define the effect as(1minus119890minus119898(119905)119896)(1minus1119890) By contrast if most friends arein abandoned state this may reduce the transmissi-bilityThemore the neighbors in abandoned state thelower the probability that the truth of the information
Define(1) Statelarr997888 current state of a node(2) 120591 larr997888 time step(3) Durationlarr997888 the period in current stateProcedure(4) for each t isin 120591 do(5) Traverse the network and fill in State of each node(6) for each node in small-world networks do(7) if State = unaware then(8) if Propagation factor then(9) Statelarr997888 interested(10) else(11) Statelarr997888 concerned(12) end if(13) else if State = concerned then(14) if Memory superposition factor then(15) Statelarr997888 interested(16) else(17) Durationlarr997888 Duration + 1(18) end if(19) else if State = interested then(20) if Information attenuation factor then(21) Statelarr997888 abandoned(22) else(23) Durationlarr997888 Duration + 1(24) end if(25) end if(26) end for(27) end for
Algorithm 1 The information spreading model pseudocode
is believed which reduces the probability of spread-ing Similarly we use the proportion of abandonedindividuals in the neighbors to calculate the impact ofherd effect and the expression is (1minus119890minus119899(119905)119896)(1minus1119890)Above all the herd mentality effect can be describedas ((1minus119890minus119898(119905)119896)minus(1minus119890minus119899(119905)119896))(1minus1119890)In this paper the propagation factor is composed ofthese two parts by the assigned 12 weight for theexpressiveness of the model and the computability ofsimulations which can be expressed in (1)Thegreaterp(t) the more possibility of transferring state fromunaware to interested
119901 (119905) = 12lowast (1 minus (1 minus 120572)119898(119905)) + 1
2
lowast(1 minus 119890minus119898(119905)119896) + (119890minus119899(119905)119896 minus 1)
1 minus 1119890
(1)
(b) If a node is in concerned state memory superpositionfactor determines state transition In our designedmodel we believe that human memory has an impacton the spread of information It is easy for anindividual in concerned state to change their stateby constantly having neighbors to spread or rejectthe message within a short period of time When
4 Complexity
different friends forward the same topic multipletimes it will promote the concerned state to becomethe interested state By contrast if not heard for along time the topic will be abandoned We definethe one who receives the information over the timeas s(t) which quantitatively describes the cumulativeeffect of the memory Initially set s(t) to 0 ie thevariable is 0 when a node changes from an unawarestate to a concerned state for the first time at tth timestep If there are new turned interested nodes aroundthe next time step then s(t+1) = s(t) +1 Otherwises(t+1) = s(t) - 1 We set threshold 120576 When memorysuperposition factor accumulates above 120576 the stateturns to interested Contrarily if s(t) lt -120576 it becomesabandoned state
(c) If the node is in interested state information atten-uation determines state transition Attenuation is acompound process here we consider both the interestdecay and time decayWe use exponential function torepresent time decay and the proportion of interestedindividuals in the neighbors to represent interestdecay Thus information attenuation factor can beexpressed as (2) We set threshold 120575 when a(t) decaysbelow the threshold the interested state node willturn to abandoned state
119886 (119905) = 119890minus119905 lowast 119898 (119905)119896
(2)
3 Design of Cellular AutomataCellular automata (CA) constitute the model of physicalsystems where space and time are discrete and interactionsare local It is an effective theoretical tool for studyingcomplex systems and at the same time can be simulatedexactly by computer program because of their intrinsicdiscreteness For those reasons we adopt CA to realize OSISmodel In CA the value of each cell presents the local statewhich can only interact with its neighbors and is incapable ofglobal communicationThe state is updated simultaneously atdiscrete time steps according to the states of their neighborsat the preceding time step Here are the elements of ourproposed cellular automata
(a) Cellular space Our model is set up in complexnetwork each node in the network consists of thecellular space
(b) Cellular neighborhood For the reason that our cellu-lar space is not the regular two-dimensional networkwe cannot adopt traditional Von Neumann or Moorestructure The neighborhood in Figure 2 illustratesthat the nodes associated with edges can be regardedas adjacent relationship which we call neighbors
(c) Cellular state According to OSIS model the stateof each cell is unaware concerned interested orabandoned
(d) State transfer rule As the methodology describedabove unaware state turns to whether interested orconcerned state is decided by propagation factor
Figure 2 The neighborhood of each cell is formed by the cellsassociated with edges
Memory superposition factor is able to determineconcerned nodersquos next state Furthermore informa-tion attenuation factor can turn interested to aban-doned state The detailed transfer relationship can befound in Figure 1
4 Experiments and Evaluation
41 Implementation of Online Social Network We use small-world network to simulate online social network in ourexperiments On the one hand small-world theory is alsocalled six degrees of separation which means the distancebetween two strangers is at most six people This feature isthe closest to the online social network On the other handon the basis of sampling and analysis on millions of leveldata excavated from the social groups in [38] online socialnetwork has small-world effect
The implementation of a small-world network comesfrom a regular lattice with a linear size L and with N =LlowastL nodes The initial node degree is k which means eachnode is connected to its k2 nearest neighbors clockwise andcounterclockwise deleting one link randomly adding a linkbetween two randomly chosen nodes with probability p andavoiding duplicate links and self-loops The link probabilityp turns the nature of the network between that of a regularnetwork (p = 0) and that of a random network (p = 1)
42 Experimental Results In our experiments we use C++programming language to build the small-world networkswith 2500 nodes and an average degree of 6The shortest dis-tance between two nodes is calculated by Dijkstrarsquos algorithmin computer graph theory The results are averaged over 100realizations
421 The Comparison with Traditional SEIR Model Todemonstrate the effectiveness of OSIS we compare ourmodel with traditional epidemics SEIR model We firstlyobserve the information spreading over time and spaceconsidering various infection rates Afterwards we illustratethe evolution Particularly unaware concerned interestedand abandoned state correspond to susceptible exposedinfected and recovered respectively in SEIR
Complexity 5
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
unawareconcerned
interestedabandoned
(a)
0 5 10 15 20 25 30
t
00
02
04
06
08
10
perc
entage
unawareconcerned
interestedabandoned
(b)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
unawareconcerned
interestedabandoned
(c)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
susceptibleexposed
infectedrecovered
(d)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
susceptibleexposed
infectedrecovered
(e)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
susceptibleexposed
infectedrecovered
(f)
Figure 3 The percentage of each state in a continuous period of time The spreading model is OSIS ((a) (b) (c)) and SEIR ((d) (e) (f))separatelyThe infection rate is 035 in (a) and (d) 055 in (b) and (f) 075 in (c) and (e)
Firstly we set up three sets of contrastive experimentsWithout generality we set the rewiring probability of thesmall-world networks 1 ie a ldquocompleterdquo small-world net-work And the threshold of the memory superposition factoris 1 and the threshold of information attenuation is 05 The
infection rate is 035 and 055 in the first and second set whichis 075 in the third one
Initially one node in the centre is infected After 30 timeperiods the percentage of each state is recorded and the resultis shown in Figure 3
6 Complexity
t
(a)
(b)
(c)
(d)
Figure 4 The information spreading evolution in small-world The spreading model is OSIS ((a) (b)) and SEIR ((c) (d)) separately (a) and(c) present the evolution of interested nodes (b) and (d) stand for abandoned nodes The infection rate is 035 The time period is 0 5 10 15
It can be seen from Figures 3(a) and 3(d) that wheninfection rate is low the percentage of nodes in final aban-doned state is 94 in SEIR which decreases as much as38 to 58 in OSIS This is because when the numberof initial infected nodes as well as information spreadingcapacity is low herd mentality inhibits the transmission ofinformation The phenomenon explains herdmentality effecton online social networks from the reverse Figures 3(b) and3(e) illustrate the peak of interested nodes is significantlydelayed which is from the 7th time step in SEIR to the 13thtime step in OSIS The result reveals that both propagationand disappearance of a topic is gradual over time or with theattention of people around which is different from epidemicFigures 3(c) and 3(f) demonstrate thatwhen the infection rateis high the number of infected nodes is increased by 72
to 057 from 033 This is due to the fact that herd mentalityand memory superposition effect promote the spreading ofinformation
Furthermore to discover the evolution of OSIS weinvestigate the information spreading over space In thisexperiment rewiring probability of the small-world networkis that 01 and 1 nodes in the centre of the network are infectedinitially It can be seen from Figure 4 that the evolution ofonline social network information is a gradual process whilethe disease transmission has the characteristics of suddenoutbreak and the one exposed to the disease eventuallydevelops antibodies Further comparing Figures 4(a) and4(c) it is evident that the disease with SEIR model spreadsvery fast and suddenly dies out By contrast the informationspreads throughout the network smoothly Meanwhile the
Complexity 7
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60 70t
num
ber o
f aba
ndon
ed
initially 1 interested initially 10 interested initially 100 interested
(a)
0
500
1000
1500
2000
2500
num
ber o
f aba
ndon
ed
0 40 50 60 7010 20 30t
initially 1 interested initially 10 interested initially 100 interested
(b)
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60 70t
num
ber o
f aba
ndon
ed
initially 1 interested initially 10 interested initially 100 interested
(c)
Figure 5 The influence of different initial infection ratio in small-word The rewiring probability is 0 01 1 in (a) (b) (c) respectively Eachexperiment randomly selects 1 10 100 initial interested nodes
contrast between Figures 4(b) and 4(d) shows that theabandoned nodes in SEIR are in the large area while thenumber of immunized nodes in OSIS is flat The results areconsistent with the above conclusions
