Optimally Influencing Complex Ising Systems

Abstract

In the study of social networks, a fundamental problem is that of influence maximization (IM): How can we maximize the collective opinion of individuals in a network given constrained marketing resources? Traditionally, the IM problem has been studied in the context of contagion models, which treat opinions as irreversible viruses that propagate through the network. To study reverberant opinion dynamics, which yield complex macroscopic behavior, the IM problem has recently been proposed in the context of the Ising model of opinion dynamics, in which individual opinions are treated as spins in an Ising system. In this paper, we are among the first to explore the \textit{Ising influence maximization (IIM)} problem, which has a natural physical interpretation as the maximization of the magnetization given a budget of external magnetic field, and we are the first to consider the IIM problem in general Ising systems with negative couplings and negative external fields. For a general Ising system, we show analytically that the optimal external field (i.e., that which maximizes the magnetization) exhibits a phase shift from intuitively focusing on high-degree nodes at high temperatures to counterintuitively focusing on "loosely-connected" nodes, which are weakly energetically bound to the ground state, at low temperatures. We also present a novel and efficient algorithm for solving IIM with provable performance guarantees for ferromagnetic systems in nonnegative external fields. We apply our algorithm on large random and real-world networks, verifying the existence of phase shifts in the optimal external fields and comparing the performance of our algorithm with the state-of-the-art mean-field-based algorithm.