Multidimensional Analysis of Complex Networks

Motivation Multidimensional Spatial analysis Growth analysis

Multidimensional analysis of complex networks

Possamai Lino

Alma Mater Studiorum Università di BolognaUniversità di Padova

Ph.D. Dissertation DefenseFebruary 21st, 2013

Possamai Lino Università di Bologna - Università di Padova

Multidimensional analysis of complex networks 1/39


Publications and conferences list

Plos One 2012 Thi Hoang, Sun, Possamai, JafariAsbagh, Patil, MenczerScholarometer: A Social Framework for Analyzing Impact across Disciplines

IPM 2012 Sun, Kaur, Possamai, MenczerAmbiguous Author Query Detection using Crowdsourced Digital Library Annotations

SocialCom11 2011 Sun, Kaur, Possamai and MenczerDetecting Ambiguous Author Names in Crowdsourced Scholarly Data

PSB2010 2010 Biasiolo, Forcato, Possamai, Ferrari, Agnelli, Lionetti, Todoerti, Neri, Marchiori et al.Critical analysis of transcriptional and post-transcriptional regulatory networks inMultiple Myeloma

Sunbelt2010 2010 Marchiori, PossamaiTelescopic analysis of complex networks

PRIB2009 2009 Forcato, Possamai, Ferrari, Agnelli, Todoerti, Lambertenghi, Bortoluzzi, Marchiori et al.Reverse Engineering and Critical Analysis of Gene Regulatory Networksin Multiple Myeloma

(under submission) 2013 Toward an optimized evolution of social networks

(under submission) 2013 Micro-macro analysis of complex networks




Outline

1 Motivation

2 MultidimensionalIntroduction

3 Spatial analysisIntroductionAlgorithmDatasetsResults

4 Growth analysisMotivationGrowth dynamicsSimulationsResults




Domain

A complex system is a network of elements thatinteracts in a non-linearly way, resulting in anoverall behavior that is difficult to predict.




Domain


The digitalization of every day’s actions allows adeeper investigation on how persons, computers,animals, companies etc interact




Domain



Networks are everywhere in Nature: from ecologyto the WWW, to food chain, to social networks, tofinance




Domain



Networks are everywhere in Nature: from ecologyto the WWW, to food chain, to social networks, tofinance

This opened up many interdisciplinary researchareas that are very active




History

Started with mathematicians Erdos–Rényi and graph theory

Watts and Strogatz, small world and 〈L〉, C metrics

Barabási-Albert first introduced the scale-free model, identified hubs and powerlaw in the degree distribution

Many other works that followed, proposed improvements in the basic statistics andin the generative models




Motivation

The aim of this Thesis was to study Complex Networks (CN) under the most importantdimensions. Key points are the following:

Currently, many studies on CN underestimate the effect of spatial constraints onthe overall evolutionMany models have been proposed in order to create CNs with the sameproperties of the observed networks

However, they are not sufficient to describe precisely how networks evolveThat is why other instincts might be at the root of the growthNo methods have been proposed to increase the commitment in users’ communities

For these reasons, we worked on a new framework that is based on these lackingfeatures. We call it multidimensional.




Introduction

So what do we mean by multidimensional?

We mean a novel framework that analyzes complexnetworks (CN) along the two fundamentalinformative axes:




Introduction



Space




Introduction



Space

Time

The study of these dimensions was performed byfreezing one axis and simulating the evolution ofthe other




Introduction

THE SPACE DIMENSION




Introduction

Space dimension

The structure of a CN is not 100% completely defined because itdepends on the level of detail with which the system is observed

For instance, biological networks could be analyzed at differentlayers. Nodes could be represented as atoms, proteins, cells,neurons and so on

Until now, no one has considered to study CN as a function of thedetail levels.

Results, properties, features that are valid in a specific level mightnot hold in other levels.




Algorithm

Spatial Analysis

So what does it means to view a network at a particularlevel?

Let us take a spatial network with information about nodes’positions over a plane.




Algorithm

Spatial Analysis



Viewing a network at different precision levels correspondsto viewing the network at a difference distance from a pointof view.