422 The Influence of Different Initial Infection Ratio Inorder to prove the influence of initial infection ratio oninformation spreading we randomly select 1 10 and 100initial interested nodes The rewiring probability in each setof experiments is 0 01 and 1 respectively and the infectionrate is 035The threshold of thememory superposition factoris 1 and the threshold of information attenuation is 05 Theabandoned state nodes are recorded
It is observable fromFigure 5(a) that the number of initialinfected nodes plays a decisive role in spreading when small-world effect is not obvious (ie rewiring probability is 0)When the number of initial interested nodes is 10 the numberof the eventually abandoned nodes is 8 times more than thatof 1 initial interested node while the multiple is 60 timeswhen the number of initial interested nodes is 100 and 1 Asshown in Figures 5(b) and 5(c) a higher rewiring probability
leads to a faster spreading and a higher steady-state infectiondensity When the rewiring probability is 01 the peak isreached at the 15th time period while it reduces to 10th timeperiodwhen the rewiring probability is 1 All the results abovereveal that the more initial infected nodes the easier it is tospread information However this effect tends to convergeas the rewiring probability increases Additionally with thestrengthening of the small-world features (ie the averagenumber of friends per person in social networks is greater)the information can spread farther and inhibited harder
423 The Effect of Memory Superposition To explore thememory superposition effect on information spreading Weperform simulations to obtain the infection density in thesteady state as a function of the effective infection rate 120572for various superposition thresholds In this experiment therewiring probability of the network is 01 and the threshold ofinformation attenuation is 05
We set up three sets of experiments The infection ratein each set is 01 05 and 07 meanwhile the superpositionthreshold is 1 3 and 5 Initially one node in network is
8 Complexity
0501 07infection rate
superposition threshold=1superposition threshold=3superposition threshold=5
0
500
1000
1500
2000
2500nu
mber o
f aba
ndon
ed
Figure 6The number of abandoned node under different superpo-sition threshold
randomly chosen to be infected After 100 time periods thenumber of nodes in abandoned state is recorded
As shown in Figure 6 the results of the experimentsall indicate that the effect of superposition is obvious andthe higher the threshold the fewer the nodes affected Forexample when the infection rate is 05 the number ofabandoned nodes with the threshold 1 is almost 4 timesmore than that with threshold 3 which even goes to 19 timeswhen compared with the result of the threshold 5The resultsaccurately explain why information spreads widely in onlinesocial networks With the explosion of Internet and we-media the spreading of information costs little investigationand individuals spread information with less filters Such lessfrequent contact to information leads to the rapid spreadingMemory superposition factor in our model stands for thenumber of contacts
5 Conclusions
In summary we introduce an OSIS model to describeinformation spreading in online social networks Firstly weproposeOSIS by combining epidemics SEIRmodelwith indi-vidualrsquos cognitive psychology Secondly a cellular automatamodel of information transmission is designed which pro-vides a computational method for information spreadingfrom a microscopic perspective Additionally we use small-world network to simulate online social network whichverifies the spreading and evolution of information Theexperimental results reveal that our model is consistent withthe propagation of information in online social networksWe find that individualrsquos cognition affects the informationspreading behavior When the infection rate is low cognitivepsychology prevents the spread of information By contrastwhen the infection rate is sufficiently high it promotes trans-mission We demonstrated that the explosion of online socialnetwork scale and the convenience of we-media can greatlyincrease the ability of information disseminationMeanwhile
our research shows that the demise of information is affectedby the dual effects of the decay of time and heat
Our current work mainly concentrates on the spreadingmodel itself As for the future work we believe that the realonline social network is not a single small-world networkDue to the vigorous development ofWeb 20 online networkssuch as Weibo Twitter and Facebook are intertwined Wedeem the information spreading in interconnected small-world networks as the promising and challenging futurework
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A L Barabasi and R E Crandall ldquoLinked the new science ofnetworksrdquo American Journal of Physics vol 71 no 4 pp 409ndash411 2003
[2] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003
[3] A Barrat M Barthlemy and A Vespignani Dynamical pro-cesses on complex networks 2008
[4] R Cohen and S Havlin ldquoComplex networks structure robust-ness and function London UKrdquo Cambridge University Press2010
[5] P Erdos and A Renyi ldquoOn the evolution of random graphsrdquoTransactions of the AmericanMathematical Society vol 286 no1 pp 257ndash274 2011
[6] D J Watts and S H Strogatz Collective Dynamics of rsquoSmall-Worldrsquo Networks 2011
[7] A Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquo Science vol 286 no 5439 pp 509ndash512 1999
[8] M Li Y Fan D Wang D Li J Wu and Z Di ldquoSmall-worldeffect induced by weight randomization on regular networksrdquoPhysics Letters A vol 364 p 488 2007
[9] Q Jin L Wang C Xia and Z Wang ldquoSpontaneous symmetrybreaking in interdependent networked gamerdquo Scientic Reportsvol 4 p 4095 2015
[10] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001
[11] N Bailey ldquoThe mathematical theory of infectious diseases andits applicationsrdquo The Journal of Immunology vol 34 no 5 pp955-956 1978
[12] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press 1991
[13] N C Grassly and C Fraser ldquoMathematicalmodels of infectiousdisease transmissionrdquo Nature Reviews Microbiology vol 6 no6 pp 477ndash487 2008
[14] R M May and A L Lloyd ldquoInfection dynamics on scale-freenetworksrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 64 no 6 Article ID 066112 2001
Complexity 9
[15] M H A Biswas L T Paiva and M D R D Pinho ldquoASEIR model for control of infectious diseases with constraintsrdquoMathematical Biosciences amp Engineering vol 11 no 4 p 7612014
[16] X Fu A Passarella D Quercia A Sala and T Strufe ldquoOnlineSocial Networksrdquo Computer Communications vol 73 pp 163ndash166 2016
[17] X Ke Z Sai C Hao et al ldquoMeasurement and analysis of onlinesocial networksrdquoChinese Journal of Computers pp 29ndash42 2014
[18] Q Ying Z Nan and Z Laijun ldquoResearch on the law of weibopublicity transmissionrdquo Journal of the China Society for Scienticand Technical Information vol 31 no 12 pp 1299ndash1304 2012(Chinese)
[19] weibo httpsweibocom[20] F Xiong Y Liu Z J Zhang J Zhu and Y Zhang ldquoAn
information diffusion model based on retweeting mechanismfor online social mediardquo Physics Letters A vol 376 p 2103 2012
[21] D Centola ldquoThe spread of behavior in an online social networkexperimentrdquo Science vol 329 no 5996 pp 1194ndash1197 2010
[22] H Wu A Arenas and S Gmez ldquoInfluence of trust in thespreading of informationrdquo Physical Review vol 95 2016
[23] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 no 11 Article ID e100096815 pages 2010
[24] L Lu D-B Chen and T Zhou ldquoSmall world yields the mosteffective information spreadingrdquo New Journal of Physics p825834 2011
[25] J X Zhang D B Chen Q Dong and Z D Zhao ldquoIdentifyinga set of influential spreaders in complex networksrdquo ScienticReports vol 6 2016
[26] W Wang M Cai and M Zheng ldquoSocial contagions oncorrelated multiplex networksrdquo Physica A Statistical Mechanicsamp Its Applications vol 499 2017
[27] W Wang X L Chen and L F Zhong ldquoSocial contagions withheterogeneous credibilityrdquo Physica A Statistical Mechanics amp ItsApplications 2018
[28] W Wang M Tang S H Eugene and L A BraunsteinldquoUnification of theoretical approaches for epidemic spreadingon complex networksrdquoReports on Progress in Physics p 0366032017
[29] W Wang M Tang H F Zhang H Gao Y Do and Z HLiu ldquoEpidemic spreading on complex networks with generaldegree and weight distributionsrdquo Physical Review E StatisticalNonlinear ampa Soft Matter Physics vol 90 p 042803 2014
[30] X Zhu W Wang S Cai and H E Stanley ldquoStanley Dynamicsof social contagions with local trend imitationrdquo Scientic Reportsvol 8 2018 Scientic Reports
[31] X Zhu W Wang S Cai and H E Stanley ldquoOptimal imitationcapacity and crossover phenomenon in the dynamics of socialcontagionsrdquo Journal of Statistical Mechanics Theory amp Experi-ment Article ID 063405 p 063405 2018
[32] A L Barabsi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 p 173 1999
[33] S A Pandit and R E Amritkar ldquoCharacterization and controlof small-world networksrdquo Physical Review E Statistical Nonlin-ear and SoftMatter Physics vol 60 no 2 pp R1119ndashR1122 1999
[34] J VonNeumann Papers of John Von Neumann on Computing ampComputer Theory vol 1 1951
[35] B Chopard and M Droz Cellular Automata for ModelingPhysics Cambridge University Press Cambridge UK 1998
[36] G J Martnez A Adamatzky F Chen and L Chua ldquoOn SolitonCollisions between Localizations in Complex Elementary Cel-lular Automata Rules 54 and 110 andBeyondrdquoComplex Systemsvol 117 2013
[37] H F Gagliardi and D Alves ldquoSmall-world effect in epidemicsusing cellular automatardquo Mathematical Population Studies AnInternational Journal of Mathematical Demography vol 17 no2 pp 79ndash90 2010
[38] X Jin J Li and L Zhang Online Social Networks Based onComplex Network Theory and Simulation Analysis SpringerInternational Publishing 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
4 Complexity
different friends forward the same topic multipletimes it will promote the concerned state to becomethe interested state By contrast if not heard for along time the topic will be abandoned We definethe one who receives the information over the timeas s(t) which quantitatively describes the cumulativeeffect of the memory Initially set s(t) to 0 ie thevariable is 0 when a node changes from an unawarestate to a concerned state for the first time at tth timestep If there are new turned interested nodes aroundthe next time step then s(t+1) = s(t) +1 Otherwises(t+1) = s(t) - 1 We set threshold 120576 When memorysuperposition factor accumulates above 120576 the stateturns to interested Contrarily if s(t) lt -120576 it becomesabandoned state
(c) If the node is in interested state information atten-uation determines state transition Attenuation is acompound process here we consider both the interestdecay and time decayWe use exponential function torepresent time decay and the proportion of interestedindividuals in the neighbors to represent interestdecay Thus information attenuation factor can beexpressed as (2) We set threshold 120575 when a(t) decaysbelow the threshold the interested state node willturn to abandoned state
