Abstract
The structure of close communication, contacts and association in social networks is studied in the form of maximal subgraphs of diameter 2 (2-clubs), corresponding to three types of close communities: hamlets, social circles and coteries. The concept of borough of a graph is defined and introduced. Each borough is a chained union of 2-clubs of the network and any 2-club of the network belongs to one borough. Thus the set of boroughs of a network, together with the 2-clubs held by them, are shown to contain the structure of close communication in a network. Applications are given with examples from real world network data.
Highlights
The structure of close communication, contacts and association in social networks is studied in the form of maximal subgraphs of diameter 2 (2-clubs), corresponding to three types of close communities: hamlets, social circles and coteries
The 2-clubs of a simple graph or network G cover the areas of close communication in that network consisting of non-inclusive, possibly mutually overlapping coteries, social circles and hamlets. This system of close communication is studied further and we show that it consists of a set of disjoint containers of nonseparable 2clubs, i.e. subgraphs that we call boroughs, each of which is formed by a set of edge-chained 2-clubs of the network G
Corollary 3 Let B be a borough of G and let B À ðu; vÞ, denote the subgraph obtained by removing an edge (u, v) from B, dmðBÞ dmðB À ðu; vÞÞ dmðBÞ þ 3: Proof Consider the set of all shortest paths containing (u, v) defining distances between pairs of nodes of B
Summary
The last decade has produced an increasing volume of methods and algorithms to analyze community structure in social and other networks, as witnessed by an abundance of recent reviews, e.g. Girvan and Newman (2002), Newman (2004), Balasundaram et al (2005), Palla et al (2005), Reichhardt and Bornholdt (2006), Blondel et al (2008), Leskovec et al (2008), Porter et al (2009), Fortunato (2010) and Xie et al (2013). We study the structure of close communication, contacts and association in networks, as represented by simple graphs. Close communication is defined here as contact between nodes at distances of at most 2, that is by direct contact or by at least one common neighboring node. The parts of a network where close communication can take place are marked by overlapping subsets of nodes, which all are neighbors of each other or have a common neighbor in the same subset. These correspond to graphs with a diameter of at most two. Mokken (1979, 2008) introduced the concept of k-clubs of a graph as maximal induced subgraphs of diameter at most k of a simple connected graph G: ’maximal’ in the sense that there is no larger induced subgraph of diameter
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