  A wide array of complex biological, social, and physical systems have
recently been shown to be quantitatively described by Ising models, which lie
at the intersection of statistical physics and machine learning. Here, we study
the fundamental question of how to optimize the state of a networked Ising
system given a budget of external influence. In the continuous setting where
one can tune the influence applied to each node, we propose a series of
approximate gradient ascent algorithms based on the Plefka expansion, which
generalizes the na\"{i}ve mean field and TAP approximations. In the discrete
setting where one chooses a small set of influential nodes, the problem is
equivalent to the famous influence maximization problem in social networks with
an additional stochastic noise term. In this case, we provide sufficient
conditions for when the objective is submodular, allowing a greedy algorithm to
achieve an approximation ratio of $1-1/e$. Additionally, we compare the
Ising-based algorithms with traditional influence maximization algorithms,
demonstrating the practical importance of accurately modeling stochastic
fluctuations in the system.