119886 (119905) = 119890minus119905 lowast 119898 (119905)119896
(2)
3 Design of Cellular AutomataCellular automata (CA) constitute the model of physicalsystems where space and time are discrete and interactionsare local It is an effective theoretical tool for studyingcomplex systems and at the same time can be simulatedexactly by computer program because of their intrinsicdiscreteness For those reasons we adopt CA to realize OSISmodel In CA the value of each cell presents the local statewhich can only interact with its neighbors and is incapable ofglobal communicationThe state is updated simultaneously atdiscrete time steps according to the states of their neighborsat the preceding time step Here are the elements of ourproposed cellular automata
(a) Cellular space Our model is set up in complexnetwork each node in the network consists of thecellular space
(b) Cellular neighborhood For the reason that our cellu-lar space is not the regular two-dimensional networkwe cannot adopt traditional Von Neumann or Moorestructure The neighborhood in Figure 2 illustratesthat the nodes associated with edges can be regardedas adjacent relationship which we call neighbors
(c) Cellular state According to OSIS model the stateof each cell is unaware concerned interested orabandoned
(d) State transfer rule As the methodology describedabove unaware state turns to whether interested orconcerned state is decided by propagation factor
Figure 2 The neighborhood of each cell is formed by the cellsassociated with edges
Memory superposition factor is able to determineconcerned nodersquos next state Furthermore informa-tion attenuation factor can turn interested to aban-doned state The detailed transfer relationship can befound in Figure 1
4 Experiments and Evaluation
41 Implementation of Online Social Network We use small-world network to simulate online social network in ourexperiments On the one hand small-world theory is alsocalled six degrees of separation which means the distancebetween two strangers is at most six people This feature isthe closest to the online social network On the other handon the basis of sampling and analysis on millions of leveldata excavated from the social groups in [38] online socialnetwork has small-world effect
The implementation of a small-world network comesfrom a regular lattice with a linear size L and with N =LlowastL nodes The initial node degree is k which means eachnode is connected to its k2 nearest neighbors clockwise andcounterclockwise deleting one link randomly adding a linkbetween two randomly chosen nodes with probability p andavoiding duplicate links and self-loops The link probabilityp turns the nature of the network between that of a regularnetwork (p = 0) and that of a random network (p = 1)
42 Experimental Results In our experiments we use C++programming language to build the small-world networkswith 2500 nodes and an average degree of 6The shortest dis-tance between two nodes is calculated by Dijkstrarsquos algorithmin computer graph theory The results are averaged over 100realizations
421 The Comparison with Traditional SEIR Model Todemonstrate the effectiveness of OSIS we compare ourmodel with traditional epidemics SEIR model We firstlyobserve the information spreading over time and spaceconsidering various infection rates Afterwards we illustratethe evolution Particularly unaware concerned interestedand abandoned state correspond to susceptible exposedinfected and recovered respectively in SEIR
Complexity 5
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
unawareconcerned
interestedabandoned
(a)
0 5 10 15 20 25 30
t
00
02
04
06
08
10
perc
entage
unawareconcerned
interestedabandoned
(b)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
unawareconcerned
interestedabandoned
(c)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
susceptibleexposed
infectedrecovered
(d)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
susceptibleexposed
infectedrecovered
(e)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
susceptibleexposed
infectedrecovered
(f)
Figure 3 The percentage of each state in a continuous period of time The spreading model is OSIS ((a) (b) (c)) and SEIR ((d) (e) (f))separatelyThe infection rate is 035 in (a) and (d) 055 in (b) and (f) 075 in (c) and (e)
Firstly we set up three sets of contrastive experimentsWithout generality we set the rewiring probability of thesmall-world networks 1 ie a ldquocompleterdquo small-world net-work And the threshold of the memory superposition factoris 1 and the threshold of information attenuation is 05 The
infection rate is 035 and 055 in the first and second set whichis 075 in the third one
Initially one node in the centre is infected After 30 timeperiods the percentage of each state is recorded and the resultis shown in Figure 3
6 Complexity
t
(a)
(b)
(c)
(d)
Figure 4 The information spreading evolution in small-world The spreading model is OSIS ((a) (b)) and SEIR ((c) (d)) separately (a) and(c) present the evolution of interested nodes (b) and (d) stand for abandoned nodes The infection rate is 035 The time period is 0 5 10 15
It can be seen from Figures 3(a) and 3(d) that wheninfection rate is low the percentage of nodes in final aban-doned state is 94 in SEIR which decreases as much as38 to 58 in OSIS This is because when the numberof initial infected nodes as well as information spreadingcapacity is low herd mentality inhibits the transmission ofinformation The phenomenon explains herdmentality effecton online social networks from the reverse Figures 3(b) and3(e) illustrate the peak of interested nodes is significantlydelayed which is from the 7th time step in SEIR to the 13thtime step in OSIS The result reveals that both propagationand disappearance of a topic is gradual over time or with theattention of people around which is different from epidemicFigures 3(c) and 3(f) demonstrate thatwhen the infection rateis high the number of infected nodes is increased by 72
to 057 from 033 This is due to the fact that herd mentalityand memory superposition effect promote the spreading ofinformation
Furthermore to discover the evolution of OSIS weinvestigate the information spreading over space In thisexperiment rewiring probability of the small-world networkis that 01 and 1 nodes in the centre of the network are infectedinitially It can be seen from Figure 4 that the evolution ofonline social network information is a gradual process whilethe disease transmission has the characteristics of suddenoutbreak and the one exposed to the disease eventuallydevelops antibodies Further comparing Figures 4(a) and4(c) it is evident that the disease with SEIR model spreadsvery fast and suddenly dies out By contrast the informationspreads throughout the network smoothly Meanwhile the
Complexity 7
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60 70t
num
ber o
f aba
ndon
ed
initially 1 interested initially 10 interested initially 100 interested
(a)
0
500
1000
1500
2000
2500
num
ber o
f aba
ndon
ed
0 40 50 60 7010 20 30t
initially 1 interested initially 10 interested initially 100 interested
(b)
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60 70t
num
ber o
f aba
ndon
ed
initially 1 interested initially 10 interested initially 100 interested
(c)
Figure 5 The influence of different initial infection ratio in small-word The rewiring probability is 0 01 1 in (a) (b) (c) respectively Eachexperiment randomly selects 1 10 100 initial interested nodes
contrast between Figures 4(b) and 4(d) shows that theabandoned nodes in SEIR are in the large area while thenumber of immunized nodes in OSIS is flat The results areconsistent with the above conclusions
422 The Influence of Different Initial Infection Ratio Inorder to prove the influence of initial infection ratio oninformation spreading we randomly select 1 10 and 100initial interested nodes The rewiring probability in each setof experiments is 0 01 and 1 respectively and the infectionrate is 035The threshold of thememory superposition factoris 1 and the threshold of information attenuation is 05 Theabandoned state nodes are recorded
It is observable fromFigure 5(a) that the number of initialinfected nodes plays a decisive role in spreading when small-world effect is not obvious (ie rewiring probability is 0)When the number of initial interested nodes is 10 the numberof the eventually abandoned nodes is 8 times more than thatof 1 initial interested node while the multiple is 60 timeswhen the number of initial interested nodes is 100 and 1 Asshown in Figures 5(b) and 5(c) a higher rewiring probability
leads to a faster spreading and a higher steady-state infectiondensity When the rewiring probability is 01 the peak isreached at the 15th time period while it reduces to 10th timeperiodwhen the rewiring probability is 1 All the results abovereveal that the more initial infected nodes the easier it is tospread information However this effect tends to convergeas the rewiring probability increases Additionally with thestrengthening of the small-world features (ie the averagenumber of friends per person in social networks is greater)the information can spread farther and inhibited harder
423 The Effect of Memory Superposition To explore thememory superposition effect on information spreading Weperform simulations to obtain the infection density in thesteady state as a function of the effective infection rate 120572for various superposition thresholds In this experiment therewiring probability of the network is 01 and the threshold ofinformation attenuation is 05
We set up three sets of experiments The infection ratein each set is 01 05 and 07 meanwhile the superpositionthreshold is 1 3 and 5 Initially one node in network is
8 Complexity
0501 07infection rate
superposition threshold=1superposition threshold=3superposition threshold=5
0
500
1000
1500
2000
2500nu
mber o
f aba
ndon
ed
Figure 6The number of abandoned node under different superpo-sition threshold
randomly chosen to be infected After 100 time periods thenumber of nodes in abandoned state is recorded
As shown in Figure 6 the results of the experimentsall indicate that the effect of superposition is obvious andthe higher the threshold the fewer the nodes affected Forexample when the infection rate is 05 the number ofabandoned nodes with the threshold 1 is almost 4 timesmore than that with threshold 3 which even goes to 19 timeswhen compared with the result of the threshold 5The resultsaccurately explain why information spreads widely in onlinesocial networks With the explosion of Internet and we-media the spreading of information costs little investigationand individuals spread information with less filters Such lessfrequent contact to information leads to the rapid spreadingMemory superposition factor in our model stands for thenumber of contacts
5 Conclusions
