  Influence maximization in social networks has typically been studied in the
context of contagion models and irreversible processes. In this paper, we
consider an alternate model that treats individual opinions as spins in an
Ising system at dynamic equilibrium. We formalize the \textit{Ising influence
maximization} problem, which has a natural physical interpretation as
maximizing the magnetization given a budget of external magnetic field. Under
the mean-field (MF) approximation, we present a gradient ascent algorithm that
uses the susceptibility to efficiently calculate local maxima of the
magnetization, and we develop a number of sufficient conditions for when the MF
magnetization is concave and our algorithm converges to a global optimum. We
apply our algorithm on random and real-world networks, demonstrating,
remarkably, that the MF optimal external fields (i.e., the external fields
which maximize the MF magnetization) shift from focusing on high-degree
individuals at high temperatures to focusing on low-degree individuals at low
temperatures. We also establish a number of novel results about the structure
of steady-states in the ferromagnetic MF Ising model on general graph
topologies, which are of independent interest.
