TY - GEN

T1 - Sample complexity bounds for influence maximization

AU - Sadeh, Gal

AU - Cohen, Edith

AU - Kaplan, Haim

N1 - Publisher Copyright:
© Gal Sadeh, Edith Cohen, and Haim Kaplan.

PY - 2020/1

Y1 - 2020/1

N2 - 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 [31]. 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 − )-approximate maximizer with confidence 1 − δ. Our main result is a surprising upper bound of O(sτ−2 ln nδ ) 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(sn−2 ln nδ ). 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 − )-approximate maximizer from simulations and applies to any submodular stochastic diffusion model that satisfies the variance bounds.

AB - 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 [31]. 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 − )-approximate maximizer with confidence 1 − δ. Our main result is a surprising upper bound of O(sτ−2 ln nδ ) 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(sn−2 ln nδ ). 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 − )-approximate maximizer from simulations and applies to any submodular stochastic diffusion model that satisfies the variance bounds.

KW - Influence maximization

KW - Sample complexity

KW - Submodular maximization

UR - http://www.scopus.com/inward/record.url?scp=85078031033&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ITCS.2020.29

DO - 10.4230/LIPIcs.ITCS.2020.29

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AN - SCOPUS:85078031033

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 11th Innovations in Theoretical Computer Science Conference, ITCS 2020

A2 - Vidick, Thomas

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 11th Innovations in Theoretical Computer Science Conference, ITCS 2020

Y2 - 12 January 2020 through 14 January 2020

ER -