In summary we introduce an OSIS model to describeinformation spreading in online social networks Firstly weproposeOSIS by combining epidemics SEIRmodelwith indi-vidualrsquos cognitive psychology Secondly a cellular automatamodel of information transmission is designed which pro-vides a computational method for information spreadingfrom a microscopic perspective Additionally we use small-world network to simulate online social network whichverifies the spreading and evolution of information Theexperimental results reveal that our model is consistent withthe propagation of information in online social networksWe find that individualrsquos cognition affects the informationspreading behavior When the infection rate is low cognitivepsychology prevents the spread of information By contrastwhen the infection rate is sufficiently high it promotes trans-mission We demonstrated that the explosion of online socialnetwork scale and the convenience of we-media can greatlyincrease the ability of information disseminationMeanwhile
our research shows that the demise of information is affectedby the dual effects of the decay of time and heat
Our current work mainly concentrates on the spreadingmodel itself As for the future work we believe that the realonline social network is not a single small-world networkDue to the vigorous development ofWeb 20 online networkssuch as Weibo Twitter and Facebook are intertwined Wedeem the information spreading in interconnected small-world networks as the promising and challenging futurework
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A L Barabasi and R E Crandall ldquoLinked the new science ofnetworksrdquo American Journal of Physics vol 71 no 4 pp 409ndash411 2003
[2] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003
[3] A Barrat M Barthlemy and A Vespignani Dynamical pro-cesses on complex networks 2008
[4] R Cohen and S Havlin ldquoComplex networks structure robust-ness and function London UKrdquo Cambridge University Press2010
[5] P Erdos and A Renyi ldquoOn the evolution of random graphsrdquoTransactions of the AmericanMathematical Society vol 286 no1 pp 257ndash274 2011
[6] D J Watts and S H Strogatz Collective Dynamics of rsquoSmall-Worldrsquo Networks 2011
[7] A Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquo Science vol 286 no 5439 pp 509ndash512 1999
[8] M Li Y Fan D Wang D Li J Wu and Z Di ldquoSmall-worldeffect induced by weight randomization on regular networksrdquoPhysics Letters A vol 364 p 488 2007
[9] Q Jin L Wang C Xia and Z Wang ldquoSpontaneous symmetrybreaking in interdependent networked gamerdquo Scientic Reportsvol 4 p 4095 2015
[10] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001
[11] N Bailey ldquoThe mathematical theory of infectious diseases andits applicationsrdquo The Journal of Immunology vol 34 no 5 pp955-956 1978
[12] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press 1991
[13] N C Grassly and C Fraser ldquoMathematicalmodels of infectiousdisease transmissionrdquo Nature Reviews Microbiology vol 6 no6 pp 477ndash487 2008
[14] R M May and A L Lloyd ldquoInfection dynamics on scale-freenetworksrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 64 no 6 Article ID 066112 2001
Complexity 9
[15] M H A Biswas L T Paiva and M D R D Pinho ldquoASEIR model for control of infectious diseases with constraintsrdquoMathematical Biosciences amp Engineering vol 11 no 4 p 7612014
[16] X Fu A Passarella D Quercia A Sala and T Strufe ldquoOnlineSocial Networksrdquo Computer Communications vol 73 pp 163ndash166 2016
[17] X Ke Z Sai C Hao et al ldquoMeasurement and analysis of onlinesocial networksrdquoChinese Journal of Computers pp 29ndash42 2014
[18] Q Ying Z Nan and Z Laijun ldquoResearch on the law of weibopublicity transmissionrdquo Journal of the China Society for Scienticand Technical Information vol 31 no 12 pp 1299ndash1304 2012(Chinese)
[19] weibo httpsweibocom[20] F Xiong Y Liu Z J Zhang J Zhu and Y Zhang ldquoAn
information diffusion model based on retweeting mechanismfor online social mediardquo Physics Letters A vol 376 p 2103 2012
[21] D Centola ldquoThe spread of behavior in an online social networkexperimentrdquo Science vol 329 no 5996 pp 1194ndash1197 2010
[22] H Wu A Arenas and S Gmez ldquoInfluence of trust in thespreading of informationrdquo Physical Review vol 95 2016
[23] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 no 11 Article ID e100096815 pages 2010
[24] L Lu D-B Chen and T Zhou ldquoSmall world yields the mosteffective information spreadingrdquo New Journal of Physics p825834 2011
[25] J X Zhang D B Chen Q Dong and Z D Zhao ldquoIdentifyinga set of influential spreaders in complex networksrdquo ScienticReports vol 6 2016
[26] W Wang M Cai and M Zheng ldquoSocial contagions oncorrelated multiplex networksrdquo Physica A Statistical Mechanicsamp Its Applications vol 499 2017
[27] W Wang X L Chen and L F Zhong ldquoSocial contagions withheterogeneous credibilityrdquo Physica A Statistical Mechanics amp ItsApplications 2018
[28] W Wang M Tang S H Eugene and L A BraunsteinldquoUnification of theoretical approaches for epidemic spreadingon complex networksrdquoReports on Progress in Physics p 0366032017
[29] W Wang M Tang H F Zhang H Gao Y Do and Z HLiu ldquoEpidemic spreading on complex networks with generaldegree and weight distributionsrdquo Physical Review E StatisticalNonlinear ampa Soft Matter Physics vol 90 p 042803 2014
[30] X Zhu W Wang S Cai and H E Stanley ldquoStanley Dynamicsof social contagions with local trend imitationrdquo Scientic Reportsvol 8 2018 Scientic Reports
[31] X Zhu W Wang S Cai and H E Stanley ldquoOptimal imitationcapacity and crossover phenomenon in the dynamics of socialcontagionsrdquo Journal of Statistical Mechanics Theory amp Experi-ment Article ID 063405 p 063405 2018
[32] A L Barabsi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 p 173 1999
[33] S A Pandit and R E Amritkar ldquoCharacterization and controlof small-world networksrdquo Physical Review E Statistical Nonlin-ear and SoftMatter Physics vol 60 no 2 pp R1119ndashR1122 1999
[34] J VonNeumann Papers of John Von Neumann on Computing ampComputer Theory vol 1 1951
[35] B Chopard and M Droz Cellular Automata for ModelingPhysics Cambridge University Press Cambridge UK 1998
[36] G J Martnez A Adamatzky F Chen and L Chua ldquoOn SolitonCollisions between Localizations in Complex Elementary Cel-lular Automata Rules 54 and 110 andBeyondrdquoComplex Systemsvol 117 2013
[37] H F Gagliardi and D Alves ldquoSmall-world effect in epidemicsusing cellular automatardquo Mathematical Population Studies AnInternational Journal of Mathematical Demography vol 17 no2 pp 79ndash90 2010
[38] X Jin J Li and L Zhang Online Social Networks Based onComplex Network Theory and Simulation Analysis SpringerInternational Publishing 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 5
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
unawareconcerned
interestedabandoned
(a)
0 5 10 15 20 25 30
t
00
02
04
06
08
10
perc
entage
unawareconcerned
interestedabandoned
(b)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
unawareconcerned
interestedabandoned
(c)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
susceptibleexposed
infectedrecovered
(d)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
susceptibleexposed
infectedrecovered
(e)
0 5 10 15 20 25 30t
00
02
04
06
08
10
perc
entage
susceptibleexposed
infectedrecovered
(f)
Figure 3 The percentage of each state in a continuous period of time The spreading model is OSIS ((a) (b) (c)) and SEIR ((d) (e) (f))separatelyThe infection rate is 035 in (a) and (d) 055 in (b) and (f) 075 in (c) and (e)
Firstly we set up three sets of contrastive experimentsWithout generality we set the rewiring probability of thesmall-world networks 1 ie a ldquocompleterdquo small-world net-work And the threshold of the memory superposition factoris 1 and the threshold of information attenuation is 05 The
infection rate is 035 and 055 in the first and second set whichis 075 in the third one
Initially one node in the centre is infected After 30 timeperiods the percentage of each state is recorded and the resultis shown in Figure 3
6 Complexity
t
(a)
(b)
(c)
(d)
Figure 4 The information spreading evolution in small-world The spreading model is OSIS ((a) (b)) and SEIR ((c) (d)) separately (a) and(c) present the evolution of interested nodes (b) and (d) stand for abandoned nodes The infection rate is 035 The time period is 0 5 10 15
It can be seen from Figures 3(a) and 3(d) that wheninfection rate is low the percentage of nodes in final aban-doned state is 94 in SEIR which decreases as much as38 to 58 in OSIS This is because when the numberof initial infected nodes as well as information spreadingcapacity is low herd mentality inhibits the transmission ofinformation The phenomenon explains herdmentality effecton online social networks from the reverse Figures 3(b) and3(e) illustrate the peak of interested nodes is significantlydelayed which is from the 7th time step in SEIR to the 13thtime step in OSIS The result reveals that both propagationand disappearance of a topic is gradual over time or with theattention of people around which is different from epidemicFigures 3(c) and 3(f) demonstrate thatwhen the infection rateis high the number of infected nodes is increased by 72
to 057 from 033 This is due to the fact that herd mentalityand memory superposition effect promote the spreading ofinformation
Furthermore to discover the evolution of OSIS weinvestigate the information spreading over space In thisexperiment rewiring probability of the small-world networkis that 01 and 1 nodes in the centre of the network are infectedinitially It can be seen from Figure 4 that the evolution ofonline social network information is a gradual process whilethe disease transmission has the characteristics of suddenoutbreak and the one exposed to the disease eventuallydevelops antibodies Further comparing Figures 4(a) and4(c) it is evident that the disease with SEIR model spreadsvery fast and suddenly dies out By contrast the informationspreads throughout the network smoothly Meanwhile the
Complexity 7
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60 70t
num
ber o
f aba
ndon
ed
initially 1 interested initially 10 interested initially 100 interested
(a)
0
500
1000
1500
2000
2500
num
ber o
f aba
ndon
ed
0 40 50 60 7010 20 30t
initially 1 interested initially 10 interested initially 100 interested
(b)
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60 70t
num
ber o
f aba
ndon
ed
initially 1 interested initially 10 interested initially 100 interested
(c)
Figure 5 The influence of different initial infection ratio in small-word The rewiring probability is 0 01 1 in (a) (b) (c) respectively Eachexperiment randomly selects 1 10 100 initial interested nodes
contrast between Figures 4(b) and 4(d) shows that theabandoned nodes in SEIR are in the large area while thenumber of immunized nodes in OSIS is flat The results areconsistent with the above conclusions
422 The Influence of Different Initial Infection Ratio Inorder to prove the influence of initial infection ratio oninformation spreading we randomly select 1 10 and 100initial interested nodes The rewiring probability in each setof experiments is 0 01 and 1 respectively and the infectionrate is 035The threshold of thememory superposition factoris 1 and the threshold of information attenuation is 05 Theabandoned state nodes are recorded
