Influence Maximization in Ising Models

TL;DR

This paper investigates influence maximization in Ising models, revealing a phase transition where the problem is efficiently solvable at high temperatures but computationally hard at low temperatures, aligning with the model's critical point.

Contribution

It establishes a sharp computational phase transition for influence maximization in Ising models, providing a linear-time algorithm in the high-temperature regime and hardness results in the low-temperature regime.

Findings

Efficient influence maximization algorithm in high-temperature regime

Hardness of influence maximization in low-temperature regime

Critical temperature aligns with the Ising model's phase transition

Abstract

Given a complex high-dimensional distribution over $\{\pm 1\}^n$, what is the best way to increase the expected number of $+1$'s by controlling the values of only a small number of variables? Such a problem is known as influence maximization and has been widely studied in social networks, biology, and computer science. In this paper, we consider influence maximization on the Ising model which is a prototypical example of undirected graphical models and has wide applications in many real-world problems. We establish a sharp computational phase transition for influence maximization on sparse Ising models under a bounded budget: In the high-temperature regime, we give a linear-time algorithm for finding a small subset of variables and their values which achieve nearly optimal influence; In the low-temperature regime, we show that the influence maximization problem cannot be solved in polynomial time under commonly-believed complexity assumption. The critical temperature coincides with the tree uniqueness/non-uniqueness threshold for Ising models which is also a critical point for other computational problems including approximate sampling and counting.