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



Social networks: Milgram’s experiment









Asymptotic behavior



In-between: Small-world networks



Size-dependence



Is that all we need ?

  • Is that all we need ?





















Tools for characterizing the various models

  • Connectivity distribution P(k)

  • Clustering

  • Assortativity

  • ...



Topological correlations: clustering

  • aij: Adjacency matrix



Topological correlations: assortativity



Assortativity

  • Assortative behaviour: growing knn(k)

  • Example: social networks

  • Large sites are connected with large sites

  • Disassortative behaviour: decreasing knn(k)

  • Example: internet

  • Large sites connected with small sites, hierarchical structure



Consequences of the topological heterogeneity

  • Robustness and vulnerability

  • Propagation of epidemics









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

  • ...



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











Perspectives: Weighted networks

  • 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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