# Codes on graphs: Models for elementary algebraic topology and statistical physics

arXiv: https://arxiv.org/abs/1707.06621 · HTML: https://mlsanity.com/paper/1707.06621/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jul 2017.

## Built on by (most cited first, 5 of 5)

- [Mappings for Marginal Probabilities with Applications to Models in Statistical Physics](https://mlsanity.com/paper/2208.05333/) (2022)
- [Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning](https://mlsanity.com/paper/2307.15778/) (2023)
- [The Exact Determinant of a Specific Class of Sparse Positive Definite Matrices](https://mlsanity.com/paper/2311.06632/) (2023)
- [Topology-Aware Exploration of Energy-Based Models Equilibrium: Toric QC-LDPC Codes and Hyperbolic MET QC-LDPC Codes](https://mlsanity.com/paper/2401.14749/) (2024)
- [Accelerated Random-Sweep Gibbs Sampling for Gaussian Graphical Models via Dual Normal Factor Graphs](https://mlsanity.com/paper/2607.28706/) (2026)

## Abstract (arXiv)

This paper is mainly a semi-tutorial introduction to elementary algebraic topology and its applications to Ising-type models of statistical physics, using graphical models of linear and group codes. It contains new material on systematic (n,k) group codes and their information sets; normal realizations of homology and cohomology spaces; dual and hybrid models; and connections with system-theoretic concepts such as observability, controllability, and input/output realizations.