Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Coloured Graphical Models and their Symmetries

Isobel Davies, Orlando Marigliano

TL;DR

Coloured graphical models define Gaussian subspaces $\mathcal{L}\subseteq \mathbb{S}^n$ determined by a coloured graph, and their reciprocal varieties $\mathcal{L}^{-1}$ carry algebraic-statistics data via the ideal $I(\mathcal{L}^{-1})$ (e.g., ML degree $\operatorname{mld}(\mathcal{L})$). The authors relate graph symmetries to the linear part of $I(\mathcal{L}^{-1})$ by showing that binomial forms from $BXB^{-1}-X$ vanish on $\mathcal{L}^{-1}$ when $B$ is a symmetry, and they develop RCOP models to capture additional linear constraints. Specializing to uniform coloured graphs, they use pencil theory and Segre symbols to connect eigenvalue multiplicities to the count of linear forms induced by symmetries, proving that the linear part is symmetry-induced precisely when the number of Aut$(G)$-orbits on $V*V$ equals the number of distinct eigenvalues $r$. They identify four families—$C_n$, $K_n$, $K_{m,m}$, and $H_m$—for which the linear part of $I(\mathcal{L}^{-1})$ is completely determined by symmetries and provide explicit generators; the work also discusses counterexamples and conjectures aiming to extend these symmetry-based descriptions to all uniform circulant graphs. Overall, the results offer symmetry-driven simplifications for understanding the algebraic structure of coloured graphical models and motivate further investigation of circulant uniform colourings and quadratic generators in $I(\mathcal{L}^{-1})$.

Abstract

Coloured graphical models are Gaussian statistical models determined by an undirected coloured graph. These models can be described by linear spaces of symmetric matrices. We outline a relationship between the symmetries of the graph and the linear forms that vanish on the reciprocal variety of the model. In particular, we give four families for which such linear forms are completely described by symmetries.

Coloured Graphical Models and their Symmetries

TL;DR

Coloured graphical models define Gaussian subspaces determined by a coloured graph, and their reciprocal varieties carry algebraic-statistics data via the ideal (e.g., ML degree ). The authors relate graph symmetries to the linear part of by showing that binomial forms from vanish on when is a symmetry, and they develop RCOP models to capture additional linear constraints. Specializing to uniform coloured graphs, they use pencil theory and Segre symbols to connect eigenvalue multiplicities to the count of linear forms induced by symmetries, proving that the linear part is symmetry-induced precisely when the number of Aut-orbits on equals the number of distinct eigenvalues . They identify four families—, , , and —for which the linear part of is completely determined by symmetries and provide explicit generators; the work also discusses counterexamples and conjectures aiming to extend these symmetry-based descriptions to all uniform circulant graphs. Overall, the results offer symmetry-driven simplifications for understanding the algebraic structure of coloured graphical models and motivate further investigation of circulant uniform colourings and quadratic generators in .

Abstract

Coloured graphical models are Gaussian statistical models determined by an undirected coloured graph. These models can be described by linear spaces of symmetric matrices. We outline a relationship between the symmetries of the graph and the linear forms that vanish on the reciprocal variety of the model. In particular, we give four families for which such linear forms are completely described by symmetries.

Paper Structure

This paper contains 6 sections, 1 theorem, 19 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Notation.
  3. Linear forms from symmetries
  4. Uniform coloured graphs
  5. Open Problems
  6. Acknowledgements.

Key Result

Theorem 3.4

Let $G$ be one of: the cycle $C_n$, the complete graph $K_n$, the complete bipartite graph $K_{m,m}$, or the hyperoctahedral graph $H_m$. Then the linear part $L$ of $I(\mathcal{L}^{-1})$ is induced by symmetries. More precisely, 1. If $G = C_n$, then $r=s=\lfloor n/2\rfloor + 1$ and $L$ is generate where all indices are taken modulo $n$ and $x_{ji}$ for $j>i$ is taken to mean $x_{ij}$. 2. If $G =

Figures (2)

  • Figure 1: Uniform coloured graphs on $6$ vertices where the linear part of $I(\mathcal{L}^{-1})$ is induced by symmetries. From left to right: the cycle $C_6$, the complete graph $K_6$, the complete bipartite graph $K_{3,3}$, and the hyperoctahedral graph $H_3$.
  • Figure 2: The uniform coloured complete bipartite graph $K_{2,4}$ and the star $K_{1,5}$. For these graphs, the linear part of $I(\mathcal{L}^{-1})$ is not induced by symmetries.

Theorems & Definitions (8)

  • proof
  • proof
  • proof
  • proof
  • Theorem 3.4
  • proof
  • proof
  • proof