Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Graph Disjointness with Applications to Reversible Markov Chains

Yang Xiang, Kevin McGoff, Andrew B. Nobel

Abstract

The correspondence between weighted undirected graphs and reversible Markov chains via vertex random walks is simple and well known. Leveraging this correspondence and ideas from the theory of dynamical systems, we study the structural discordance of graphs and Markov chains by means of graph joinings. Informally, a joining of graphs $G$ and $H$ is a graph on the product of their vertex sets giving rise to a coupling of their random walks. Graphs $G$ and $H$ are strongly disjoint if their only joining is the tensor product, and they are weakly disjoint if the degree function of every joining is equal to the degree function of the tensor product. We establish close connections between graph joinings, disjointness, and graph factors. Our first principal result characterizes weak disjointness of graphs in terms of the spectral overlap of their Markov transition matrices. The second establishes that two graphs without self loops are strongly disjoint if and only if they are weakly disjoint and exactly one of the graphs is a tree. The third shows that the strong or weak disjointness of graphs is essentially determined by their vertex and edge sets, without regard to edge weights. Translating these results into the language of Markov chains yields new insights into the rigidity and structure of reversible couplings of reversible Markov chains.

Graph Disjointness with Applications to Reversible Markov Chains

Abstract

The correspondence between weighted undirected graphs and reversible Markov chains via vertex random walks is simple and well known. Leveraging this correspondence and ideas from the theory of dynamical systems, we study the structural discordance of graphs and Markov chains by means of graph joinings. Informally, a joining of graphs and is a graph on the product of their vertex sets giving rise to a coupling of their random walks. Graphs and are strongly disjoint if their only joining is the tensor product, and they are weakly disjoint if the degree function of every joining is equal to the degree function of the tensor product. We establish close connections between graph joinings, disjointness, and graph factors. Our first principal result characterizes weak disjointness of graphs in terms of the spectral overlap of their Markov transition matrices. The second establishes that two graphs without self loops are strongly disjoint if and only if they are weakly disjoint and exactly one of the graphs is a tree. The third shows that the strong or weak disjointness of graphs is essentially determined by their vertex and edge sets, without regard to edge weights. Translating these results into the language of Markov chains yields new insights into the rigidity and structure of reversible couplings of reversible Markov chains.
Paper Structure (64 sections, 64 theorems, 142 equations, 2 figures, 1 table)

This paper contains 64 sections, 64 theorems, 142 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theoretical Results
  3. Scope and Contributions
  4. Organization of the Paper
  5. Related Work
  6. Definition and Properties of Graph Joinings
  7. Basic Definitions
  8. Examples of Graph Joinings
  9. Isomorphic Graphs
  10. Vertex and Edge Preservation
  11. Optimal Graph Joinings
  12. Disjointness
  13. Definitions of graph disjointness
  14. Examples of Disjointness
  15. Graph Factors
  16. ...and 49 more sections

Key Result

Proposition 2.4

Let $G=(U,\alpha)$ and $H=(V,\beta)$ be graphs with degree functions $p$ and $q$, respectively. If $G$ and $H$ are connected, then any weight function $\gamma$ satisfying condition (b) of Definition Def:weight_joining also satisfies condition (a).

Figures (2)

  • Figure 1: (A) and (B) Two distinct graph joinings of the same pair of uniformly weighted graphs (with self-loops); (C) The only graph joining between this pair of graphs is the product joining; (D) An example of a weighted joining between two graphs with self-loops.
  • Figure 2: A graph joining of $C_3$ and $C_4$ that is not the tensor product $C_3\otimes C_4$.

Theorems & Definitions (122)

  • Definition 2.1: Weight function and degree function
  • Definition 2.2: Weighted undirected graphs
  • Definition 2.3: Weight Joining
  • Definition 2.4: Graph Joining
  • Proposition 2.4
  • Lemma 2.4
  • Definition 3.1: Graph disjointness
  • Proposition 3.1
  • Remark 3.2
  • Lemma 3.2
  • ...and 112 more