Internet C. Elegans Large Complex Networks: Deteministic Models (Recursive Clique-Trees) WWW Erdös number Air routes http://www.caida.org/tools/visualization/plankton/ Francesc Comellas Departament de Matemàtica Aplicada IV, Proteins Universitat Politècnica de Catalunya, Barcelona
[email protected] Power grid Real networks very often are Most “real” networks are Large Complex systems small-world scale-free self-similar Small-world Different elements (nodes) Interaction among elements (links) small diameter log(|V|), large clustering Scale-free power law degree distribution ( “hubs” ) Complex networks Self-similar / fractal Mathematical model: Graphs Deterministic models Small diameter (logarithmic) Power law Fractal (degrees) Based on cliques Milgram 1967 Song, Havlin & (hierarchical graphs, recursive High clustering Barabási & Makse Watts & Albert 1999 2005,2006 clique-trees, Apollonian graphs) Strogatz 1998 6 degrees of separation ! Stanley Milgram (1967) Main parameters (invariants) 160 letters Omaha -Nebraska- -> Boston Diameter – average distance Degree Δ degree. Small-world networks P(k): Degree distribution. small diameter (or average dist.) Clustering Are neighbours of a vertex also neighbours among high clustering them? Small world phenomenon in social networks What a small-world ! Structured graph Small-world graph Random graph Network characteristics •highclustering •highclustering • small clustering • large diameter • small diameter • small diameter # of links among neighbors •regular Clustering C(v) = Erdös number •almostregular n(n-1)/2 http://www.oakland.edu/enp/ Diameter or Average distance Maximum communication delay 1- 509 Degree distribution 2- 7494 Resilience N= 268.000 Jul 2004 |V| = 1000 Δ =10 |V|=1000 Δ =8-13 |V|=1000 Δ = 5-18 Real life networks are clustered, large C(p), but have small (connected component) D = 100 d =49.51 D = 14 d = 11.1 D = 5 d = 4.46 average distance L(p ).