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CS728-2008 Lecture 6 Generative Graph Models Part II

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Review of Generative Models Waxman’s – locality model in plane Configuration – specifies degree sequence Price’s – cumulative advantage Barabasi and Albert - preferential attachment Copying Model Watts and Strogatz Beta Model – link rewiring in clustered, organized network Temporal Evolution models - Densification Today’s lecture

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Densification – Possible Explanations Generative models to capture the Densification Power Law and Shrinking diameters 2 proposed models: –Community Guided Attachment – obeys Densification –Forest Fire model – obeys Densification, Shrinking diameter (and Power Law degree distribution)

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Community structure Assume organized community structure Expect many within- group friendships and fewer cross- group ones How hard is it to cross communities? Self-similar university community structure CS Math DramaMusic Science Arts University

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cross-community linking probability of nodes at tree-distance h is scale-free linking probability: prob(x – y) = c -h where: c ≥ 1 … the Difficulty constant h … tree-distance of x,y Cross-community Linking Assumption

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Densification Power Law (1) Theorem: The Community Guided Attachment leads to Densification Power Law with exponent a … densification exponent b … community structure branching factor c … difficulty constant

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Theorem: Gives any non-integer Densification exponent If c = 1: easy to cross communities –Then: a=2, quadratic growth of edges – near clique If c = b: hard to cross communities –Then: a=1, linear growth of edges – constant out-degree Difficulty Constant

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Room for Improvement Community Guided Attachment explains Densification Power Law Issues: –Requires explicit Community structure –Does not obey Shrinking Diameters

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Ingredients for “Forest Fire” model –“Rich get richer” preferential attachment process, to get power-law in-degrees –“Copying” model, to lead to communities –Community Guided Attachment, to produce Densification Power Law

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“Forest Fire” model – Intuition How do authors identify references? 1.Find first paper and cite it 2.Copy a few citations from first 3.Continue recursively 4.From time to time use bibliographic tools (e.g. CiteSeer) and chase back-links

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“Forest Fire” – the Model A node arrives Randomly chooses an “ambassador” Starts burning nodes (with probability p) and adds links to burned nodes “Fire” spreads recursively, with exponential decay

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Forest Fire in Action Forest Fire generates graphs that Densify and have Shrinking Diameter densification diameter 1.21 N(t) E(t) N(t) diameter

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Forest Fire in Action Forest Fire also generates graphs with heavy-tailed degree distribution in-degreeout-degree count vs. in-degreecount vs. out-degree

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Forest Fire model – Justification Densification Power Law: –Similar to Community Guided Attachment –The probability of linking decays exponentially with the distance – Densification Power Law Power law out-degrees: –From time to time we get large fires Power law in-degrees: –The fire is more likely to burn hubs

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Forest Fire – the Formal Model 2 parameters: –p … forward burning probability –r … backward burning ratio Nodes arrive one at a time New node v attaches to a random node – the ambassador Then v begins burning ambassador’s neighbors: –Burn X links, where X is binomially distributed with mean 1/1-p –Choose in-links with probability r times less than out- links Fire spreads recursively Node v attaches to all nodes that got burned

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Forest Fire – Phase plots Exploring the Forest Fire parameter space Sparse graph Dense graph Increasing diameter Shrinking diameter

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Densification and Shrinking Diameter Are the Densification and Shrinking Diameter two different observations of the same phenomena? No! Forest Fire can generate: –(1) Sparse graphs with increasing diameter –Sparse graphs with decreasing diameter –(2) Dense graphs with decreasing diameter 1 2

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Next time: Searchable Networks Questions: Social: How does a person in a small world find their soul mate? Comp Sci: How does the notion of long and short edges in a “random” network impact ability to find key nodes?

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