Power Law Structure Weili Wu Ding-Zhu Du Univ of Texas at Dallas Lecture 1-5 Power Law Structure Weili Wu Ding-Zhu Du Univ of Texas at Dallas First, I want to thank you for you presence. ********In this presentation I will try to introduce The social network which is a theoretical structure to study relationships between individuals, groups, organizations, or even entire societies.  It is related to a wide range of disciplines. These disciplines include, but are not limited to information science, biology, economics, geography, communication studies, and so on.. The study of social networks begins with the late eighteenth century, two sociologists (Émile [ei'mi:l] Durkheim and Ferdinand ['fɝdənænd] Fer迪南de Tönnies) foreshadowed the idea of social networks in their theories and research of social groups. Nowadays, we study social networks using network analysis to identify social communities, pick influential person, and design good software. lidong.wu@utdallas.edu

“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?

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

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

Power Law Nodes with high degrees may have “butterfly effect”. Small number Big influence a small change at one place can result in large differences in a later state.

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.

What is Power Law Graph?

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

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,

Why still NP-hard in Power-law?

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.

Why approximate easily in Power-law?

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

Modularity Maximization Modularity Function (Newman 2006)

Modularity Maximization

Idea Lower-degree nodes follow higher-degree nodes.

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

Low-Degree Following Algorithm

Low-Degree Following Algorithm Choice of d0

Low-Degree Following Algorithm Theorem

Idea of Proof

Lower bound for positive part

Upper bound for negative part

Can we get same results if not do so? 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?

References

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