Maximum entropy temporal networks

TL;DR

The paper develops a principled, continuous-time framework for temporal networks by casting them as marked point processes and applying a maximum-entropy (MaxEnt) principle to constrain both temporal dynamics and edge structure. The key result is a time–mark factorization of the edge intensity, $\lambda_{ij}(t)=\phi_{r(i,j)}(t)\,w_{ij}$, which yields tractable log-likelihoods and closed-form expressions for edge counts, degrees, unique edges, and motif statistics, while allowing Poisson, renewal, and Hawkes time layers and block-structured partitions. The authors introduce practical fitting procedures (IPFP) and extensions like block-pair partitions and frozen-path NHPP surrogates, along with explicit formulas for burstiness and higher-order motifs, enabling principled null-model construction and significance testing. Empirical evaluation on diverse datasets (e.g., Enron) demonstrates improvements in likelihood with time layers and shows that maximum-entropy marks capture degree and clustering patterns, offering a scalable, interpretable baseline for complex, bursty temporal networks and a basis for incorporating neural kernels and Hawkes calibration in future work.

Abstract

Temporal networks consist of timestamped 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, probability. This time-edge labels factorization yields closed-form log-likelihoods, degree, clustering and motif expectations, and yields a whole class of effective generative models. We provide maximum-entropy derivation of an inhomogeneous Poisson edge intensity for temporal networks via functional optimization over path entropy, connecting NHPP modeling to maximum-entropy network ensembles. NHPP 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.

Figures (14)

• Figure 1: Time-dependent motif ratio for reciprocated links in the Enron dataset versus the maximum-entropy temporal network ensembles obtained combining Poisson/Exp-Hawkes (GH-Exp)/PL-Hawkes (GH-PL) temporal profiles and configuration model (CM) and block-configuration model (Block-CM) edge structures.
• Figure 2: Frozen-path NHPP runs approximating the theoretical Hawkes auto-covariance.
• Figure 3: Degree-related statistics for Enron TRAIN (blockpair--PL model). observed vs. expected out-degree
• Figure 4: Burstiness vs. degree for the Enron TRAIN dataset. The plot compares empirical observations with the blockpair--PL model, showing how local transitivity scales with node degree.
• Figure 5: Log of event counts per edge (left) and Burstiness per edge (right) for the Enron TRAIN dataset.