Influence Maximization in Ising Models
Zongchen Chen, Elchanan Mossel
TL;DR
The paper studies influence maximization on Ising models, formulating the problem as selecting a seed set of size at most $k$ and a partial assignment to maximize $\mathbb{E}_\mu[\sum_{v\in V} X_v]$ under the constraint. It shows a sharp computational phase transition at the tree-uniqueness threshold: efficient linear-time algorithms exist in the high-temperature regime while hardness results hold in the low-temperature regime, with the critical temperature $\beta_c(\Delta)$ governing tractability. The method localizes global influence to finite-radius neighborhoods using correlation decay and spectral independence, approximates local influences $\Phi^{(r)}$, and reduces the optimization to a max-weight independent set on a bounded auxiliary graph, achieving $O(n)\,(1/\varepsilon)^{O(1)}$ time for fixed $k$. These results link a fundamental statistical-physics phase transition to computational feasibility for influence control in correlated binary systems.
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.