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Irreducible Markov Chains on spaces of graphs with fixed degree-color sequences

Félix Almendra-Hernández, Jesús A. De Loera, Sonja Petrović

TL;DR

This work develops a colored generalization of the switch Markov chain for sampling graphs with a fixed degree sequence and a color-derived statistic called the color-degree sequence, within the β-SBM framework. It establishes a quadratic Markov basis $\mathcal{M}_{n,z}$ consisting of 4-cycles with two opposite vertices sharing a color as a connectivity tool for the space of multigraphs, and links this basis to a Gröbner-basis description of the associated toric ideal through a specially constructed monomial order. The authors show that while such quadratic moves suffice for multigraph connectivity, simple-graph connectivity can fail dramatically as the number of colors grows, with lower bounds on move size and no universal constants bounding 1-norms. They further demonstrate that the quadratic Markov basis forms a Gröbner basis, yielding a unimodular regular triangulation of the design polytope and enabling Ehrhart/Hilbert-polynomial analyses, while highlighting open questions about the combinatorics for general colorings and higher color counts.

Abstract

We study a colored generalization of the famous simple-switch Markov chain for sampling the set of graphs with a fixed degree sequence. Here we consider the space of graphs with colored vertices, in which we fix the degree sequence and another statistic arising from the vertex coloring, and prove that the set can be connected with simple color-preserving switches or moves. These moves form a basis for defining an irreducible Markov chain necessary for testing statistical model fit to block-partitioned network data. Our methods further generalize well-known algebraic results from the 1990s: namely, that the corresponding moves can be used to construct a regular triangulation for a generalization of the second hypersimplex. On the other hand, in contrast to the monochromatic case, we show that for simple graphs, the 1-norm of the moves necessary to connect the space increases with the number of colors.

Irreducible Markov Chains on spaces of graphs with fixed degree-color sequences

TL;DR

This work develops a colored generalization of the switch Markov chain for sampling graphs with a fixed degree sequence and a color-derived statistic called the color-degree sequence, within the β-SBM framework. It establishes a quadratic Markov basis consisting of 4-cycles with two opposite vertices sharing a color as a connectivity tool for the space of multigraphs, and links this basis to a Gröbner-basis description of the associated toric ideal through a specially constructed monomial order. The authors show that while such quadratic moves suffice for multigraph connectivity, simple-graph connectivity can fail dramatically as the number of colors grows, with lower bounds on move size and no universal constants bounding 1-norms. They further demonstrate that the quadratic Markov basis forms a Gröbner basis, yielding a unimodular regular triangulation of the design polytope and enabling Ehrhart/Hilbert-polynomial analyses, while highlighting open questions about the combinatorics for general colorings and higher color counts.

Abstract

We study a colored generalization of the famous simple-switch Markov chain for sampling the set of graphs with a fixed degree sequence. Here we consider the space of graphs with colored vertices, in which we fix the degree sequence and another statistic arising from the vertex coloring, and prove that the set can be connected with simple color-preserving switches or moves. These moves form a basis for defining an irreducible Markov chain necessary for testing statistical model fit to block-partitioned network data. Our methods further generalize well-known algebraic results from the 1990s: namely, that the corresponding moves can be used to construct a regular triangulation for a generalization of the second hypersimplex. On the other hand, in contrast to the monochromatic case, we show that for simple graphs, the 1-norm of the moves necessary to connect the space increases with the number of colors.
Paper Structure (4 sections, 18 theorems, 17 equations, 8 figures)

This paper contains 4 sections, 18 theorems, 17 equations, 8 figures.

Table of Contents

  1. Introduction
  2. A quadratic Markov basis for the $\beta$-SBM matrix
  3. Restriction to simple graphs
  4. A quadratic Gröbner basis

Key Result

Theorem 1.4

The set of quadratic moves $\mathcal{M}_{n,z}$ is a Markov basis for $\mathrel{\ooalign{$D$\cr$C$}}_{\!n,z}$. These are the moves in $\ker_\mathbb{Z}\!\mathrel{\ooalign{$D$\cr$C$}}_{\!n,z}$ of minimal 1-norm.

Figures (8)

  • Figure 1: Graph $G$ with vertices 1 and 2 colored blue, 3 and 4 red, and 5 green.
  • Figure 2: Simple graphs $\gamma_1$ and $\gamma_2$ being connected with switches by leaving the simple-graph fiber. The switches used in each step are highlighted in orange.
  • Figure 3: Monomial walk $m=[1,5,2,4,5,3,2,4]$.
  • Figure 4: Reconnecting with a 4-cycle. On the left of the equality is the graph $\gamma$, while on the right are the $4$-cycle corresponding to $x_{12}x_{78}-x_{17}x_{28}$ and $\gamma'$.
  • Figure 5: Peeling off a 4-cycle. On the left of the equality is the graph $\gamma$, while on the right are the $4$-cycle corresponding to $x_{17}x_{46}-x_{14}x_{67}$ and $\gamma'$.
  • ...and 3 more figures

Theorems & Definitions (42)

  • Definition 1.1
  • Example 1.2
  • Example 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Example 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Example 2.1
  • Lemma 2.2
  • ...and 32 more