It is observable fromFigure 5(a) that the number of initialinfected nodes plays a decisive role in spreading when small-world effect is not obvious (ie rewiring probability is 0)When the number of initial interested nodes is 10 the numberof the eventually abandoned nodes is 8 times more than thatof 1 initial interested node while the multiple is 60 timeswhen the number of initial interested nodes is 100 and 1 Asshown in Figures 5(b) and 5(c) a higher rewiring probability
leads to a faster spreading and a higher steady-state infectiondensity When the rewiring probability is 01 the peak isreached at the 15th time period while it reduces to 10th timeperiodwhen the rewiring probability is 1 All the results abovereveal that the more initial infected nodes the easier it is tospread information However this effect tends to convergeas the rewiring probability increases Additionally with thestrengthening of the small-world features (ie the averagenumber of friends per person in social networks is greater)the information can spread farther and inhibited harder
423 The Effect of Memory Superposition To explore thememory superposition effect on information spreading Weperform simulations to obtain the infection density in thesteady state as a function of the effective infection rate 120572for various superposition thresholds In this experiment therewiring probability of the network is 01 and the threshold ofinformation attenuation is 05
We set up three sets of experiments The infection ratein each set is 01 05 and 07 meanwhile the superpositionthreshold is 1 3 and 5 Initially one node in network is
8 Complexity
0501 07infection rate
superposition threshold=1superposition threshold=3superposition threshold=5
0
500
1000
1500
2000
2500nu
mber o
f aba
ndon
ed
Figure 6The number of abandoned node under different superpo-sition threshold
randomly chosen to be infected After 100 time periods thenumber of nodes in abandoned state is recorded
As shown in Figure 6 the results of the experimentsall indicate that the effect of superposition is obvious andthe higher the threshold the fewer the nodes affected Forexample when the infection rate is 05 the number ofabandoned nodes with the threshold 1 is almost 4 timesmore than that with threshold 3 which even goes to 19 timeswhen compared with the result of the threshold 5The resultsaccurately explain why information spreads widely in onlinesocial networks With the explosion of Internet and we-media the spreading of information costs little investigationand individuals spread information with less filters Such lessfrequent contact to information leads to the rapid spreadingMemory superposition factor in our model stands for thenumber of contacts
5 Conclusions
In summary we introduce an OSIS model to describeinformation spreading in online social networks Firstly weproposeOSIS by combining epidemics SEIRmodelwith indi-vidualrsquos cognitive psychology Secondly a cellular automatamodel of information transmission is designed which pro-vides a computational method for information spreadingfrom a microscopic perspective Additionally we use small-world network to simulate online social network whichverifies the spreading and evolution of information Theexperimental results reveal that our model is consistent withthe propagation of information in online social networksWe find that individualrsquos cognition affects the informationspreading behavior When the infection rate is low cognitivepsychology prevents the spread of information By contrastwhen the infection rate is sufficiently high it promotes trans-mission We demonstrated that the explosion of online socialnetwork scale and the convenience of we-media can greatlyincrease the ability of information disseminationMeanwhile
our research shows that the demise of information is affectedby the dual effects of the decay of time and heat
Our current work mainly concentrates on the spreadingmodel itself As for the future work we believe that the realonline social network is not a single small-world networkDue to the vigorous development ofWeb 20 online networkssuch as Weibo Twitter and Facebook are intertwined Wedeem the information spreading in interconnected small-world networks as the promising and challenging futurework
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A L Barabasi and R E Crandall ldquoLinked the new science ofnetworksrdquo American Journal of Physics vol 71 no 4 pp 409ndash411 2003
[2] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003
[3] A Barrat M Barthlemy and A Vespignani Dynamical pro-cesses on complex networks 2008
[4] R Cohen and S Havlin ldquoComplex networks structure robust-ness and function London UKrdquo Cambridge University Press2010
[5] P Erdos and A Renyi ldquoOn the evolution of random graphsrdquoTransactions of the AmericanMathematical Society vol 286 no1 pp 257ndash274 2011
[6] D J Watts and S H Strogatz Collective Dynamics of rsquoSmall-Worldrsquo Networks 2011
[7] A Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquo Science vol 286 no 5439 pp 509ndash512 1999
[8] M Li Y Fan D Wang D Li J Wu and Z Di ldquoSmall-worldeffect induced by weight randomization on regular networksrdquoPhysics Letters A vol 364 p 488 2007
[9] Q Jin L Wang C Xia and Z Wang ldquoSpontaneous symmetrybreaking in interdependent networked gamerdquo Scientic Reportsvol 4 p 4095 2015
[10] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001
[11] N Bailey ldquoThe mathematical theory of infectious diseases andits applicationsrdquo The Journal of Immunology vol 34 no 5 pp955-956 1978
[12] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press 1991
[13] N C Grassly and C Fraser ldquoMathematicalmodels of infectiousdisease transmissionrdquo Nature Reviews Microbiology vol 6 no6 pp 477ndash487 2008
[14] R M May and A L Lloyd ldquoInfection dynamics on scale-freenetworksrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 64 no 6 Article ID 066112 2001
Complexity 9
[15] M H A Biswas L T Paiva and M D R D Pinho ldquoASEIR model for control of infectious diseases with constraintsrdquoMathematical Biosciences amp Engineering vol 11 no 4 p 7612014
[16] X Fu A Passarella D Quercia A Sala and T Strufe ldquoOnlineSocial Networksrdquo Computer Communications vol 73 pp 163ndash166 2016
[17] X Ke Z Sai C Hao et al ldquoMeasurement and analysis of onlinesocial networksrdquoChinese Journal of Computers pp 29ndash42 2014
[18] Q Ying Z Nan and Z Laijun ldquoResearch on the law of weibopublicity transmissionrdquo Journal of the China Society for Scienticand Technical Information vol 31 no 12 pp 1299ndash1304 2012(Chinese)
[19] weibo httpsweibocom[20] F Xiong Y Liu Z J Zhang J Zhu and Y Zhang ldquoAn
information diffusion model based on retweeting mechanismfor online social mediardquo Physics Letters A vol 376 p 2103 2012
[21] D Centola ldquoThe spread of behavior in an online social networkexperimentrdquo Science vol 329 no 5996 pp 1194ndash1197 2010
[22] H Wu A Arenas and S Gmez ldquoInfluence of trust in thespreading of informationrdquo Physical Review vol 95 2016
[23] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 no 11 Article ID e100096815 pages 2010
[24] L Lu D-B Chen and T Zhou ldquoSmall world yields the mosteffective information spreadingrdquo New Journal of Physics p825834 2011
[25] J X Zhang D B Chen Q Dong and Z D Zhao ldquoIdentifyinga set of influential spreaders in complex networksrdquo ScienticReports vol 6 2016
[26] W Wang M Cai and M Zheng ldquoSocial contagions oncorrelated multiplex networksrdquo Physica A Statistical Mechanicsamp Its Applications vol 499 2017
[27] W Wang X L Chen and L F Zhong ldquoSocial contagions withheterogeneous credibilityrdquo Physica A Statistical Mechanics amp ItsApplications 2018
[28] W Wang M Tang S H Eugene and L A BraunsteinldquoUnification of theoretical approaches for epidemic spreadingon complex networksrdquoReports on Progress in Physics p 0366032017
[29] W Wang M Tang H F Zhang H Gao Y Do and Z HLiu ldquoEpidemic spreading on complex networks with generaldegree and weight distributionsrdquo Physical Review E StatisticalNonlinear ampa Soft Matter Physics vol 90 p 042803 2014
[30] X Zhu W Wang S Cai and H E Stanley ldquoStanley Dynamicsof social contagions with local trend imitationrdquo Scientic Reportsvol 8 2018 Scientic Reports
[31] X Zhu W Wang S Cai and H E Stanley ldquoOptimal imitationcapacity and crossover phenomenon in the dynamics of socialcontagionsrdquo Journal of Statistical Mechanics Theory amp Experi-ment Article ID 063405 p 063405 2018
[32] A L Barabsi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 p 173 1999
[33] S A Pandit and R E Amritkar ldquoCharacterization and controlof small-world networksrdquo Physical Review E Statistical Nonlin-ear and SoftMatter Physics vol 60 no 2 pp R1119ndashR1122 1999
[34] J VonNeumann Papers of John Von Neumann on Computing ampComputer Theory vol 1 1951
[35] B Chopard and M Droz Cellular Automata for ModelingPhysics Cambridge University Press Cambridge UK 1998
[36] G J Martnez A Adamatzky F Chen and L Chua ldquoOn SolitonCollisions between Localizations in Complex Elementary Cel-lular Automata Rules 54 and 110 andBeyondrdquoComplex Systemsvol 117 2013
[37] H F Gagliardi and D Alves ldquoSmall-world effect in epidemicsusing cellular automatardquo Mathematical Population Studies AnInternational Journal of Mathematical Demography vol 17 no2 pp 79ndash90 2010
[38] X Jin J Li and L Zhang Online Social Networks Based onComplex Network Theory and Simulation Analysis SpringerInternational Publishing 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
6 Complexity
t
(a)
(b)
(c)
(d)
Figure 4 The information spreading evolution in small-world The spreading model is OSIS ((a) (b)) and SEIR ((c) (d)) separately (a) and(c) present the evolution of interested nodes (b) and (d) stand for abandoned nodes The infection rate is 035 The time period is 0 5 10 15
It can be seen from Figures 3(a) and 3(d) that wheninfection rate is low the percentage of nodes in final aban-doned state is 94 in SEIR which decreases as much as38 to 58 in OSIS This is because when the numberof initial infected nodes as well as information spreadingcapacity is low herd mentality inhibits the transmission ofinformation The phenomenon explains herdmentality effecton online social networks from the reverse Figures 3(b) and3(e) illustrate the peak of interested nodes is significantlydelayed which is from the 7th time step in SEIR to the 13thtime step in OSIS The result reveals that both propagationand disappearance of a topic is gradual over time or with theattention of people around which is different from epidemicFigures 3(c) and 3(f) demonstrate thatwhen the infection rateis high the number of infected nodes is increased by 72
to 057 from 033 This is due to the fact that herd mentalityand memory superposition effect promote the spreading ofinformation
Furthermore to discover the evolution of OSIS weinvestigate the information spreading over space In thisexperiment rewiring probability of the small-world networkis that 01 and 1 nodes in the centre of the network are infectedinitially It can be seen from Figure 4 that the evolution ofonline social network information is a gradual process whilethe disease transmission has the characteristics of suddenoutbreak and the one exposed to the disease eventuallydevelops antibodies Further comparing Figures 4(a) and4(c) it is evident that the disease with SEIR model spreadsvery fast and suddenly dies out By contrast the informationspreads throughout the network smoothly Meanwhile the
