What is expected out degree of a node x how many

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What is expected out degree of a node x? How many nodes are at distance h? A l t Analyze separate cases: 10/27/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 36 Can also generalize the model to get power law degrees and densification [seeTKDD 07]
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Claim: The Community Guided Attachment Claim: The leads to Densification Power Law with exponent: d ifi ti t a … densification exponent b … community tree branching factor c difficulty constant 1 c b … difficulty constant, 1 c 10/27/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 37
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DPL: Gives any non integer Densification exponent If c = 1 : easy to cross communities Then: a=2 quadratic growth of edges near , quadratic growth of edges – near clique If c = b : hard to cross communities c b Then: a=1 , linear growth of edges – constant out degree 10/27/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 38
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[Leskovec et al.TKDD 07] But, we do not want to have explicit communities Want to model graphs that density and have shrinking diameters Intuition: How do we meet friends at a party? H d id tif f h iti ? How do we identify references when writing papers? v w 10/27/2010 39
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[Leskovec et al.TKDD 07] The Forest Fire model has 2 parameters: p … forward burning probability r … backward burning probability h d l The model: Each turn a new node v arrives Uniformly at random chooses an “ambassador” w Flip 2 geometric coins to determine the b f i d t li k f t f ll number of in and out links of w to follow Fire spreads recursively until it dies New node v links to all burned nodes Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 40 10/27/2010
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Forest Fire generates graphs that densify and have shrinking diameter E(t) densification diameter 1.32 meter diam Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 41 N(t) N(t) 10/27/2010
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Forest Fire also generates graphs with Power Law degree distribution i d t d in degree out degree Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 42 log count vs. log in-degree log count vs. log out-degree 10/27/2010
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Fix backward b bilit d probability r and vary forward burning probability Clique like graph p Notice a sharp Increasing diameter Constant di t transition between sparse and clique like graphs Sparse graph Decreasing d diameter Sweet spot is very narrow diameter Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 43 10/27/2010
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  • Fall '09
  • Graph Theory, Jure Leskovec, Information Network Analysis, Stanford CS224W

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