1. lidong.wu@utdallas.edu Lecture 1-5 Power Law Structure Weili Wu Ding-Zhu Du Univ of Texas at Dallas

2. 2 is a type of mathematical graph in which most nodes are not neighbors of one another, but most nodes can be reached from every other by a small number of hops or steps.” “The small world network Why small distance and large size can stay together?

3. During the evolution and growth of a network, the great majority of new edges are to nodes with an already high degree. Power Law 3

4. Power law distribution: f(x) ~ x –α Log-log scale: log f(x) ~ –αlog x Power-law distribution 4

5. Nodes with high degrees may have “butterfly effect”. Small number Big influence Power Law 5

6. Important Facts on Power-law 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.

7. What is Power Law Graph?

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9. Warning In study on Power-law Graph, a lot of real numbers are treated as integers!!! 9

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11. 11 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, SIGCOMM’99,

12. Why still NP-hard in Power-law? 12

13. 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. 13

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15. Why approximate easily in Power- law? 15

16. More nodes with low degree Less nodes with high degree Size of opt solution is often determined by # of nodes with low degree. 16

17. Modularity Maximization 17 Modularity Function (Newman 2006)

18. Modularity Maximization 18

19. Lower-degree nodes follow higher-degree nodes. Idea 19

20. Low-Degree Following Algorithm 20 i i j i i t T.N. Dinh & M.T. Thai, 2013

21. Low-Degree Following Algorithm 21

22. Low-Degree Following Algorithm 22 Choice of d 0

23. Low-Degree Following Algorithm 23 Theorem

24. Idea of Proof 24

25. Lower bound for positive part 25

26. Upper bound for negative part 26

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28. Warning In study on Power-law Graph, a lot of real numbers are treated as integers!!! 28 Can we get same results if not do so?

29. References 29

30. THANK YOU!