Colored Markov Random Fields for Probabilistic Topological Modeling
Lorenzo Marinucci, Leonardo Di Nino, Gabriele D'Acunto, Mario Edoardo Pandolfo, Paolo Di Lorenzo, Sergio Barbarossa
TL;DR
This work addresses probabilistic modeling of Gaussian edge signals defined on topological domains where standard PGMs are insufficient. It proposes Colored Markov Random Fields (CMRFs) that augment Gaussian Markov Fields with link coloring to encode both conditional dependencies via connectivity and marginal independencies via topology-induced colors on edge variables, grounded in discrete Hodge theory. Theoretical results show that CMRFs preserve the global Markov property while enabling additional marginal independence statements; a distributed estimation case study demonstrates significant gains over baselines with partial topological information. The approach enables topology-aware probabilistic inference on complexes and suggests directions toward non-Gaussian extensions and data-driven topology learning.
Abstract
Probabilistic Graphical Models (PGMs) encode conditional dependencies among random variables using a graph -nodes for variables, links for dependencies- and factorize the joint distribution into lower-dimensional components. This makes PGMs well-suited for analyzing complex systems and supporting decision-making. Recent advances in topological signal processing highlight the importance of variables defined on topological spaces in several application domains. In such cases, the underlying topology shapes statistical relationships, limiting the expressiveness of canonical PGMs. To overcome this limitation, we introduce Colored Markov Random Fields (CMRFs), which model both conditional and marginal dependencies among Gaussian edge variables on topological spaces, with a theoretical foundation in Hodge theory. CMRFs extend classical Gaussian Markov Random Fields by including link coloring: connectivity encodes conditional independence, while color encodes marginal independence. We quantify the benefits of CMRFs through a distributed estimation case study over a physical network, comparing it with baselines with different levels of topological prior.