Complex networks: community detectation and virus propagation

Complex networks: community detection,virus propagation and immunization

Eliezer de Souza da Silva

[email protected]

Friday 27th November, 2015

Communitydetection and

viruspropagation

E.S. Silva

CommunityDetectionClustering coefficient

Detectingcommunities

ViruspropagationandimmunizationTerminology andbasic models

In social network

References

Summary

1 Community DetectionClustering coefficientDetecting communities

2 Virus propagation and immunizationTerminology and basic modelsIn social network

3 References


viruspropagation

E.S. Silva




In social network

References

Introduction

basic intuition

A community is a set of nodes with more connectionwithin the set than outside. Elements of a communitymay:

Share common properties;Play similar roles;Compress/summarize collective information;


viruspropagation

E.S. Silva




In social network

References

Introduction


viruspropagation

E.S. Silva




In social network

References

Introduction

main question

How to formalize this basic intuition?

intra-cluster density and extra-cluster density;clustering coefficient;connectedness, centrality and edge-betweenness;graph partitioning;Authorities and hubs;...


viruspropagation

E.S. Silva




In social network

References

Clustering coefficient

Global: measure of how elements of the graph tend toform clusterLocal: measure of “transitivity” of the neighbourhood ofone vertex. The probability of “friends of friends” beingconnected.


viruspropagation

E.S. Silva




In social network

References

Local Clustering coefficient

Given a vertex vi , with ki neighbours with ni edges betweenthe set of neighbours we define the local clusteringcoefficient [1] for direct (Eq 1) and undirected graphs (Eq 2)for ki > 1 as:

Ci =2ni

ki(ki − 1)(1)

Ci =ni

ki(ki − 1)(2)

If ki = 0,1 then Ci = 0


viruspropagation

E.S. Silva




In social network

References

Global Clustering coefficient

For a graph with N vertices:Watts and Strogatz (Eq 3)Counting triangles and triples (Eq 4)

C =1N

N∑i=1

Ci (3)

C =3 × number of triangles in the graph

number of connected triples in the graph(4)


viruspropagation

E.S. Silva




In social network

References

Detecting communities

Dendograms;Edge-betweenness (centrality);Max-flow min-cut;Graph partitioning;Bipartite cores;Spectral methods;cross-association: minimum description length andcompressability;Random walks;Metric embeddings.


viruspropagation

E.S. Silva




In social network

References

Virus propagation and immunization

Long standing tradition of mathematical models inepidemiology.Traditional models are based on dividing the populationin a small set of compartments with few differentialequations describing the transition between thecompartments – many analytical results derived fromthis framework.Application in distinct emerging areas such asinformation spreading, social contagion, marketing andanalysis of online social networks depends onexpanding this framework to more complex networktopologies using more computational intensive tools.


viruspropagation

E.S. Silva




In social network

References

Terminology

basic terminology

S: Susceptible/healthyI: Infected (and infectious)R: Removed/recovered – the node has immunityfor life (or is deceased)V: Vigilant – the node can not be infected (butmay lose it’s immunity, depending on the VPM)E: Exposed – the node is not infectious, but it is acarrier of the virus, and it will eventually evolve tothe “Infected/Infectious” state.


viruspropagation

E.S. Silva




In social network

References

Basic models

(a) SIR and SEIV (b) SIR,SIS,SIRS andSEIR


viruspropagation

E.S. Silva




In social network

References

Main results


viruspropagation

E.S. Silva




In social network

References

Main result

main result

For S ∗ I2V∗ model with arbitrar undirect graph withadjancy matrix A the sufficient condition for stability is:

s < 1

where s is s = λ1 × CVPM


viruspropagation

E.S. Silva




In social network

References

Modelling approaches

Markov theory;Mean Field (individual, density based);non-markovian simulations;


viruspropagation

E.S. Silva




In social network

References

Contagion in social networks


viruspropagation

E.S. Silva




In social network

References

References I

Duncan J Watts and Steven H Strogatz.Collective dynamics of ‘small-world’networks.nature, 393(6684):440–442, 1998.

Data & Analytics

Complex networks: community detectation and virus propagation