Algorithm

Spatial Analysis



Viewing a network at different precision levels correspondsto viewing the network at a difference distance from a pointof view.

This process is modeled utilizing a concept that comesfrom the human eyes ability to distinguish two points atsome distance from the observer.

The points are nodes of the network with x , y coordinatesover a plane.




Algorithm

Spatial Analysis

Generally, the telescopic algorithm is a function t : (G × f ) → G′ that takes asinput:

a graph Gfuzziness f (distance)and produces a resulting graph G′




Algorithm

Spatial Analysis

Generally, the telescopic algorithm is a function t : (G × f ) → G′ that takes asinput:

a graph Gfuzziness f (distance)and produces a resulting graph G′

In order to emulate the network abstraction capability, we placed a virtual grid ontop of the input graph.

Cell’s dimensions depend on the fuzziness value.




Algorithm

Spatial Analysis

All the nodes belonging to the same cell are collapsed andrepresented by a unique node in the new graph.

If there is an edge from at least one node of the i cell to atleast one of the j cell then the (i , j) edge exists in the newgraph G′.

With these rules, the long range edges are preserved.




Algorithm

Spatial Analysis

By repeatedly applying this function we create afuzziness-varying family of graphs T = {G0,G1, . . .Gp}where p is the number of precision levels.

G0 is the micro view and Gp is the macro view.

This novel analysis then allows creating the telescopicspectrum of a network, and study, wrt each property ofinterest, what changes in the micro-macro shift (in[Sunbelt2010]).




Datasets

Tracking properties

To characterize the structural properties during the abstraction process, weconsider several features widely used in network literature

Number of nodes, edges, kmax , kmean, standard deviation of k

Physical, topological and metrical diameter

Topological and metrical efficiency:

E tglob =

1

n(n − 1)

∑

i 6=j

1

hijEm

glob =1

n(n − 1)

∑

i 6=j

1

δij

Topological and metrical local efficiency

Topological and metrical costs:

Ct =|E|

n(n − 1)/2Cm =

∑

i 6=j aij lij∑

i 6=j lij

Homophily (degree correlation)




Datasets

Network datasets

Two different classes of networks are considered:




Datasets

Network datasets


Four subway networks are considered: two fromthe U.S., Boston and New York and two fromEurope, Paris and Milan




Datasets

Network datasets



The US airline network




Datasets

Network datasets



The US airline network

The VirtualTourist online social network (*)

They all are undirected networks.




Results

Global Efficiency

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Top

olog

ical

Egl

ob

Fuzziness

BosNYCParMil

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Met

rical

Egl

ob

Fuzziness

BosNYCParMil

We found different results by considering topological and metrical efficiency

Topological: networks with high efficiency at macro level might have low Eglob atmicro

Metrical: stable under detail levels variation.




Results

Global Efficiency

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Top

olog

ical

Egl

ob

Fuzziness

ITUKNLAUINAir

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Met

rical

Egl

ob

Fuzziness

ITUKNLAUINAir

All the curves start at higher values because of the better structure of SM-SFnetworks

Both subways and SM-SF networks will be simpler as f increases, more efficient,but indistinguishable




Results

Local Efficiency

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Top

olog

ical

Elo

c

Fuzziness

BosParMil

NYC

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Top

olog

ical

Elo

c

Fuzziness

ITUKNLAUINAir

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Met

rical

Elo

c

Fuzziness

BosParMil

NYC

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Met

rical

Elo

c

Fuzziness

ITUKNLAUINAir

Eloc is stable under our telescopic framework. Low values of local clusteringmaintained throughout the spectrumResults strongly differ from subways. This clearly means that the abstractionprocess is able to distinguish the two different principles that guided the evolution




Results

Cost

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Ct

Fuzziness

BosNYCParMil

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Cm

Fuzziness

BosNYCParMil

It might be counterintuitive that simple (abstracted) networks are expensive

The cost is directly connected to the efficiency of a network




Results

Cost

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Ct

Fuzziness

ITUKNLAUINAir

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Cm

Fuzziness

ITUKNLAUINAir

However, when compared to SM-SF networks turn out that the inborn economicprinciples that characterize subways are maintained




