The Barabási-Albert [BA] model (1999) ER Model Look at the distribution of degrees ER ModelWS Model actorspower grid www The probability of finding a highly connected node decreases exponentially with k Random graphs, Watts-Strogatz graphs are homogeneous graphs (small fluctuations of the degree k).

Most real world networks have the same internal structure: Scale-free networks Why? What does it mean?

Traditional modeling: Network as a static graph Given a network with N nodes and L links Create a graph with statistically identical topology RESULT: model the static network topology PROBLEM: Real networks are dynamical systems!  Evolving networks OBJECTIVE: capture the network dynamics METHOD : identify the processes that contribute to the network topology develop dynamical models that capture these processes BONUS: get the topology correctly. 

SCALE-FREE NETWORKS (1) The number of nodes (N) is NOT fixed. Networks continuously expand by the addition of new nodes Examples: WWW : addition of new documents Citation : publication of new papers (2) The attachment is NOT uniform. A node is linked with higher probability to a node that already has a large number of links. Examples : WWW : new documents link to well known sites (CNN, YAHOO, NewYork Times, etc) Citation : well cited papers are more likely to be cited again Origins SF

H.A. Simon (1955). Power laws in distributions: –Scientists by number of papers published –Cities by population –Income by size -> “rich get richer” growth-like stochastic process Barabasi et al. (1999). Power laws in WWW –in-degree & out-degree -> growth processes

Scale-free model (1) GROWTH : A t every timestep we add a new node with m edges (connected to the nodes already present in the system). (2) PREFERENTIAL ATTACHMENT : The probability Π that a new node will be connected to node i depends on the connectivity k i of that node A.-L.Barabási, R. Albert, Science 286, 509 (1999) P(k) ~k -3 BA model