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Finding Statistically Significant Communities in Networks

Figure 11

Test on random graphs.

We plot the fraction of vertices belonging to non-trivial clusters (i.e. to clusters with more than one and less than vertices, where is as usual the size of the graph), as a function of the average degree of the graph. The curves correspond to Erdös-Rényi graphs (diamonds) and scale-free networks (circles). All graphs have vertices. The only parameter needed to build Erdös-Rényi graphs is the probability that a pair of vertices is connected, which is determined by the average degree . The scale-free networks were built with the configuration model [39], starting from a fixed degree sequence for the vertices obeying the predefinite power law distribution. The parameters of the distribution are: degree exponent , maximum degree .

Figure 11