Complex networks A. Barrat, lpt, Université Paris-Sud, France Complex networks: examples

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Complex networks A. Barrat, LPT, Université Paris-Sud, France
Complex networks: examples Complex networks: examples Small-world networks Scale-free networks: evidences, modeling , tools for characterization Consequences of SF structure Perspectives: weighted complex networks
Examples of complex networks Internet WWW Transport networks Food webs Social networks ...
Social networks: Milgram’s experiment
Asymptotic behavior
In-between: Small-world networks
Size-dependence
Is that all we need ?
Tools for characterizing the various models Connectivity distribution P(k) Clustering Assortativity ...
Topological correlations: clustering
Topological correlations: assortativity
Assortativity Assortative behaviour: growing knn(k) Example: social networks Large sites are connected with large sites Disassortative behaviour: decreasing knn(k) Large sites connected with small sites, hierarchical structure
Consequences of the topological heterogeneity Robustness and vulnerability Propagation of epidemics
Most connected nodes Nodes with largest betweenness Removal of links linked to nodes with large k Cascades ...
Betweenness measures the “centrality” of a node i: for each pair of nodes (l,m) in the graph, there are bi is the sum of ilm / lm over all pairs (l,m)
Other attack strategies Most connected nodes Nodes with largest betweenness Removal of links linked to nodes with large k Removal of links with largest betweenness Cascades ...
What about computer viruses? Very long average lifetime (years!) compared to the time scale of the antivirus Small prevalence in the endemic case
SIS model on SF networks SIS= Susceptible – Infected – Susceptible Mean-Field usual approximation: all nodes are “equivalent” (same connectivity) => existence of an epidemic threshold 1/ for the order parameter density of infected nodes) Scale-free structure => necessary to take into account the strong heterogeneity of connectivities => k=density of infected nodes of connectivity k
Scientific collaborations Internet Emails Airports' network Finance, economic networks ...
Weights: examples Scientific collaborations:
Weights Weights: heterogeneous (broad distributions)? Correlations between topology and traffic ? Effects of the weights on the dynamics ?
Weights: recent works and perspectives Empirical studies (airport network; collaboration network: PNAS 2004) New tools (PNAS 2004) strength weighted clustering coefficient (vs. clustering coefficient) weighted assortativity (vs. assortativity) New models (PRL 2004) New effects on dynamics (resilience, epidemics...) on networks (work in progress)
Alain.Barrat@th.u-psud.fr http://www.th.u-psud.fr/
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