# Colored Markov Random Fields for Probabilistic Topological Modeling

arXiv: https://arxiv.org/abs/2512.03727 · HTML: https://mlsanity.com/paper/2512.03727/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 20 references that are in the map.
- First posted on arXiv 3 Dec 2025.

## Builds on (most cited first, 8 of 8)

- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Two's company, three (or more) is a simplex: Algebraic-topological tools for understanding higher-order structure in neural data](https://mlsanity.com/paper/1601.01704/) (2016)
- [Hodge-Compositional Edge Gaussian Processes](https://mlsanity.com/paper/2310.19450/) (2023)
- [Gaussian Processes on Cellular Complexes](https://mlsanity.com/paper/2311.01198/) (2023)
- [HLSAD: Hodge Laplacian-based Simplicial Anomaly Detection](https://mlsanity.com/paper/2505.24534/) (2025)
- [Simplicial Gaussian Models: Representation and Inference](https://mlsanity.com/paper/2510.12983/) (2025)
- [Diffusion Adaptation Strategies for Distributed Estimation over Gaussian Markov Random Fields](https://mlsanity.com/paper/1407.3698/) (2014)

## Abstract (arXiv)

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.