1. 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

2. Background Traditional approach - random graph theory of Erdos and Renyi  Rarely tested in real world Current technology allows analysis of large complex networks (Ex: WWW, citation patterns in science, etc)

3. 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.

4. Complex network examples Actor collaborationWWWPower grid data Citations in published papers:  cite = 3

5. 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

6. 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.

7. 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

8. Barabasi’s Experiment Start with small number of vertices: m O At each time step, add new vertex with m(<=m O ) edges that link new vertex to m previous vertices Probability  that a new vertex will be connected to vertex i depends on connectivity k i of that vertex  k i ) = k i /  j k j (Preferential attachment) Demo in Matlab

9. 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.

10. 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.