ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011 NETWORK MODELING.

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

NETWORK MODELING IN INTERNATIONAL COUNTERACTING GLOBAL THREATS

A. Tikhomirov , International Informatization Academy, Moscow , RF

A.Trufanov , Irkutsk State Technical University, Irkutsk, RF,e-mail: troufan.istu.edu

A.Caruso, Court of Auditors, Regional Chamber of Control , Milan, Italy

A.Rossodivita , San Raffaele Hospital Scientific Foundation, Milan, Italy

E. Shubnikov,Institute of Internal Medicine, Novosibirsk, RF

R.Umerov, Crimean Engineering and Pedagogical University, Simferopol, Ukraine

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Networking in counteracting global threats: policy, research, education and practice

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Why do we care networking and networks ?

Because networks are of great value for Disasters and Emergencies

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Graph Theory originated in the moment when Leonhard Euler, Swiss, German and Russian mathematician, decided to prove that a passerby can not get around Konigsberg (modern Kaliningrad), using only one each of the seven city bridges.

Its key conclusion is: structural characteristics of graphs (networks) define a potential for their use.

The first example of using the methods of modern algebra in graph theory accounts for the work of the physicist Gustav Robert Kirchhoff, in 1845 he formulated so called Kirchhoff's laws to calculate voltages and currents in electrical circuits.

Advances in theory and practice of networks

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Introduction of probabilistic methods in graph theory, especially in research of Paul Erdős and Alfréd Rényi

on asymptotic probabilities of graphs created another branch known as theory of random graphs

Mathematician Dénes Kőnig published in 1936 a book titled "Theory of finite and infinite graphs” - the first textbook in the field of graph theory

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Four structural models for networks

• Regular networks ( e.g. crystal lattice)

• Random networks

• Small-world networks

• Scale-free networks

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Efficiency of a Network

The network efficiency E (G) is a measure to quantify how efficiently the nodes of the network exchange information.To define efficiency of G first we calculate the shortest path lengths {dij} between two nodes i and j. Suppose that every node sends information along the network, through its edges. The efficiency ij in the communication between vertex i and j is inversely proportional to the shortest distance dij: ij = 1/dij

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Current classification of complex networks

Three important complex network models:

• random graph model

• small-world network

• scale-free network model

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Erdös-Renyi Random graphs

Paul Erdös (1913-1996)

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Erdös-Renyi Random/Exponential / Homogeneous Graphs

• The Gn,p model

– n : the number of vertices– 0 ≤ p ≤ 1– for each pair (i,j),

generate the edge (i,j)

independently

with probability p

Exponential Graphs (Networks)

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Small world phenomena• Small worlds: networks

with short paths

Psychologist Stanley Milgram (1933-1984): “The man who shocked the world”

Measuring the small world phenomenon

dij = shortest path between i and j nodes

ijji,

dmaxd

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011Watts and Strogatz model – WS that analyses Milgram’s theory

• Start with a ring, where every node is connected to the next k nodes (regular network)

• With probability p, rewire every edge (or, add a shortcut) to a uniformly chosen destination.

order randomness

p = 0 p = 10 < p < 1

Small World

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Properties of Small-world Graphs

– Large networks (n >> 1)– Sparse connectivity (avg degree k << n)

– No central node (kmax << n)

– Large clustering coefficient (larger than in random graphs of same size)

– Short average paths (~log n, close to those of random graphs of the same size)

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

The next step

Barabasi-Albert (BA) model

( Barabasi model/ Scale Free model)

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

P(k) k

32 These networks have no natural average number of edges and are called scale-free

Typical range for

Power – law distributionfor evolving self-organized networks was

proposed by Barabasi, Albert and collaborators

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Two types of network connectivity

1. Homogeneous network connectivity 2. Inhomogeneous network connectivity

Red nodes are most connected nodes ( cluster centers )

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Two types of network connectivity

1. Bad Workshops 2. ASI Slavonski Brod structure

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Power law tail kkP )(

kekP )( Exponential tail

random graphs (Erdös-Réyni) model

Exponential: Power law:

HUBS

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Scale free model :• The degree distributions of most real-life networks follow

a power law• there is a non-negligible fraction of nodes that has very

high degree (hubs)• scale-free: no characteristic scale, average is not

informative

Contrary random model :• highly concentrated around the mean• the probability of very high degree nodes is exponentially

small

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Two types of network connectivity

1. Homogeneous network connectivity 2. Inhomogeneous network connectivity

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Two types of network connectivity, but…

In real life we may encounter Variations on the Barabasi-Albert Model

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Truncated power-law

is the cut-off, s. t. the number of connections is less than expected for pure scale-free networks for

and the behaviour is approximately scale-

free within the range

0 1 2 3 4

02

46

8

log(1:l)

log(tabula

te(lin

ks1))

)/exp(~)( ckkkkp

ck

ckk

ckk 1

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

One can consider that, in real networks

Link costThe cost of hosting new link increases with the number of linksE.g., for a Web site this implies adding more computational power, for a router this means buying a new powerful router

Node AgingThe possibility of hosting new links decreased with the “age” of the nodeE.g. nodes get tired or out-of date

Aging and Cost explain the “exponential cut-off” in power law

networks

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

BA model advances

Power law distribution with Scale –Free properties ( that means that these networks have no specific scale contrary to random /exponential/ ones )

Preferential Attachment and Growth of a Network / Dynamic

Simple and clear terminology for all interested societies

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Power law distribution with Scale –Free properties ( that means that these networks have no specific scale contrary to random (exponential) ones

This implies that scale-free networks are self-similar, i.e. any part of the network is statistically similar to the whole network and parameters are assumed to be independent of the system size.

BA model advances

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Preferential Attachment and Growth of a Network ( dynamics )

• At every time step t, – A new node is connected to node i

• depends on the connectivity ki of node i

– The probability • ∏i = ki / ∑j kj

BA model advances

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Example: In practice sophisticated terms of Theory of Graphs are similar to Chinese ABC

Simple and clear terminology for all interested societies: nodes and links instead vertexes and edges of Theory of Graphs )

BA model advances

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Vulnerability of Networks: these are not left alone

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

The error tolerance/ robustness of the network

Threat-Attack-Damage• Evaluation

– The changes in diameter when a small fraction f of the nodes is removed.

– The absence of any node in general increases the distances between the remaining nodes.

• How to remove nodes– Failure/ unintentional attack;

• Any node is removed with the equivalent probability– Attack/ intentional attack;

• The node which has the most connectivity is removed first.

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Attack• Attack / intentional attack (initiated by human beings)

– Will be on the most connected node rather than randomly

• Attack model– Remove the most connected node, – Continue selecting and removing nodes in decreasing order of

their connectivity k.

1 2

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Fragmentation

• Fragmentation– When nodes are removed from a network,

• Clusters of nodes may be cut off (fragmented) from the main cluster.

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Evaluation

• Attack Evaluation – one more metrics– S; The size of the largest cluster

• Divided by the initial total system size to normalize.

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

So there are two important complex network models to explore

• Regular networks

• Random networks

• Small-world networks

• Scale-free networks

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

• Exponential model attack

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

The size of the largest cluster - exponential modelFailure/ unintentional attack; Attack/ intentional attack

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

• Scale free model attack

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

The size of the largest cluster- scale free modelFailure/ unintentional attack; Attack/ intentional attack

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Complex Network tools have been successfully applied to understanding and counteracting such threats as infection diseases spread and terrorist activity.

Martin Rosvall† and Carl T. Bergstrom . An information-theoretic framework for resolving community structure in complex networks. PNAS . May 1, 2007, vol. 104 , N 18 , 7327–7331

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Complexity of a real problem

• Diversity of attacks–The impact of failures and attacks on the

network structure

BA model in not enough to explore all aspect of attacks

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Regular Model

Watts and Strogatz (WS)

Erdös-Renyi (ER)

Barabasi-Albert (BA)

Rossodivita- Trufanov (RT)

Caruso- Rossodivita- Shubnikov-Tikhomirov-Trufanov -Umerov

Levitin G, Hausken K.