Complexity 7
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60 70t
num
ber o
f aba
ndon
ed
initially 1 interested initially 10 interested initially 100 interested
(a)
0
500
1000
1500
2000
2500
num
ber o
f aba
ndon
ed
0 40 50 60 7010 20 30t
initially 1 interested initially 10 interested initially 100 interested
(b)
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60 70t
num
ber o
f aba
ndon
ed
initially 1 interested initially 10 interested initially 100 interested
(c)
Figure 5 The influence of different initial infection ratio in small-word The rewiring probability is 0 01 1 in (a) (b) (c) respectively Eachexperiment randomly selects 1 10 100 initial interested nodes
contrast between Figures 4(b) and 4(d) shows that theabandoned nodes in SEIR are in the large area while thenumber of immunized nodes in OSIS is flat The results areconsistent with the above conclusions
422 The Influence of Different Initial Infection Ratio Inorder to prove the influence of initial infection ratio oninformation spreading we randomly select 1 10 and 100initial interested nodes The rewiring probability in each setof experiments is 0 01 and 1 respectively and the infectionrate is 035The threshold of thememory superposition factoris 1 and the threshold of information attenuation is 05 Theabandoned state nodes are recorded
It is observable fromFigure 5(a) that the number of initialinfected nodes plays a decisive role in spreading when small-world effect is not obvious (ie rewiring probability is 0)When the number of initial interested nodes is 10 the numberof the eventually abandoned nodes is 8 times more than thatof 1 initial interested node while the multiple is 60 timeswhen the number of initial interested nodes is 100 and 1 Asshown in Figures 5(b) and 5(c) a higher rewiring probability
leads to a faster spreading and a higher steady-state infectiondensity When the rewiring probability is 01 the peak isreached at the 15th time period while it reduces to 10th timeperiodwhen the rewiring probability is 1 All the results abovereveal that the more initial infected nodes the easier it is tospread information However this effect tends to convergeas the rewiring probability increases Additionally with thestrengthening of the small-world features (ie the averagenumber of friends per person in social networks is greater)the information can spread farther and inhibited harder
423 The Effect of Memory Superposition To explore thememory superposition effect on information spreading Weperform simulations to obtain the infection density in thesteady state as a function of the effective infection rate 120572for various superposition thresholds In this experiment therewiring probability of the network is 01 and the threshold ofinformation attenuation is 05
We set up three sets of experiments The infection ratein each set is 01 05 and 07 meanwhile the superpositionthreshold is 1 3 and 5 Initially one node in network is
8 Complexity
0501 07infection rate
superposition threshold=1superposition threshold=3superposition threshold=5
0
500
1000
1500
2000
2500nu
mber o
f aba
ndon
ed
Figure 6The number of abandoned node under different superpo-sition threshold
randomly chosen to be infected After 100 time periods thenumber of nodes in abandoned state is recorded
As shown in Figure 6 the results of the experimentsall indicate that the effect of superposition is obvious andthe higher the threshold the fewer the nodes affected Forexample when the infection rate is 05 the number ofabandoned nodes with the threshold 1 is almost 4 timesmore than that with threshold 3 which even goes to 19 timeswhen compared with the result of the threshold 5The resultsaccurately explain why information spreads widely in onlinesocial networks With the explosion of Internet and we-media the spreading of information costs little investigationand individuals spread information with less filters Such lessfrequent contact to information leads to the rapid spreadingMemory superposition factor in our model stands for thenumber of contacts
5 Conclusions
In summary we introduce an OSIS model to describeinformation spreading in online social networks Firstly weproposeOSIS by combining epidemics SEIRmodelwith indi-vidualrsquos cognitive psychology Secondly a cellular automatamodel of information transmission is designed which pro-vides a computational method for information spreadingfrom a microscopic perspective Additionally we use small-world network to simulate online social network whichverifies the spreading and evolution of information Theexperimental results reveal that our model is consistent withthe propagation of information in online social networksWe find that individualrsquos cognition affects the informationspreading behavior When the infection rate is low cognitivepsychology prevents the spread of information By contrastwhen the infection rate is sufficiently high it promotes trans-mission We demonstrated that the explosion of online socialnetwork scale and the convenience of we-media can greatlyincrease the ability of information disseminationMeanwhile
our research shows that the demise of information is affectedby the dual effects of the decay of time and heat
Our current work mainly concentrates on the spreadingmodel itself As for the future work we believe that the realonline social network is not a single small-world networkDue to the vigorous development ofWeb 20 online networkssuch as Weibo Twitter and Facebook are intertwined Wedeem the information spreading in interconnected small-world networks as the promising and challenging futurework
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A L Barabasi and R E Crandall ldquoLinked the new science ofnetworksrdquo American Journal of Physics vol 71 no 4 pp 409ndash411 2003
[2] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003
[3] A Barrat M Barthlemy and A Vespignani Dynamical pro-cesses on complex networks 2008
[4] R Cohen and S Havlin ldquoComplex networks structure robust-ness and function London UKrdquo Cambridge University Press2010
[5] P Erdos and A Renyi ldquoOn the evolution of random graphsrdquoTransactions of the AmericanMathematical Society vol 286 no1 pp 257ndash274 2011
[6] D J Watts and S H Strogatz Collective Dynamics of rsquoSmall-Worldrsquo Networks 2011
[7] A Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquo Science vol 286 no 5439 pp 509ndash512 1999
[8] M Li Y Fan D Wang D Li J Wu and Z Di ldquoSmall-worldeffect induced by weight randomization on regular networksrdquoPhysics Letters A vol 364 p 488 2007
[9] Q Jin L Wang C Xia and Z Wang ldquoSpontaneous symmetrybreaking in interdependent networked gamerdquo Scientic Reportsvol 4 p 4095 2015
[10] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001
[11] N Bailey ldquoThe mathematical theory of infectious diseases andits applicationsrdquo The Journal of Immunology vol 34 no 5 pp955-956 1978
[12] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press 1991
[13] N C Grassly and C Fraser ldquoMathematicalmodels of infectiousdisease transmissionrdquo Nature Reviews Microbiology vol 6 no6 pp 477ndash487 2008
[14] R M May and A L Lloyd ldquoInfection dynamics on scale-freenetworksrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 64 no 6 Article ID 066112 2001
Complexity 9
[15] M H A Biswas L T Paiva and M D R D Pinho ldquoASEIR model for control of infectious diseases with constraintsrdquoMathematical Biosciences amp Engineering vol 11 no 4 p 7612014
[16] X Fu A Passarella D Quercia A Sala and T Strufe ldquoOnlineSocial Networksrdquo Computer Communications vol 73 pp 163ndash166 2016
[17] X Ke Z Sai C Hao et al ldquoMeasurement and analysis of onlinesocial networksrdquoChinese Journal of Computers pp 29ndash42 2014
[18] Q Ying Z Nan and Z Laijun ldquoResearch on the law of weibopublicity transmissionrdquo Journal of the China Society for Scienticand Technical Information vol 31 no 12 pp 1299ndash1304 2012(Chinese)
[19] weibo httpsweibocom[20] F Xiong Y Liu Z J Zhang J Zhu and Y Zhang ldquoAn
information diffusion model based on retweeting mechanismfor online social mediardquo Physics Letters A vol 376 p 2103 2012
[21] D Centola ldquoThe spread of behavior in an online social networkexperimentrdquo Science vol 329 no 5996 pp 1194ndash1197 2010
[22] H Wu A Arenas and S Gmez ldquoInfluence of trust in thespreading of informationrdquo Physical Review vol 95 2016
[23] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 no 11 Article ID e100096815 pages 2010
[24] L Lu D-B Chen and T Zhou ldquoSmall world yields the mosteffective information spreadingrdquo New Journal of Physics p825834 2011
[25] J X Zhang D B Chen Q Dong and Z D Zhao ldquoIdentifyinga set of influential spreaders in complex networksrdquo ScienticReports vol 6 2016
[26] W Wang M Cai and M Zheng ldquoSocial contagions oncorrelated multiplex networksrdquo Physica A Statistical Mechanicsamp Its Applications vol 499 2017
[27] W Wang X L Chen and L F Zhong ldquoSocial contagions withheterogeneous credibilityrdquo Physica A Statistical Mechanics amp ItsApplications 2018
[28] W Wang M Tang S H Eugene and L A BraunsteinldquoUnification of theoretical approaches for epidemic spreadingon complex networksrdquoReports on Progress in Physics p 0366032017
[29] W Wang M Tang H F Zhang H Gao Y Do and Z HLiu ldquoEpidemic spreading on complex networks with generaldegree and weight distributionsrdquo Physical Review E StatisticalNonlinear ampa Soft Matter Physics vol 90 p 042803 2014
[30] X Zhu W Wang S Cai and H E Stanley ldquoStanley Dynamicsof social contagions with local trend imitationrdquo Scientic Reportsvol 8 2018 Scientic Reports
[31] X Zhu W Wang S Cai and H E Stanley ldquoOptimal imitationcapacity and crossover phenomenon in the dynamics of socialcontagionsrdquo Journal of Statistical Mechanics Theory amp Experi-ment Article ID 063405 p 063405 2018
[32] A L Barabsi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 p 173 1999
[33] S A Pandit and R E Amritkar ldquoCharacterization and controlof small-world networksrdquo Physical Review E Statistical Nonlin-ear and SoftMatter Physics vol 60 no 2 pp R1119ndashR1122 1999
[34] J VonNeumann Papers of John Von Neumann on Computing ampComputer Theory vol 1 1951
[35] B Chopard and M Droz Cellular Automata for ModelingPhysics Cambridge University Press Cambridge UK 1998
[36] G J Martnez A Adamatzky F Chen and L Chua ldquoOn SolitonCollisions between Localizations in Complex Elementary Cel-lular Automata Rules 54 and 110 andBeyondrdquoComplex Systemsvol 117 2013
[37] H F Gagliardi and D Alves ldquoSmall-world effect in epidemicsusing cellular automatardquo Mathematical Population Studies AnInternational Journal of Mathematical Demography vol 17 no2 pp 79ndash90 2010
[38] X Jin J Li and L Zhang Online Social Networks Based onComplex Network Theory and Simulation Analysis SpringerInternational Publishing 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 7
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60 70t
num
ber o
f aba
ndon
ed
initially 1 interested initially 10 interested initially 100 interested
(a)
0
500
1000
1500
2000
2500
num
ber o
f aba
ndon
ed
