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is one. Thus, betweenness centrality for an edge e can be expressed as B(e) = σe(u, v) (2.6) 28 u∈V,v∈V σ(u, v) where σ(u,v) represents the number of shortest paths between u and v, and σe(u,v) represents the number of shortest paths between u and v that include e. The be- tweenness centrality of an edge can be viewed as a metric for the importance of an edge in a graph, as edges with a higher betweenness centrality fall on more shortest paths, and are therefore more important for the structure of the graph. 2.3.9 Modularity When examining communities in networks, one often requires an objective metric to evaluate how “good” a particular division of the network into communities is. One such metric is the the modularity measure proposed by Newman [118]. Consider a community structure of k communities. Let e be a symmetric k × k matrix, whose element eij is the fraction of edges in the network that connect vertices in community i to community j by considering all the edges in the original network. Also, we define ai = j eij be the fraction of edges that touch vertices in community i. Then, the trace of the matrix Tr e = i eii gives the fraction of edges in the network within the same community. Hence, modularity is defined as Q=(eii −a2i)=Tre−||e2|| (2.7) i where ||y|| indicates the sum of elements of matrix y. Modularity is then a measure of the fraction of intra-community edges minus the expected value of the same quan-PDF Image | Online Social Networks: Measurement, Analysis, and Applications to Distributed Information Systems
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