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Entropy and the growth rate of universal covering trees

Idan Eisner, Shlomo Hoory

Abstract

This work studies the relation between two graph parameters, $ρ$ and $Λ$. For an undirected graph $G$, $ρ(G)$ is the growth rate of its universal covering tree, while $Λ(G)$ is a weighted geometric average of the vertex degree minus one, corresponding to the rate of entropy growth for the non-backtracking random walk (NBRW). It is well known that $ρ(G) \geq Λ(G)$ for all graphs, and that graphs with $ρ=Λ$ exhibit some special properties. In this work we derive an easy to check, necessary and sufficient condition for the equality to hold. Furthermore, we show that the variance of the number of random bits used by a length $\ell$ NBRW is $O(1)$ if $ρ= Λ$ and $Ω(\ell)$ if $ρ> Λ$. As a consequence we exhibit infinitely many non-trivial examples of graphs with $ρ= Λ$.

Entropy and the growth rate of universal covering trees

Abstract

This work studies the relation between two graph parameters, and . For an undirected graph , is the growth rate of its universal covering tree, while is a weighted geometric average of the vertex degree minus one, corresponding to the rate of entropy growth for the non-backtracking random walk (NBRW). It is well known that for all graphs, and that graphs with exhibit some special properties. In this work we derive an easy to check, necessary and sufficient condition for the equality to hold. Furthermore, we show that the variance of the number of random bits used by a length NBRW is if and if . As a consequence we exhibit infinitely many non-trivial examples of graphs with .

Paper Structure

This paper contains 18 sections, 19 theorems, 55 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Main Players
  3. Some Motivation
  4. Our Results
  5. Preliminaries
  6. The Non-Backtracking Random Walk and Adjacency Matrix
  7. The Universal Covering Tree and its Growth Rate $\rho$
  8. The Average Growth Rate $\Lambda$ and its Connection to $\rho$
  9. The Main Theorem
  10. The suspended path condition implies $\rho = \Lambda$
  11. $\rho = \Lambda$ implies the cycle condition
  12. Cycle condition implies the suspended path condition
  13. Graphs satisfying $\rho = \Lambda$
  14. Variance of the Number of Random Bits
  15. Variance Dichotomy for General Graphs
  16. ...and 3 more sections

Key Result

Lemma 6

Given an NB-irreducible graph $G$, the following definitions for $\rho(G)$ are equivalent: where $B$ denotes the non-backtracking adjacency matrix, and $\rho(B)$ is the largest (Perron) eigenvalue of $B$. Equality holds regardless of the choice of $v \in V(\tilde{G})$ and $e \in \vec{E}$, where applicable.

Figures (4)

  • Figure 1: Three graphs with $\rho = \Lambda$ that are not regular, bipartite bi-regular, or obtained from such graphs by replacing each edge by a length $k$ path. It can be verified that the suspended path condition holds with $\Lambda$ equals $\sqrt{2}$, $2$ and $\sqrt{2}$ for the graphs (a), (b) and (c), respectively.
  • Figure 2: A cycle with a vertex of degree more than two.
  • Figure 3: $K_4$ minus an edge.
  • Figure 4: The number of bits per step, $R_\ell/\ell$, for the length $\ell=1000$ NBRW on $K_4$ minus an edge. The black dots are the exact PDF, the yellow line is a best-fit normal distribution, the green line is at $\log_2\Lambda$ and the orange line is at $\log_2\rho$ which is the limit on $\ell$ of $\frac{1}{\ell} \log_2 \mathbb{E}\!\left[2^{R_\ell}\right]$.

Theorems & Definitions (47)

  • Definition 1
  • Remark 2
  • Claim 3
  • proof
  • Claim 4: glover2021non proposition 3.3
  • proof
  • Definition 5
  • Lemma 6: angel2015nonhoory2024girth
  • Definition 7
  • Lemma 8
  • ...and 37 more