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lidong.wu@utdallas.edu Lecture 3 Power Law Structure Ding-Zhu Du Univ of Texas at Dallas

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Why study Power-law? Many Social Networks are power-law graphs. Many NP-hard network problems are still NP- hard in power-law graphs. While they have no good approximation in general, they have constant-approximation in power-law graphs.

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What is Power Law Graph?

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During the evolution and growth of a network, the great majority of new edges are to nodes with an already high degree. Power Law 4

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Power law distribution: f(x) ~ x Log-log scale: log f(x) ~ log x Power-law distribution 5

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Nodes with high degrees may have butterfly effect. Small number Big influence Power Law 6

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Warning In study on Power-law Graph, a lot of real numbers are treated as integers!!! 8

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10 A.L. Barabasi, et al., Evolution of the social network of scientific collaborations, Physica A, vol. 311, 2002. R. Albert, et al., Erro and attack tolerance of complex networks, Nature, vol. 406, M. Faloutsos, et al., On power-law relationship of the internet topology, SIGCOMM99,

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Why still NP-hard in Power-law? 11

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Proof Techniques NP-hard in graph with constant degree, e.g., the Vertex-Cover is NP-hard in cubic graphs. Embedding a constant-degree graph into a power-law graph. 12

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Why approximate easily in Power- law? 14

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More nodes with low degree Less nodes with high degree Size of opt solution is often determined by # of nodes with low degree. 15

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Modularity Maximization 16 Modularity Function (Newman 2006)

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Modularity Maximization 17

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Low-Degree Following Algorithm 18 i i j i i j T.N. Dinh & M.T. Thai, 2013

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Low-Degree Following Algorithm 19

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Low-Degree Following Algorithm 20 Choice of d 0

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Low-Degree Following Algorithm 21 Theorem

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Idea of Proof 22

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Lower bound for positive part 23

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Upper bound for negative part 24

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Warning In study on Power-law Graph, a lot of real numbers are treated as integers!!! 26 Can we get same results if not do so?

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THANK YOU!