Speaker: Francesco Iafrate (University of Hamburg)
Title: Statistical Guarantees for Learning Network Stochastic Differential Equations
Abstract: We propose a novel framework for Network Stochastic Differential Equations (N-SDE), where each node in a network is governed by an SDE influenced by interactions with its neighbors. The evolution of each node is driven by the interplay of three key components: the node's intrinsic dynamics (momentum effect), feedback from neighboring nodes (network effect), and a stochastic volatility term modeled by Brownian motion. Our objectives are parameter estimation of the N-SDE system, for which we provide non-asymptotic error bounds, as well as graph recovery.
The motivation behind this model lies in its ability to analyze very high-dimensional time series by leveraging the inherent sparsity of the underlying network graph. We consider two distinct scenarios:
- Known network structure: the graph is fully specified, and we establish conditions under which the parameters can be identified, considering the linear growth of the parameter space with the number of edges.
- Unknown network structure: the graph must be inferred from the data. For this, we develop an iterative procedure using adaptive Lasso, tailored to a specific subclass of N-SDE models.
We assume the network graph is oriented, paving the way for novel applications of SDEs in causal inference, enabling the study of cause-effect relationships in dynamic systems. Through extensive simulation studies, we demonstrate the performance of our estimators across various graph topologies in high-dimensional settings. We also showcase the framework's applicability to real-world datasets, highlighting its potential for advancing the analysis of complex networked systems.
We consider both the frameworks of high frequency observations for an ergodic diffusion [1] as well as the case of continuous observations in a small-diffusion non-ergodic setting [2].
References
[1] Iafrate, F., Iacus, S. M. (2024) Ergodic network stochastic differential equations,
arXiv preprint arXiv:2412.17779.
[2] Iafrate, F., Iacus, S. M. (2025) Small diffusion network stochastic differential equations, working paper.
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