ALARM: A logistic auto-regressive model for binary processes on networks

We introduce the ALARM model, a logistic autoregressive model for discrete-time binary processes on networks, and describe a technique for learning the graph structure underlying the model from observations. Using only a small number of parameters, the proposed ALARM can describe a wide range of dynamic behavior on graphs, such as the contact process, voter process, and even some epidemic processes. Under ALARM, at each time step, the probability of a node having value 1 is determined by the values taken by its neighbors in the past; specifically, its probability is given by the logistic function evaluated at a linear combination of its neighbors' past values (within a fixed time window) plus a bias term. We examine the behavior of this model for 1D and 2D lattice graphs, and observe a phase transition in the steady state for 2D lattices. We then study the problem of learning a graph from ALARM observations. We show how a regularizer promoting group sparsity can be used to efficiently learn the parameters of the model from a realization, and demonstrate the resulting ability to reconstruct the underlying network from the data.