0 40 50 60 7010 20 30t
initially 1 interested initially 10 interested initially 100 interested
(b)
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60 70t
num
ber o
f aba
ndon
ed
initially 1 interested initially 10 interested initially 100 interested
(c)
Figure 5 The influence of different initial infection ratio in small-word The rewiring probability is 0 01 1 in (a) (b) (c) respectively Eachexperiment randomly selects 1 10 100 initial interested nodes
contrast between Figures 4(b) and 4(d) shows that theabandoned nodes in SEIR are in the large area while thenumber of immunized nodes in OSIS is flat The results areconsistent with the above conclusions
422 The Influence of Different Initial Infection Ratio Inorder to prove the influence of initial infection ratio oninformation spreading we randomly select 1 10 and 100initial interested nodes The rewiring probability in each setof experiments is 0 01 and 1 respectively and the infectionrate is 035The threshold of thememory superposition factoris 1 and the threshold of information attenuation is 05 Theabandoned state nodes are recorded
It is observable fromFigure 5(a) that the number of initialinfected nodes plays a decisive role in spreading when small-world effect is not obvious (ie rewiring probability is 0)When the number of initial interested nodes is 10 the numberof the eventually abandoned nodes is 8 times more than thatof 1 initial interested node while the multiple is 60 timeswhen the number of initial interested nodes is 100 and 1 Asshown in Figures 5(b) and 5(c) a higher rewiring probability
leads to a faster spreading and a higher steady-state infectiondensity When the rewiring probability is 01 the peak isreached at the 15th time period while it reduces to 10th timeperiodwhen the rewiring probability is 1 All the results abovereveal that the more initial infected nodes the easier it is tospread information However this effect tends to convergeas the rewiring probability increases Additionally with thestrengthening of the small-world features (ie the averagenumber of friends per person in social networks is greater)the information can spread farther and inhibited harder
423 The Effect of Memory Superposition To explore thememory superposition effect on information spreading Weperform simulations to obtain the infection density in thesteady state as a function of the effective infection rate 120572for various superposition thresholds In this experiment therewiring probability of the network is 01 and the threshold ofinformation attenuation is 05
We set up three sets of experiments The infection ratein each set is 01 05 and 07 meanwhile the superpositionthreshold is 1 3 and 5 Initially one node in network is
8 Complexity
0501 07infection rate
superposition threshold=1superposition threshold=3superposition threshold=5
0
500
1000
1500
2000
2500nu
mber o
f aba
ndon
ed
Figure 6The number of abandoned node under different superpo-sition threshold
randomly chosen to be infected After 100 time periods thenumber of nodes in abandoned state is recorded
As shown in Figure 6 the results of the experimentsall indicate that the effect of superposition is obvious andthe higher the threshold the fewer the nodes affected Forexample when the infection rate is 05 the number ofabandoned nodes with the threshold 1 is almost 4 timesmore than that with threshold 3 which even goes to 19 timeswhen compared with the result of the threshold 5The resultsaccurately explain why information spreads widely in onlinesocial networks With the explosion of Internet and we-media the spreading of information costs little investigationand individuals spread information with less filters Such lessfrequent contact to information leads to the rapid spreadingMemory superposition factor in our model stands for thenumber of contacts
5 Conclusions
In summary we introduce an OSIS model to describeinformation spreading in online social networks Firstly weproposeOSIS by combining epidemics SEIRmodelwith indi-vidualrsquos cognitive psychology Secondly a cellular automatamodel of information transmission is designed which pro-vides a computational method for information spreadingfrom a microscopic perspective Additionally we use small-world network to simulate online social network whichverifies the spreading and evolution of information Theexperimental results reveal that our model is consistent withthe propagation of information in online social networksWe find that individualrsquos cognition affects the informationspreading behavior When the infection rate is low cognitivepsychology prevents the spread of information By contrastwhen the infection rate is sufficiently high it promotes trans-mission We demonstrated that the explosion of online socialnetwork scale and the convenience of we-media can greatlyincrease the ability of information disseminationMeanwhile
our research shows that the demise of information is affectedby the dual effects of the decay of time and heat
Our current work mainly concentrates on the spreadingmodel itself As for the future work we believe that the realonline social network is not a single small-world networkDue to the vigorous development ofWeb 20 online networkssuch as Weibo Twitter and Facebook are intertwined Wedeem the information spreading in interconnected small-world networks as the promising and challenging futurework
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A L Barabasi and R E Crandall ldquoLinked the new science ofnetworksrdquo American Journal of Physics vol 71 no 4 pp 409ndash411 2003
[2] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003
[3] A Barrat M Barthlemy and A Vespignani Dynamical pro-cesses on complex networks 2008
[4] R Cohen and S Havlin ldquoComplex networks structure robust-ness and function London UKrdquo Cambridge University Press2010
[5] P Erdos and A Renyi ldquoOn the evolution of random graphsrdquoTransactions of the AmericanMathematical Society vol 286 no1 pp 257ndash274 2011
[6] D J Watts and S H Strogatz Collective Dynamics of rsquoSmall-Worldrsquo Networks 2011
[7] A Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquo Science vol 286 no 5439 pp 509ndash512 1999
[8] M Li Y Fan D Wang D Li J Wu and Z Di ldquoSmall-worldeffect induced by weight randomization on regular networksrdquoPhysics Letters A vol 364 p 488 2007
[9] Q Jin L Wang C Xia and Z Wang ldquoSpontaneous symmetrybreaking in interdependent networked gamerdquo Scientic Reportsvol 4 p 4095 2015
[10] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001
[11] N Bailey ldquoThe mathematical theory of infectious diseases andits applicationsrdquo The Journal of Immunology vol 34 no 5 pp955-956 1978
[12] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press 1991
[13] N C Grassly and C Fraser ldquoMathematicalmodels of infectiousdisease transmissionrdquo Nature Reviews Microbiology vol 6 no6 pp 477ndash487 2008
[14] R M May and A L Lloyd ldquoInfection dynamics on scale-freenetworksrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 64 no 6 Article ID 066112 2001
Complexity 9
[15] M H A Biswas L T Paiva and M D R D Pinho ldquoASEIR model for control of infectious diseases with constraintsrdquoMathematical Biosciences amp Engineering vol 11 no 4 p 7612014
[16] X Fu A Passarella D Quercia A Sala and T Strufe ldquoOnlineSocial Networksrdquo Computer Communications vol 73 pp 163ndash166 2016
[17] X Ke Z Sai C Hao et al ldquoMeasurement and analysis of onlinesocial networksrdquoChinese Journal of Computers pp 29ndash42 2014
[18] Q Ying Z Nan and Z Laijun ldquoResearch on the law of weibopublicity transmissionrdquo Journal of the China Society for Scienticand Technical Information vol 31 no 12 pp 1299ndash1304 2012(Chinese)
[19] weibo httpsweibocom[20] F Xiong Y Liu Z J Zhang J Zhu and Y Zhang ldquoAn
information diffusion model based on retweeting mechanismfor online social mediardquo Physics Letters A vol 376 p 2103 2012
[21] D Centola ldquoThe spread of behavior in an online social networkexperimentrdquo Science vol 329 no 5996 pp 1194ndash1197 2010
[22] H Wu A Arenas and S Gmez ldquoInfluence of trust in thespreading of informationrdquo Physical Review vol 95 2016
[23] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 no 11 Article ID e100096815 pages 2010
[24] L Lu D-B Chen and T Zhou ldquoSmall world yields the mosteffective information spreadingrdquo New Journal of Physics p825834 2011
[25] J X Zhang D B Chen Q Dong and Z D Zhao ldquoIdentifyinga set of influential spreaders in complex networksrdquo ScienticReports vol 6 2016
[26] W Wang M Cai and M Zheng ldquoSocial contagions oncorrelated multiplex networksrdquo Physica A Statistical Mechanicsamp Its Applications vol 499 2017
[27] W Wang X L Chen and L F Zhong ldquoSocial contagions withheterogeneous credibilityrdquo Physica A Statistical Mechanics amp ItsApplications 2018
[28] W Wang M Tang S H Eugene and L A BraunsteinldquoUnification of theoretical approaches for epidemic spreadingon complex networksrdquoReports on Progress in Physics p 0366032017
[29] W Wang M Tang H F Zhang H Gao Y Do and Z HLiu ldquoEpidemic spreading on complex networks with generaldegree and weight distributionsrdquo Physical Review E StatisticalNonlinear ampa Soft Matter Physics vol 90 p 042803 2014
[30] X Zhu W Wang S Cai and H E Stanley ldquoStanley Dynamicsof social contagions with local trend imitationrdquo Scientic Reportsvol 8 2018 Scientic Reports
[31] X Zhu W Wang S Cai and H E Stanley ldquoOptimal imitationcapacity and crossover phenomenon in the dynamics of socialcontagionsrdquo Journal of Statistical Mechanics Theory amp Experi-ment Article ID 063405 p 063405 2018
[32] A L Barabsi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 p 173 1999
[33] S A Pandit and R E Amritkar ldquoCharacterization and controlof small-world networksrdquo Physical Review E Statistical Nonlin-ear and SoftMatter Physics vol 60 no 2 pp R1119ndashR1122 1999
[34] J VonNeumann Papers of John Von Neumann on Computing ampComputer Theory vol 1 1951
[35] B Chopard and M Droz Cellular Automata for ModelingPhysics Cambridge University Press Cambridge UK 1998
[36] G J Martnez A Adamatzky F Chen and L Chua ldquoOn SolitonCollisions between Localizations in Complex Elementary Cel-lular Automata Rules 54 and 110 andBeyondrdquoComplex Systemsvol 117 2013
[37] H F Gagliardi and D Alves ldquoSmall-world effect in epidemicsusing cellular automatardquo Mathematical Population Studies AnInternational Journal of Mathematical Demography vol 17 no2 pp 79ndash90 2010
[38] X Jin J Li and L Zhang Online Social Networks Based onComplex Network Theory and Simulation Analysis SpringerInternational Publishing 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
8 Complexity
0501 07infection rate
superposition threshold=1superposition threshold=3superposition threshold=5
0
500
1000
1500
2000
2500nu
mber o
f aba
ndon
ed
Figure 6The number of abandoned node under different superpo-sition threshold
randomly chosen to be infected After 100 time periods thenumber of nodes in abandoned state is recorded
