Synopsis of “Emergence

of Scaling in

Random Networks”*

*Albert-Laszlo Barabasi and Reka Albert, Science, Vol 286, 15 October 1999

Presentation for ENGS 112

Doug Madory

Wed, 27 APR 05

Background

Traditional approach - random graph theory of Erdos and RenyiRarely tested in real world

Current technology allows analysis of large complex networks (Ex: WWW, citation patterns in science, etc)

Barabasi’s Claim

Independent of system and identity of its constituents, the probability P(k) that a vertex in the network interacts with k other vertices decays as a power law, following:

P(k) ~ k-

Existing network models fail to incorporate growth and preferential attachment, two key features of real networks.

Complex network examplesActor collaboration WWW Power grid data

Citations in published papers: cite = 3

Problems with other theories

Erdos-Renyi & Watts-Strogatz theories suggest probability of finding a highly connected vertex (large k) decreases exponentially with k Vertices with large k are practically absent

Barabasi - power-law tail characterizing P(k) for networks studied indicates that highly connected (large k) vertices have a large chance of occurring and dominating the connectivity

Problems with other theories

Erdos-Renyi & Watts-Strogatz assume fixed number (N) of vertices

Barabasi - real world networks form by continuous addition of new vertices, thus N increases throughout lifetime of network.

Problems with other theories

Erdos-Renyi & Watts-Strogatz - probability that two vertices are connected is random and uniform

Barabasi - real networks exhibit preferential connectivity New actor cast supporting established one New webpage will link to established pages

Barabasi’s Experiment Start with small number of vertices: mO At each time step, add new vertex with m(<=mO)

edges that link new vertex to m previous vertices Probability that a new vertex will be connected

to vertex i depends on connectivity ki of that vertex

ki) = ki/jkj (Preferential attachment)

Demo in Matlab

The “rich get richer” theory Similar mechanisms could explain the origin of social

and economic disparities governing competitive systems, because scale-free inhomogeneities are the inevitable consequence of self-organization due to local decisions made by individual vertices, based on information that is biased toward more visible (richer) vertices, irrespective of the nature and origin of this visibility.

Summary

Common property of many large networks is vertex connectivities follow a scale-free power-law distribution.

Consequence of two generic mechanisms:

(i) networks expand continuously by the addition of new vertices, and

(ii) new vertices attach preferentially to sites that are already well connected.