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Power Law Networks: Small Distance, Large Size

Explore the relationship between small distance and large size in power law networks, where most nodes can be reached from each other with a small number of hops. Discover the implications of power-law distribution, the butterfly effect, and the importance of nodes with high degrees. Learn about the challenges and advantages of working with power-law graphs and why many NP-hard problems remain NP-hard in this context. Explore proof techniques and approximation methods that make it easier to work with power-law networks.

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Power Law Networks: Small Distance, Large Size

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

  2. “The small world network 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.” Why small distance and large size can stay together?

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

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

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

  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?

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

  9. 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,

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

  11. 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. Why approximate easily in Power-law?

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

  14. Modularity Maximization Modularity Function (Newman 2006)

  15. Modularity Maximization

  16. Idea • Lower-degree nodes follow higher-degree nodes.

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

  18. Low-Degree Following Algorithm

  19. Low-Degree Following Algorithm Choice of d0

  20. Low-Degree Following Algorithm Theorem

  21. Idea of Proof

  22. Lower bound for positive part

  23. Upper bound for negative part

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

  25. References

  26. THANK YOU!

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