Physics – Physics and Society
Scientific paper
2012-01-03
Physics
Physics and Society
16 pages, 5 figures
Scientific paper
Much effort has gone into understanding the modular nature of complex networks. Communities, also known as clusters or modules, are densely interconnected groups of nodes that are only sparsely connected to other groups in the network. Discovering high quality communities is a difficult and important problem in a number of areas. The most popular approach is the objective function known as Modularity, used to both discover communities and measure their strength. To understand the modular structure of networks it is then crucial to know how such functions evaluate different topologies, what features they account for and what implicit assumptions they may make. We show that trees and treelike networks can have unexpectedly and often arbitrarily high values of modularity. This is surprising since trees are maximally sparse connected graphs and are not typically considered to possess modular structure, yet the non-local null model used by modularity assigns low probabilities, and thus high significance, to the densities of these sparse tree communities. We further study the practical performance of popular methods on model trees and on a genealogical dataset, and find that the discovered communities also have very high modularity, often approaching its maximum value.
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