Distance-Based Influence in Networks: Computation and Maximization

Abstract

A premise at a heart of network analysis is that entities in a network derive utilities from their connections. The {\em influence} of a seed set $S$ of nodes is defined as the sum over nodes $u$ of the {\em utility} of $S$ to $u$. {\em Distance-based} utility, which is a decreasing function of the distance from $S$ to $u$, was explored in several successful research threads from social network analysis and economics: Network formation games [Bloch andJackson 2007], Reachability-based influence [Richardson and Domingos 2002, Kempe et al. 2003], "threshold" influence [Gomez-Rodriguez et al. 2011], and {\em closeness centrality} [Bavelas 1948]. We formulate a model that unifies and extends this previous work and address the two fundamental computational problems in this domain: {\em Influence oracles} and {\em influence maximization} (IM). An oracle performs some preprocessing, after which influence queries for arbitrary seed sets can be efficiently computed. With IM, we seek a set of nodes of a given size with maximum influence. Since the IM problem is computationally hard, we instead seek a {\em greedy sequence} of nodes, with each prefix having influence that is at least $1-1/e$ of that of the optimal seed set of the same size. We present the first highly scalable algorithms for both problems, providing statistical guarantees on approximation quality and near-linear worst-case bounds on the computation. We perform an experimental evaluation which demonstrates the effectiveness of our designs on networks with hundreds of millions of edges.