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Sample Complexity Bounds for Influence Maximization

Authors Gal Sadeh, Edith Cohen , Haim Kaplan

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Author Details

Gal Sadeh
  • Tel Aviv University, Israel
Edith Cohen
  • Google Research, Mountain View, CA, USA
  • Tel Aviv University, Israel
Haim Kaplan
  • Google Research, Tel Aviv, Israel
  • Tel Aviv University, Israel

Funding

This research is partially supported by the Israel Science Foundation (Grant No. 1841/14 and Grant No. 1595/19).

Cite AsGet BibTex

Gal Sadeh, Edith Cohen, and Haim Kaplan. Sample Complexity Bounds for Influence Maximization. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 29:1-29:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ITCS.2020.29

Abstract

Influence maximization (IM) is the problem of finding for a given s ≥ 1 a set S of |S|=s nodes in a network with maximum influence. With stochastic diffusion models, the influence of a set S of seed nodes is defined as the expectation of its reachability over simulations, where each simulation specifies a deterministic reachability function. Two well-studied special cases are the Independent Cascade (IC) and the Linear Threshold (LT) models of Kempe, Kleinberg, and Tardos [Kempe et al., 2003]. The influence function in stochastic diffusion is unbiasedly estimated by averaging reachability values over i.i.d. simulations. We study the IM sample complexity: the number of simulations needed to determine a (1-ε)-approximate maximizer with confidence 1-δ. Our main result is a surprising upper bound of O(s τ ε^{-2} ln (n/δ)) for a broad class of models that includes IC and LT models and their mixtures, where n is the number of nodes and τ is the number of diffusion steps. Generally τ ≪ n, so this significantly improves over the generic upper bound of O(s n ε^{-2} ln (n/δ)). Our sample complexity bounds are derived from novel upper bounds on the variance of the reachability that allow for small relative error for influential sets and additive error when influence is small. Moreover, we provide a data-adaptive method that can detect and utilize fewer simulations on models where it suffices. Finally, we provide an efficient greedy design that computes an (1-1/e-ε)-approximate maximizer from simulations and applies to any submodular stochastic diffusion model that satisfies the variance bounds.

Subject Classification

ACM Subject Classification
  • Theory of computation → Machine learning theory
Keywords
  • Sample complexity
  • Influence maximization
  • Submodular maximization

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