Submodularity of Influence in Social Networks: From Local to Global

Abstract

Social networks are often represented as directed graphs, where the nodes are individuals and the edges indicate a form of social relationship. A simple way to model the diffusion of ideas, innovative behavior, or “word-of-mouth” effects on such a graph is to consider an increasing process of “infected” (or active) nodes: each node becomes infected once an activation function of the set of its infected neighbors crosses a certain threshold value. Such a model was introduced by Kempe, Kleinberg, and Tardos (KKT) in [Maximizing the spread of influence through a social network, in Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2003, pp. 137–146] and [Influential nodes in a diffusion model for social networks, in Proceedings of the 32nd International Colloquium on Automata, Languages and Programming (ICALP), 2005], where the authors also impose several natural assumptions: the threshold values are random and the activation functions are monotone and submodular. The monotonicity condition indicates that a node is more likely to become active if more of its neighbors are active, while the submodularity condition indicates that the marginal effect of each neighbor is decreasing when the set of active neighbors increases. For an initial set of active nodes S, let $\sigma(S)$ denote the expected number of active nodes at termination. Here, we prove a conjecture of KKT: we show that the function $\sigma(S)$ is submodular under the assumptions above. We prove the same result for the expected value of any monotone, submodular function of the set of active nodes at termination. Roughly, our results demonstrate that “local” submodularity is preserved “globally” under this diffusion process. This is of natural computational interest, as many optimization problems have good approximation algorithms for submodular functions.