Closeness Centrality for Networks with Overlapping Community Structure

Certain real-life networks have a community structure in which communities overlap. For example, a typical bus network includes bus stops (nodes), which belong to one or more bus lines (communities) that often overlap. Clearly, it is important to take this information into account when measuring the centrality of a bus stop—how important it is to the functioning of the network. For example, if a certain stop becomes inaccessible, the impact will depend in part on the bus lines that visit it. However, existing centrality measures do not take such information into account. Our aim is to bridge this gap. We begin by developing a new game-theoretic solution concept, which we call the Configuration semivalue, in order to have greater flexibility in modelling the community structure compared to previous solution concepts from cooperative game theory. We then use the new concept as a building block to construct the first extension of Closeness centrality to networks with community structure (overlapping or otherwise). Despite the computational complexity inherited from the Configuration semivalue, we show that the corresponding extension of Closeness centrality can be computed in polynomial time. We empirically evaluate this measure and our algorithm that computes it by analysing the Warsaw public transportation network. Introduction One of the key problems in network science involves identifying the most important (or central) nodes (Freeman 1979; Dezső and Barabási 2002; Keinan et al. 2004; Page et al. 1999). The four best-known centrality measures are Degree, Betweenness, Closeness and Eigenvector centralities (Bonacich 1972; Freeman 1979), each of which views centrality from a different perspective, focusing on certain traits that make nodes important, or, “central,” to the functioning of a network (Brandes and Erlebach 2005; Koschutzki et al. 2005). Our focus in this paper is on Closeness centrality. This measure considers the important nodes to be those that are relatively close to all other nodes in the network: the closer a node is to the others, the higher its centrality. Closeness centrality has many applications, from coauthorship networks (Yan and Ding 2009), through tourism (Shih 2006), to social networks (Barabasi 2003; Karinthy 2006). Copyright c © 2016, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. One aspect of networks that has been largely ignored in the literature on centrality is the fact that certain real-life networks have a predefined community structure. In public transportation networks, for example, bus stops are typically grouped by the bus lines (or routes) that visit them. In coauthorship networks, the various venues where authors publish can be interpreted as communities (Szczepański, Michalak, and Wooldridge 2014). In social networks, individuals grouped by similar interests can be thought of as members of a community. Clearly for such networks, it is desirable to have a centrality measure that accounts for the predefined community structure. Yet, to the best of our knowledge, only one such measure has been developed to date (Szczepański, Michalak, and Wooldridge 2014), which extends Degree centrality to networks with community structure. Despite this recent development, one important aspect of real-life networks remains missing from existing centrality measures: the ability to consider overlapping communities. Take social networks, for example, where such overlaps are widespread due to the various affiliations and interests of the individuals involved (Kelley et al. 2012). Likewise, in our example of transportation networks, a bus stop may be on the route of multiple (i.e., overlapping) bus lines. If such a stop becomes inaccessible, then all the bus lines that visit it would no longer function properly. As such, the importance of a bus stop clearly depends (at least partially) on the importance of the bus lines to which it belongs. In an attempt to define a centrality measure that accounts for overlapping communities, we focused on game-theoretic centrality measures.1 The inspiration behind this line of research comes from solution concepts in cooperative game theory. In essence, given a set of players, a cooperative solution concept typically defines a payoff for each player by comparing his or her contribution to the various groups of players (more on this in the next section). The rich repository of solution concepts has been extensively refined and expanded over the past decades, making it an ideal toolkit for quantifying the importance of individuals in a setting where those individuals co-exist and operate in groups. In the context of game-theoretic network centrality, the indiSee www.game-theoretic-centrality.com and www.network-centrality.com for an overview of this line of research. Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence (AAAI-16)