As shown in Figure 6 the results of the experimentsall indicate that the effect of superposition is obvious andthe higher the threshold the fewer the nodes affected Forexample when the infection rate is 05 the number ofabandoned nodes with the threshold 1 is almost 4 timesmore than that with threshold 3 which even goes to 19 timeswhen compared with the result of the threshold 5The resultsaccurately explain why information spreads widely in onlinesocial networks With the explosion of Internet and we-media the spreading of information costs little investigationand individuals spread information with less filters Such lessfrequent contact to information leads to the rapid spreadingMemory superposition factor in our model stands for thenumber of contacts
5 Conclusions
In summary we introduce an OSIS model to describeinformation spreading in online social networks Firstly weproposeOSIS by combining epidemics SEIRmodelwith indi-vidualrsquos cognitive psychology Secondly a cellular automatamodel of information transmission is designed which pro-vides a computational method for information spreadingfrom a microscopic perspective Additionally we use small-world network to simulate online social network whichverifies the spreading and evolution of information Theexperimental results reveal that our model is consistent withthe propagation of information in online social networksWe find that individualrsquos cognition affects the informationspreading behavior When the infection rate is low cognitivepsychology prevents the spread of information By contrastwhen the infection rate is sufficiently high it promotes trans-mission We demonstrated that the explosion of online socialnetwork scale and the convenience of we-media can greatlyincrease the ability of information disseminationMeanwhile
our research shows that the demise of information is affectedby the dual effects of the decay of time and heat
Our current work mainly concentrates on the spreadingmodel itself As for the future work we believe that the realonline social network is not a single small-world networkDue to the vigorous development ofWeb 20 online networkssuch as Weibo Twitter and Facebook are intertwined Wedeem the information spreading in interconnected small-world networks as the promising and challenging futurework
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A L Barabasi and R E Crandall ldquoLinked the new science ofnetworksrdquo American Journal of Physics vol 71 no 4 pp 409ndash411 2003
[2] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003
[3] A Barrat M Barthlemy and A Vespignani Dynamical pro-cesses on complex networks 2008
[4] R Cohen and S Havlin ldquoComplex networks structure robust-ness and function London UKrdquo Cambridge University Press2010
[5] P Erdos and A Renyi ldquoOn the evolution of random graphsrdquoTransactions of the AmericanMathematical Society vol 286 no1 pp 257ndash274 2011
[6] D J Watts and S H Strogatz Collective Dynamics of rsquoSmall-Worldrsquo Networks 2011
[7] A Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquo Science vol 286 no 5439 pp 509ndash512 1999
[8] M Li Y Fan D Wang D Li J Wu and Z Di ldquoSmall-worldeffect induced by weight randomization on regular networksrdquoPhysics Letters A vol 364 p 488 2007
[9] Q Jin L Wang C Xia and Z Wang ldquoSpontaneous symmetrybreaking in interdependent networked gamerdquo Scientic Reportsvol 4 p 4095 2015
[10] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001
[11] N Bailey ldquoThe mathematical theory of infectious diseases andits applicationsrdquo The Journal of Immunology vol 34 no 5 pp955-956 1978
[12] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press 1991
[13] N C Grassly and C Fraser ldquoMathematicalmodels of infectiousdisease transmissionrdquo Nature Reviews Microbiology vol 6 no6 pp 477ndash487 2008
[14] R M May and A L Lloyd ldquoInfection dynamics on scale-freenetworksrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 64 no 6 Article ID 066112 2001
Complexity 9
[15] M H A Biswas L T Paiva and M D R D Pinho ldquoASEIR model for control of infectious diseases with constraintsrdquoMathematical Biosciences amp Engineering vol 11 no 4 p 7612014
[16] X Fu A Passarella D Quercia A Sala and T Strufe ldquoOnlineSocial Networksrdquo Computer Communications vol 73 pp 163ndash166 2016
[17] X Ke Z Sai C Hao et al ldquoMeasurement and analysis of onlinesocial networksrdquoChinese Journal of Computers pp 29ndash42 2014
[18] Q Ying Z Nan and Z Laijun ldquoResearch on the law of weibopublicity transmissionrdquo Journal of the China Society for Scienticand Technical Information vol 31 no 12 pp 1299ndash1304 2012(Chinese)
[19] weibo httpsweibocom[20] F Xiong Y Liu Z J Zhang J Zhu and Y Zhang ldquoAn
information diffusion model based on retweeting mechanismfor online social mediardquo Physics Letters A vol 376 p 2103 2012
[21] D Centola ldquoThe spread of behavior in an online social networkexperimentrdquo Science vol 329 no 5996 pp 1194ndash1197 2010
[22] H Wu A Arenas and S Gmez ldquoInfluence of trust in thespreading of informationrdquo Physical Review vol 95 2016
[23] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 no 11 Article ID e100096815 pages 2010
[24] L Lu D-B Chen and T Zhou ldquoSmall world yields the mosteffective information spreadingrdquo New Journal of Physics p825834 2011
[25] J X Zhang D B Chen Q Dong and Z D Zhao ldquoIdentifyinga set of influential spreaders in complex networksrdquo ScienticReports vol 6 2016
[26] W Wang M Cai and M Zheng ldquoSocial contagions oncorrelated multiplex networksrdquo Physica A Statistical Mechanicsamp Its Applications vol 499 2017
[27] W Wang X L Chen and L F Zhong ldquoSocial contagions withheterogeneous credibilityrdquo Physica A Statistical Mechanics amp ItsApplications 2018
[28] W Wang M Tang S H Eugene and L A BraunsteinldquoUnification of theoretical approaches for epidemic spreadingon complex networksrdquoReports on Progress in Physics p 0366032017
[29] W Wang M Tang H F Zhang H Gao Y Do and Z HLiu ldquoEpidemic spreading on complex networks with generaldegree and weight distributionsrdquo Physical Review E StatisticalNonlinear ampa Soft Matter Physics vol 90 p 042803 2014
[30] X Zhu W Wang S Cai and H E Stanley ldquoStanley Dynamicsof social contagions with local trend imitationrdquo Scientic Reportsvol 8 2018 Scientic Reports
[31] X Zhu W Wang S Cai and H E Stanley ldquoOptimal imitationcapacity and crossover phenomenon in the dynamics of socialcontagionsrdquo Journal of Statistical Mechanics Theory amp Experi-ment Article ID 063405 p 063405 2018
[32] A L Barabsi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 p 173 1999
[33] S A Pandit and R E Amritkar ldquoCharacterization and controlof small-world networksrdquo Physical Review E Statistical Nonlin-ear and SoftMatter Physics vol 60 no 2 pp R1119ndashR1122 1999
[34] J VonNeumann Papers of John Von Neumann on Computing ampComputer Theory vol 1 1951
[35] B Chopard and M Droz Cellular Automata for ModelingPhysics Cambridge University Press Cambridge UK 1998
[36] G J Martnez A Adamatzky F Chen and L Chua ldquoOn SolitonCollisions between Localizations in Complex Elementary Cel-lular Automata Rules 54 and 110 andBeyondrdquoComplex Systemsvol 117 2013
[37] H F Gagliardi and D Alves ldquoSmall-world effect in epidemicsusing cellular automatardquo Mathematical Population Studies AnInternational Journal of Mathematical Demography vol 17 no2 pp 79ndash90 2010
[38] X Jin J Li and L Zhang Online Social Networks Based onComplex Network Theory and Simulation Analysis SpringerInternational Publishing 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 9
[15] M H A Biswas L T Paiva and M D R D Pinho ldquoASEIR model for control of infectious diseases with constraintsrdquoMathematical Biosciences amp Engineering vol 11 no 4 p 7612014
[16] X Fu A Passarella D Quercia A Sala and T Strufe ldquoOnlineSocial Networksrdquo Computer Communications vol 73 pp 163ndash166 2016
[17] X Ke Z Sai C Hao et al ldquoMeasurement and analysis of onlinesocial networksrdquoChinese Journal of Computers pp 29ndash42 2014
[18] Q Ying Z Nan and Z Laijun ldquoResearch on the law of weibopublicity transmissionrdquo Journal of the China Society for Scienticand Technical Information vol 31 no 12 pp 1299ndash1304 2012(Chinese)
[19] weibo httpsweibocom[20] F Xiong Y Liu Z J Zhang J Zhu and Y Zhang ldquoAn
information diffusion model based on retweeting mechanismfor online social mediardquo Physics Letters A vol 376 p 2103 2012
[21] D Centola ldquoThe spread of behavior in an online social networkexperimentrdquo Science vol 329 no 5996 pp 1194ndash1197 2010
[22] H Wu A Arenas and S Gmez ldquoInfluence of trust in thespreading of informationrdquo Physical Review vol 95 2016
[23] A L Hill D G Rand M A Nowak and N A ChristakisldquoInfectious disease modeling of social contagion in networksrdquoPLoS Computational Biology vol 6 no 11 Article ID e100096815 pages 2010
[24] L Lu D-B Chen and T Zhou ldquoSmall world yields the mosteffective information spreadingrdquo New Journal of Physics p825834 2011
[25] J X Zhang D B Chen Q Dong and Z D Zhao ldquoIdentifyinga set of influential spreaders in complex networksrdquo ScienticReports vol 6 2016
[26] W Wang M Cai and M Zheng ldquoSocial contagions oncorrelated multiplex networksrdquo Physica A Statistical Mechanicsamp Its Applications vol 499 2017
[27] W Wang X L Chen and L F Zhong ldquoSocial contagions withheterogeneous credibilityrdquo Physica A Statistical Mechanics amp ItsApplications 2018
[28] W Wang M Tang S H Eugene and L A BraunsteinldquoUnification of theoretical approaches for epidemic spreadingon complex networksrdquoReports on Progress in Physics p 0366032017
[29] W Wang M Tang H F Zhang H Gao Y Do and Z HLiu ldquoEpidemic spreading on complex networks with generaldegree and weight distributionsrdquo Physical Review E StatisticalNonlinear ampa Soft Matter Physics vol 90 p 042803 2014
[30] X Zhu W Wang S Cai and H E Stanley ldquoStanley Dynamicsof social contagions with local trend imitationrdquo Scientic Reportsvol 8 2018 Scientic Reports
[31] X Zhu W Wang S Cai and H E Stanley ldquoOptimal imitationcapacity and crossover phenomenon in the dynamics of socialcontagionsrdquo Journal of Statistical Mechanics Theory amp Experi-ment Article ID 063405 p 063405 2018
[32] A L Barabsi R Albert and H Jeong ldquoMean-field theory forscale-free random networksrdquo Physica A vol 272 p 173 1999
[33] S A Pandit and R E Amritkar ldquoCharacterization and controlof small-world networksrdquo Physical Review E Statistical Nonlin-ear and SoftMatter Physics vol 60 no 2 pp R1119ndashR1122 1999
[34] J VonNeumann Papers of John Von Neumann on Computing ampComputer Theory vol 1 1951
[35] B Chopard and M Droz Cellular Automata for ModelingPhysics Cambridge University Press Cambridge UK 1998
[36] G J Martnez A Adamatzky F Chen and L Chua ldquoOn SolitonCollisions between Localizations in Complex Elementary Cel-lular Automata Rules 54 and 110 andBeyondrdquoComplex Systemsvol 117 2013
[37] H F Gagliardi and D Alves ldquoSmall-world effect in epidemicsusing cellular automatardquo Mathematical Population Studies AnInternational Journal of Mathematical Demography vol 17 no2 pp 79ndash90 2010
[38] X Jin J Li and L Zhang Online Social Networks Based onComplex Network Theory and Simulation Analysis SpringerInternational Publishing 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Related Documents
