This paper proposes a new measure of node centrality in social networks, the Harmonic Influence Centrality, which emerges naturally in the study of social influence over networks. Using an intuitive analogy between social and electrical networks, we introduce a distributed message passing algorithm to compute the Harmonic Influence Centrality of each node. Although its design is based on theoretical results which assume the network to have no cycle, the algorithm can also be successfully applied on general graphs.
In social science and network science there is a large literature on defining and computing centrality measures @cite_5 . Among the most popular definitions, we mention degree centrality, node betweenness, information centrality @cite_13 , and Bonacich centrality @cite_1 , which is related to the well-studied Google PageRank algorithm. These notions have proven useful in a range of applications but are not universally the appropriate concepts by any means. Our interest in opinion dynamics thus motivates the choice to define a centrality measure for our purposes. The problem of computing the HIC has previously been solved, via a centralized solution, in @cite_3 for a special case of the opinion dynamics model considered here.
@cite_1 @cite_5 @cite_13 @cite_3
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Although network centrality is generally assumed to produce power, recent research shows that this is not the case in exchange networks. This paper proposes a generalization of the concept of centrality that accounts for both the usual positive relationship between power and centrality and 's recent exceptional results. Three measures of actors' network centrality are derived from an elementary rocess model of social influence. The measures are closely related to, and cast new light on, widely used measures of actors' centrality; for example, the essential social organization of status that has been assumed by Hubbell, Bonacich, Coleman, and Burt appears as a deducible outcome of this social influence process. Unlike previous measures, which have been viewed as competing alternatives, the present measures are complementary and, in their juxtaposition, provide for a rich description of social structure. The complementary indicates a degree of theoretical unification in the work on network centrality that was heretofore unsuspected. Abstract A new model of centrality is proposed for networks. The centrality measure is based on the “information” contained in all possible paths between pairs of points. The method does not require path enumeration and is not limited to the shortest paths or geodesies. We apply this measure to two examples: a network of homosexual men diagnosed with AIDS, and observations on a colony of baboons. Comparisons are made with “betweenness” and “closeness” centrality measures. The processes by which structural changes in networks occur over time are also discussed. We study discrete opinion dynamics in a social network with "stubborn agents" who influence others but do not change their opinions. We generalize the classical voter model by introducing nodes (stubborn agents) that have a fixed state. We show that the presence of stubborn agents with opposing opinions precludes convergence to consensus; instead, opinions converge in distribution with disagreement and fluctuations. In addition to the first moment of this distribution typically studied in the literature, we study the behavior of the second moment in terms of network properties and the opinions and locations of stubborn agents. We also study the problem of "optimal placement of stubborn agents" where the location of a fixed number of stubborn agents is chosen to have the maximum impact on the long-run expected opinions of agents.