Aminova - Rossodivita- Tikhomirov-Trufanov

Xiao, Xiao and Cheng

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

S. Xiao, G. Xiao ,T. H. Cheng Tolerance of local information-based intentional attacks in complex networks.J. Phys. A: Math. Theor. 43 (2010) 335101Distributed attacks basically target on some or all of the live nodes adjacent to the

crashed nodes in each step, and the selections of the targets depend on only thelocal network-topology information.

Xiao models

Division of Communication Engineering, School of Electrical and Electronic Engineering,Nanyang, Singapore

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

S. Xiao, G. Xiao On Degree-Based Decentralized Search in Complex NetworksarXiv e-print (arXiv:cs/0610173)

Decentralized search aims to find the target node in a large network by using only local information

Xiao, Xiao and Cheng models

Division of Communication Engineering, School of Electrical and Electronic Engineering,Nanyang, Singapore

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Caruso- Rossodivita- Shubnikov-Tikhomirov-Trufanov -Umerov models

Real life attacks : mixture of Failures and Attacks

Combined attacks model :

• Sequence of Failures and Sequence of Attacks

• Sequence of Attacks and Sequence of Failures

Failure/ unintentional attack; Attack/ intentional attack

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Failures+Attacks / Attacks + Failures

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Network connectivity

- Inhomogeneous network connectivity

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Real life : protection of nodes and links

Node Protection model :

•Protection barriers are constructed for network nodes with “thickness” d.d is sum of traditional protection measures:Ethical; Legal; Organizational; Technological; Physical; Math• Attenuation of any attack is proportional to exp(-µd) , where µ is a coefficient;µd=(µd)E+ (µd)L +(µd)O + (µd)T+ (µd)P+ (µd)M

Failure/ unintentional attack; Attack/ intentional attack

Caruso- Rossodivita- Shubnikov-Tikhomirov-Trufanov -Umerov models

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Real life : protection of nodes and links

Node Protection model :

•Metrics of d: Investments (Money)

d ~ Funding

Failure/ unintentional attack; Attack/ intentional attack

Caruso- Rossodivita- Shubnikov-Tikhomirov-Trufanov -Umerov models

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

d0

d1

d2

d0<d1<d2

F. Galindo, N.V.Dmitrienko, A.Caruso, A. Rossodivita, A.A.Tikhomirov, A. I.Trufanov, E. V. Shubnikov, Modeling of Aggregate Attacks on Complex Networks. Information Security Technologies , Moscow – 2010, N3, P.115-121

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Real life : protection strategy of nodes and links

Node Protection Strategy model :

• how Investments should be distributed among different nodes

1.d ~ Const Funding 2.d ~ Funding (k)3.d ~ Funding (k2)

Failure/ unintentional attack; Attack/ intentional attack

Caruso- Rossodivita- Shubnikov-Tikhomirov-Trufanov -Umerov models

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

d0

dC

dk

dk2

F. Galindo, N.V.Dmitrienko, A.Caruso, A. Rossodivita, A.A.Tikhomirov, A. I.Trufanov, E. V. Shubnikov, Modeling of Aggregate Attacks on Complex Networks. Information Security Technologies , Moscow – 2010, N3, P.115-121

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

ConclusionER, BA & XXC, RT, СRSTTU - ( protected network,

aggregated attacks)

These models show how to built a Robust Network

RT model is complex and close to real secure Networking

CRSTTU - model is under developing…

ASI “Applying Lessons Learned and Sharing Best Practices in Addressing Influenza Pandemics and Catastrophic Events ”, Slavonski Brod, 2011

Authors express their gratitude and sincere respect to NATO SPS Program

(with its ASI and ARW Institutions had led by Dr. F.Linkov, 2005;

Dr. P.Rumm and Prof. E.Stikova,2006; Prof. J.-G.Fontaine, 2010;

Dr. E.Gursky and Dr. B.Hreckovski, 2011 )