Results

Randomized fuzziness-varying graphs

In order to understand how the topological and metrical structure of CNs isaffected by the spatial analysis, we used also null models in our simulations




Results

Randomized fuzziness-varying graphs

In order to understand how the topological and metrical structure of CNs isaffected by the spatial analysis, we used also null models in our simulationsIn particular, we provided four models that account for different perturbations

+n, shuffling nodes’ positions+a, rewiring edges+r, that is the union of +n and +a+s, scale-free structure (using BA model)




Results

Evolution on randomized networks

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Top

olog

ical

Egl

ob

Fuzziness

BostonBoston

Norm+r+a+n+s

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Met

rical

Egl

ob

Fuzziness

Boston

Norm+r+a+n+s

In E tglob, randomizations increase the efficiency because they create the right

shortcuts that drop L

Conversely, randomness in a spatial context destroys the global efficiency. Indeed,when f > 0.3 all the networks will be indistinguishable.




Results

Evolution on randomized networks

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Top

olog

ical

Egl

ob

Fuzziness

Aus

Norm+r+a+n+s

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Met

rical

Egl

ob

Fuzziness

Aus

Norm+r+a+n+s

Random perturbations do not alter Eglob because random networks are bydefinition very efficient

The destroying effect found in subways is also present but constrained to smallvalues of f in metrical efficiency

SM-SF are robust because the randomizations do not alter considerably thenetworks on the spectrum




Motivation

THE TIME DIMENSION




Motivation

Time analysis

Many researches in the literature have dealt with proposing generative modelsthat uncover the key ingredients of network evolution

These are based on simple and advanced local rules that produce a globalbehavior that is similar to the steady-state target’s network

Since many of them are based on social systems, we also concentrate on thesetypes of CNs




Growth dynamics

Growth rule I

The random rule assumes that:

Definition

Nodes of the networks randomly connect each other withuniform probability

pij = k

Empirical tests discovered that real world networks are far frombeing random




Growth dynamics

Growth rule II

The rule of Preferential attachment assumes that:

Definition

Older nodes are more likely to acquire new linkscompared to new ones.

Π(ki ) =ki

∑

j kj




Growth dynamics

Growth rule III

The Social rule assumes that:

Definition

if two people have a friend in common then there is an increasedlikelihood that they will become friend in the future

This rule is at the root of the local clustering property (found inmany networks)

Clearly, these rules are not sufficient to completely describe theevolution of social networks.

There must be some other instincts that trigger the networkevolution




Growth dynamics

Settings with special nodes

The contribution of this Thesis is to understandwhether new instincts on top of the previous growthmodels can leverage the users’ commitment innetworks

Insight on network evolution with special nodes




Growth dynamics




m = number of sirens (6,12)

a = attractiveness

d = activation time span

m a d




Growth dynamics




m = number of sirens (6,12)

a = attractiveness

d = activation time span

configurations ci = (m, a, d)

configurations cost Cs = m · a · d

m a d




Simulations

Simulations

Both sequential and simultaneous simulations are considered

The network evolves according to one of the following rules random, aristocratic orsocial both at the users and sirens levels

The entire system dynamics is accounted by two almost independent user andsiren subprocesses that evolve according to the previous local rules

In both cases, the future evolution Gt+1 will depend on Gt




Simulations

Simulations

Sirens are used for a limited time span (d) after that the system will evolve by itself

Sirens acquire new links constantly over time as

es = |V s| · |V | · a

a is the attractiveness of the sirens

a(s) =q(s)

∑

u∈V∪V s q(u)q(u) = 10 ∀u ∈ V s q(u) = 1 ∀u ∈ V

In simultaneous simulations, many edges can be created and this number variesas a function of Eglob

et = 1 +

⌊

C ·E(Gt−1)

E(Gideal)· (nart−1 − 1)

⌋




Results

Results and Datasets

At this point, based on the framework we provided, we are now able to answer thefollowing set of fundamental questions:

Are the sirens effective in leveraging users’ commitment in new on line social networks?




