Maximizing Influence in an Ising Network: A Mean-Field Optimal Solution
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 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.
The paper proposes an Ising network based model for opinion dynamics and proposes the Ising influence maximization method based on it. Under mean field approximation, it develops several sufficient conditions for when the problem is convex and solvable and a projected gradient descent algorithm. It also establishes several results about the structure of steady-states in the ferromagnetic MF Ising model. It conducts experiments on both synthetic and real world authorship datasets.
The paper studies the influence maximization using the Ising network model. However, the motivation for using Ising network model for opinion dynamics is not well motivated. Why do we need to use the Ising model? It is less convincing by just saying that this model can capture the steady-state of the opinion dynamics. Some other model can also consider the steady state. For example, how does it compare with prior work such as [1]. In [1], it also considers modeling the steady state of the opinion dynamics. Moreover, how does the discrete-time Ising compare with the continuous-time model in [1]? Some prior work is missing and the paper should have a paragraph clearly stating the stand of this method with respect to other work.  As for experiments, it would be good to compare with other standard baselines in terms of the influence maximization. In the experiments it only compares with three heuristics, which is less convincing. Moreover, it would be good to add the scalability experiments on large networks.  As for writing, proof sketch could be moved to the appendix. Moreover, the theorems stating sufficient conditions for existence of steady state are less connected to influence maximization, hence makes section 4.1 less connected to other sections. Some terms are also not well defined. For example, what does the susceptibility mean in equation (8)? Any citation on how to obtain equation (5)?   Several important literatures are missing: [1]. Learning Opinion Dynamics in Social Networks [2]. Influence maximization in continuous time diffusion networks [3]. Scalable Influence Estimation in Continuous-Time Diffusion Networks [4]. Scalable influence maximization in social networks under the linear threshold model [5]. A data-based approach to social influence maximization