Maximum entropy temporal networks

Temporal networks consist of time-stamped directed interactions that may appear continuously in time, yet few studies have directly tackled the continuous-time modeling of networks. Here, we introduce a maximum-entropy approach to temporal networks and with basic assumptions on constraints, the corresponding network ensembles admit a modular and interpretable representation: a set of global time processes and a static maximum-entropy edge, e.g., node pair or probability. This time-edge labels factorization yields closed-form log-likelihoods, degree, clustering, and motif expectations, and yields an entire class of effective generative models. We provide the maximum-entropy derivation for the nonhomogeneous Poisson process (NHPP) intensities governing the probability of directed edges in temporal networks via the functional optimization over path entropy, connecting NHPP modeling to maximum-entropy network ensembles. NHPPs consistently improve log-likelihood over generic Poisson processes, while the maximum-entropy edge labels recover strength constraints and reproduce expected unique-degree curves. We discuss the limitations of this framework and how it can be integrated with multivariate Hawkes calibration procedures, renewal theory, and neural kernel estimation in graph neural networks.