Complex networks



  1. Introduction.
    • History of complex networks. Sociology and Mathematics.
    • Examples of networks. Biological, social, technological networks.
    • Basic concepts of graph theory.
  2. Random networks and Regular Networks.
    • The Erdős-Rényi model.
  3. Small-world networks.
    • Diameter and clustering. Empirical evidence.
    • Watts-Strogatz model.
  4. Scale-free networks.
    • Distribution of degree. Empirical evidence.
    • Barabasi-Albert model.
    • The Configurational model.
  5. Characterization of networks.
    • Degree correlations. Assortativity.
    • Centrality measures: Closeness, Degree, Eigenvector, PageRank, Betweenness.
    • Communities. Detection of communities.
    • Motifs.
    • Spectral properties: spectra of the Adjacency matrix. Graph Laplacian.
  6. Resilience of complex networks.
    • Percolation theory.
    • Tolerance to errors and attacks.
  7. Dynamics on networks.
    • Diffusion on networks: random walks and levy flights.
    • Spreading processes: SIS and SIR dynamics on networks. Homogenous, Heterogenous and Quenched mean-field approximations.
    • Evolutionary Games on Networks. Two-players games: Prisoner’s Dilemma. N-players games: public good games.
  8. Advanced topics.
    • Temporal Networks
    • Multilayer and Multiplex Networks
    • Higher-Order Interactions: Simplicial Complexes, Hypergraphs
  9. Computational modeling of complex networks.
    • Introduction to Python's NetworkX
    • Visualization with Gephi, Cytoscape