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29 tity in a network with the same community divisions, but with edges placed without regard for communities. Modularity therefore ranges from -1 to 1, with 0 representing no more community structure than would be expected in a random graph, and signif- icantly positive values representing the presence of community structure. In practice, a modularity over 0.3 or higher is observed in real-world networks with significant community structure [118]. 2.3.10 Connected components Finally, we discuss the notion of connected components in graphs. For an undirected graph, a connected component as a subset of the nodes such that there is a path in the network between all pairs of nodes in the set. For a directed graph, we distin- guish between a strongly connected component and a weakly connected component. A strongly connected component (SCC) is defined as a set of nodes such that there is a path in the network between all pairs of nodes in the set. In contrast, a weakly connected component (WCC) is defined as a set of nodes such that there is a path in the network between all pairs of nodes in set if the all links in the network were viewed as undirected. Studies of real-world networks, such as the Web and the Internet topology, has shown that there often exists a single, dominating SCC which is orders of magnitude larger than all other SCCs [23]. In this thesis, we refer to this dominating component as the dominant SCC .PDF Image | Online Social Networks: Measurement, Analysis, and Applications to Distributed Information Systems
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