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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

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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)

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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.

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

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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

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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.

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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

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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

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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.

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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.

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