[Paper Review] A combinatorial proof of Aldous-Broder theorem for general Markov chains

Luis Fredes, Jean‐François Marckert|arXiv (Cornell University)|Feb 16, 2021

Markov Chains and Monte Carlo Methods7 references4 citations

TL;DR

This paper presents two new proofs of the generalized Aldous–Broder theorem for irreducible, non-reversible Markov chains on finite graphs: a probabilistic adaptation of the classical argument and a novel combinatorial proof using a new object called 'golf sequences'. The key contribution is establishing that the distribution of the first-entrance tree sampled via a Markov chain is proportional to the product of the reversed transition probabilities, extending the original theorem beyond the reversible case.

ABSTRACT

Aldous-Broder algorithm is a famous algorithm used to sample a uniform spanning tree of any finite connected graph $G$, but it is more general: given an irreducible and reversible Markov chain $M$ on $G$ started at $r$, the tree rooted at $r$ formed by the first entrance steps in each node (different from the root) has a probability proportional to $\\prod_{e=(e^-,e^+)\\in {\\sf Edges}(t,r)} M_{e^{-},e^+}$, where the edges are directed toward $r$. In this paper we give proofs of Aldous-Broder theorem in the general case, where the kernel $M$ is irreducible but not assumed to be reversible (this generalized version appeared recently in Hu, Lyons and Tang )

Motivation & Objective

• To extend the Aldous–Broder theorem beyond reversible Markov chains to general irreducible chains.
• To provide a combinatorial proof of the generalized theorem, filling a gap in the literature where such a proof was previously missing.
• To introduce and formalize the concept of 'golf sequences' as a new combinatorial tool for analyzing first-entrance trees in Markov chains.
• To clarify the role of time reversal in the Aldous–Broder process, showing that the reversed kernel naturally emerges in the distributional characterization.
• To unify the understanding of spanning tree sampling via Markov chains by demonstrating that Wilson’s algorithm with the reversed kernel produces the same distribution as Aldous–Broder in the non-reversible case.

Proposed method

• Adapting the classical probabilistic argument of Aldous–Broder to the non-reversible case by incorporating the time-reversed kernel $\overleftarrow{M}$.
• Introducing 'golf sequences'—a novel combinatorial structure that encodes the sequence of first-entrance steps in a covering path, enabling a direct counting argument.
• Using the balance equation for a Markov chain on the space of rooted spanning trees, where transition weights are given by $\overleftarrow{M}_{r,r'}$ for root transitions.
• Establishing that the stationary distribution on the tree space is proportional to $\prod_{e \in E(t,r)} \overleftarrow{M}_e$, which implies the main result.
• Leveraging the duality between first-entrance trees and last-exit trees under time reversal, showing that $\mathbb{P}[\text{LastExitTree}(Y_{\leq 0}) = (t,r)] = \text{Const} \cdot \prod_{e \in E(t,r)} \overleftarrow{M}_e$.
• Proving that the distribution of the first-entrance tree under a Markov chain with kernel $M$ is equal to the distribution of the last-exit tree under the reversed chain $\overleftarrow{M}$, under stationarity.

Experimental results

Research questions

• RQ1 What is the correct generalization of the Aldous–Broder theorem when the Markov chain is not reversible?
• RQ2 Can a purely combinatorial proof be constructed for the generalized Aldous–Broder theorem, independent of probabilistic coupling arguments?
• RQ3 How do golf sequences relate to the structure of first-entrance trees in covering paths of Markov chains?
• RQ4 Why does the reversed kernel $\overleftarrow{M}$ naturally appear in the distribution of first-entrance trees for non-reversible chains?
• RQ5 Is there a duality between first-entrance and last-exit trees that explains the appearance of $\overleftarrow{M}$ in the distribution?

Key findings

• The paper establishes that for any irreducible Markov chain with kernel $M$ and invariant distribution $\rho$, the probability of obtaining a rooted spanning tree $(t,r)$ via the first-entrance process is proportional to $\prod_{e \in E(t,r)} \overleftarrow{M}_e / \rho(r)$.
• The combinatorial proof using 'golf sequences' provides a direct, non-probabilistic derivation of the generalized Aldous–Broder theorem, offering new insight into the structure of first-entrance trees.
• The stationary distribution on the space of rooted spanning trees induced by the first-entrance process is proportional to $\prod_{e \in E(t,r)} \overleftarrow{M}_e$, which confirms the role of the reversed kernel.
• The result implies that Wilson’s algorithm, when run with the reversed kernel $\overleftarrow{M}$, produces the same distribution of uniform spanning trees as Aldous–Broder, even in the non-reversible case.
• The paper confirms that the time-reversed chain $\overleftarrow{M}$ is not just a technical tool but a fundamental component of the theorem’s structure, as shown by the duality between first-entrance and last-exit trees.
• The balance equation for the tree Markov chain is satisfied when the stationary measure is set to $P(t,r) = \prod_{e \in E(t,r)} \overleftarrow{M}_e$, proving the correctness of the distributional characterization.