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137 To calculate the expected value of K for a random graph, we need to calculate the expected values of eAA and eAB for a graph with the same community division and degree distribution, but with the links placed without regard for the communities. We define eA = eAA +eAB and eB = eBB +eAB, with eA denoting the number of edges that reach vertices within A, and eB giving the same quantity for B. In a random graph, we would expect that eXY = eXeY . Thus, our normalized conductance metric C can be written as C= eAA − eAeA (7.4) eAA + eAB eAeA + eAeB The metric C ranges between -1 and 1. Similar to modularity, a value of 1 indicates a significant community structure in A, a value of 0 indicates no more community struc- ture than a random graph, and a value of -1 indicates less community structure than a random graph. One particularly useful property of this definition of conductance is that it is comparable across communities of different sizes and densities. Previous definitions generally only use the ratio of intra-community links to inter-community links, which is naturally biased towards very large communities. Algorithm We now describe our algorithm for detecting a single community, using the normalized conductance metric C. We assume the algorithm is given as input a subset of users S in a community and the social network graph G = (V,E). The algorithm then returns the other members of the community. Similar to the approach that was takenPDF Image | Online Social Networks: Measurement, Analysis, and Applications to Distributed Information Systems
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