Maximizing Influence-based Group Shapley Centrality

TL;DR

This paper investigates a realistic influence maximization problem considering unknown pre-existing seed users, models it with Group Shapley value, and establishes its computational hardness while proposing a near-optimal greedy solution for bounded seed sets.

Contribution

It introduces a new influence maximization problem accounting for unknown existing seed users, analyzes its computational hardness, and provides a greedy approximation algorithm with provable guarantees.

Findings

The problem is hard to approximate within 1/n^{o(1)} under the Gap Exponential Time Hypothesis.

A greedy algorithm achieves a (1-1/e)/k - epsilon approximation for bounded seed sets.

Standard influence maximization approximations do not apply to this new problem setting.

Abstract

One key problem in network analysis is the so-called influence maximization problem, which consists in finding a set $S$ of at most $k$ seed users, in a social network, maximizing the spread of information from $S$. This paper studies a related but slightly different problem: We want to find a set $S$ of at most $k$ seed users that maximizes the spread of information, when $S$ is added to an already pre-existing - but unknown - set of seed users $T$. We consider such scenario to be very realistic. Assume a central entity wants to spread a piece of news, while having a budget to influence $k$ users. This central authority may know that some users are already aware of the information and are going to spread it anyhow. The identity of these users being however completely unknown. We model this optimization problem using the Group Shapley value, a well-founded concept from cooperative game theory. While the standard influence maximization problem is easy to approximate within a factor $1-1/e-\epsilon$ for any $\epsilon>0$, assuming common computational complexity conjectures, we obtain strong hardness of approximation results for the problem at hand in this paper. Maybe most prominently, we show that it cannot be approximated within $1/n^{o(1)}$ under the Gap Exponential Time Hypothesis. Hence, it is unlikely to achieve anything better than a polynomial factor approximation. Nevertheless, we show that a greedy algorithm can achieve a factor of $\frac{1-1/e}{k}-\epsilon$ for any $\epsilon>0$, showing that not all is lost in settings where $k$ is bounded.