A1068
Title: Learning continuous-time network dynamics via network and sheaf SDEs
Authors: Francesco Iafrate - University of Hamburg (Germany) [presenting]
Abstract: Learning and inference is investigated for continuous-time network dynamics and their topological generalizations. First, a class of network stochastic differential equations(N-SDEs) is proposed, in which every node of a directed graph follows an SDE driven by: (i) a momentum term capturing the nodes own past, (ii) a network effect that aggregates feedback from its neighbors, and (iii) stochastic volatility from Brownian noise. Non-asymptotic error bounds are provided for joint parameter estimation and graph recovery in a high-frequency (ergodic) observation scheme, and design an adaptive-lasso routine that learns the graph when it is unknown. The possibility of exploiting graph sparsity and modeling oriented edges makes the model attractive for causal inference. Motivated by the growing success of topological methods in data science, N-SDEs are then extended to cellular sheaves, obtaining the Sheaf-SDEs. A cellular sheaf equips every node and edge with a vector space and restriction maps; embedding SDE dynamics in this structure lets multivariate signals evolve under local self-interaction, cross-variable coupling, and higher-order influences that propagate through the sheaf, capturing relationships that ordinary graph models cannot express. Synthetic experiments validate exact structural recovery, while a case study on pollutant diffusion over spatial networks illustrates how the framework allows for powerful location-dependent models without sacrificing interpretability.
