Graph Topology Identification Based on Covariance Matching
Yongsheng Han, Raj Thilak Rajan, Geert Leus
TL;DR
This paper tackles graph topology identification (GTI) from nodal observations when the underlying graph is hidden. It proposes CovMatch, a covariance-matching framework that aligns the empirical covariance $\mathbf{C}_x$ with the theoretical covariance $\boldsymbol{\Sigma}_x$—with $\boldsymbol{\Sigma}_x=\mathbf{H}\boldsymbol{\Sigma}_e\mathbf{H}^T$ and $\mathbf{H}=(\mathbf{I}-\mathbf{S})^{-1}$—to infer graph structure. By reparameterizing for undirected graphs (via eigen decomposition) and directed graphs (via SVD), CovMatch yields tractable reformulations: a conic mixed-integer program for undirected graphs and an orthogonal-rotation optimization for directed graphs, both augmented with structural priors like hollowness and sparsity. Empirical results on synthetic data (undirected, DAGs, cyclic directed graphs) and real T-cell data show CovMatch attains near-zero, or highly accurate, topology recovery asymptotically and robustly with moderate samples, often outperforming or matching specialized baselines without enforcing hard acyclicity. The work offers a scalable, assumption-light alternative to log-determinant or Bayesian GTI methods, broadening the applicability of GTI in complex networks.
Abstract
Graph topology identification (GTI) is a central challenge in networked systems, where the underlying structure is often hidden, yet nodal data are available. Conventional solutions to address these challenges rely on probabilistic models or complex optimization formulations, commonly suffering from non-convexity or requiring restrictive assumptions on acyclicity or positivity. In this paper, we propose a novel covariance matching (CovMatch) framework that directly aligns the empirical covariance of the observed data with the theoretical covariance implied by an underlying graph. We show that as long as the data-generating process permits an explicit covariance expression, CovMatch offers a unified route to topology inference. We showcase our methodology on linear structural equation models (SEMs), showing that CovMatch naturally handles both undirected and general sparse directed graphs - whether acyclic or positively weighted - without explicit knowledge of these structural constraints. Through appropriate reparameterizations, CovMatch simplifies the graph learning problem to either a conic mixed integer program for undirected graphs or an orthogonal matrix optimization for directed graphs. Numerical results confirm that, even for relatively large graphs, our approach efficiently recovers the true topology and outperforms standard baselines in accuracy. These findings highlight CovMatch as a powerful alternative to log-determinant or Bayesian methods for GTI, paving the way for broader research on learning complex network topologies with minimal assumptions.