Results



Are the sirens effective in leveraging users’ commitment in new on line social networks?What are the best parameters for the same cost configurations?




Results



Are the sirens effective in leveraging users’ commitment in new on line social networks?What are the best parameters for the same cost configurations?Is the benefit of sirens proportional to the amount of money involved?




Results



Are the sirens effective in leveraging users’ commitment in new on line social networks?What are the best parameters for the same cost configurations?Is the benefit of sirens proportional to the amount of money involved?

We were particularly interested in on line social networks like VirtualTourist andCommunities




Results

Q1: Effectiveness

In order to understand whether sirens are effective we compare the simulationswith and without sirens

0

0.05

0.1

0.15

0.2

0.25

0 600 1200 1800 2400 3000

Egl

ob

Step

CM

rndari

soc

0

0.05

0.1

0.15

0.2

0.25

0 20 40 60 80 100 120 140

Egl

ob

Step

CM + Sir

rndari

soc




Results

Q2: Best parameter

What are the best parameters in the siren configurations ci = (m, a, d)?

The configurations that have the higher value of attractiveness are the ones thatperform best

Results are valid for all the rules and networks considered

0

0.05

0.1

0.15

0.2

0.25

0 10 20 30 40 50 60 70 80

Egl

ob

Step

aristocraticCs = 1200

(12,10,10)(6,10,20)(6,20,10)

0

0.05

0.1

0.15

0.2

0.25

0 10 20 30 40 50 60 70

Egl

ob

Step

aristocraticCs = 2400

(12,10,20)(12,20,10)

(6,20,20)




Results

Q3: Benefit

We set the number of sirens and see how the other configuration parametersinfluence the growth behavior

We clearly see that the benefit increases, as the cost gets higher. In fact, it is notproportional to Cs.

0

0.05

0.1

0.15

0.2

0.25

0 40 80 120 160 200

Egl

ob

Step

aristocratic

CM+Sir

(6,10,10)(6,10,20)(6,20,10)(6,20,20) 0

1000

2000

3000

4000

40 60 80 100 120 140

Cs

Tmin

rnd prefari pref

soc pref




Results

Recap of contributions

We introduced a new framework in which we consider the two most importantinformative axes along with a CN evolves




Results


We introduced a new framework in which we consider the two most importantinformative axes along with a CN evolvesThe first, spatial analysis, deals with analyzing a network under different detaillevels

Subway networks indexes tend to be more stable under the telescopic variationsNetwork properties change in the telescopic spectrum: their micro and macro behaviorare different




Results


We introduced a new framework in which we consider the two most importantinformative axes along with a CN evolvesThe first, spatial analysis, deals with analyzing a network under different detaillevels

Subway networks indexes tend to be more stable under the telescopic variationsNetwork properties change in the telescopic spectrum: their micro and macro behaviorare different

The second, time analysis, models the growth of social networks by using a set ofprivileged nodes that promote network evolution

These special nodes are an effective way to increase network efficiencyThe benefit increases as cost increases, however it is not proportionalInvest on attractiveness




Results

Referees reports

From leading expert in the Complex System areaJesús Gómez Gardeñes (University of Zaragoza)

Overall positive feedbackAcknowledged contributions to state-of-the-art




Results

Closing remarks and ongoing activities

Consider more spatial networks in order to have a broader coverage and testwhether our findings are still valid

Study force-based network permutations such as Kamada-Kawai andFruchterman-Reingold




Results

Closing remarks and ongoing activities

Consider more spatial networks in order to have a broader coverage and testwhether our findings are still valid

Study force-based network permutations such as Kamada-Kawai andFruchterman-Reingold

Define network growth that consider mixed rules instead of independent ones

Study the evolution by simultaneously varying the two axes

Continue the work done at Indiana University and in particular verify whether theidea of “duplex” networked systems can be extended to digital libraries




Results

Thank you



Multidimensional Analysis of Complex Networks

Education

complex networks cn

bologna universit

complex system

observed networks

multidimensional introduction

overall behavior

activepossamai linouniversit

interactpossamai linouniversit