Statistics Papers

Document Type

Journal Article

Date of this Version

2007

Publication Source

Proceedings of the Thirty-ninth Annual ACM Symposium on Theory of Computing

Volume

STOC '07

Start Page

128

Last Page

134

DOI

10.1145/1250790.1250811

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 [KKT03, KKT05] 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 σ(S) denote the expected number of active nodes at termination. Here we prove a conjecture of KKT: we show that the function σ(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.

Comments

At the time of publication, author Elchanan Mossel was affiliated with University of California, Berkeley. Currently, he is a faculty member at the Statistics Department at the University of Pennsylvania.

Keywords

coupling, social networks, submodularity, viral marketing

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Date Posted: 27 November 2017