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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?
• During the evolution and growth of a network, the great majority of new edges are to nodes with an already high degree.
Power Law
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Power law distribution:f(x) ~ x–α
Log-log scale:log f(x) ~ –αlog x
Power-law distribution
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• Nodes with high degrees may have “butterfly effect”.
• Small number• Big influence
Power Law
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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?
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.45.21.2 networks,ion collaborat Scientific
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?
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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.
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Why approximate easily in Power-law?
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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.
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Modularity Maximization
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Idea
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Low-Degree Following Algorithm
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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?
References
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(2013) 1006-997 :31(6)
,modularity maximizing
for algorithmsion approximat :networks free-scalein
detection Community :Thai T.My Dinh, N. Thang .1
onsommunicatiAreas in C
ctedal on SeleIEEE Journ
THANK YOU!
