[Paper Review] Realization of finite-state mixing Markov chain as a random walk subject to a synchronizing road coloring

Kouji Yano, Kenji Yasutomi|arXiv (Cornell University)|Jun 3, 2010

Cellular Automata and Applications18 references3 citations

TL;DR

This paper proves that any finite-state mixing Markov chain can be realized as a random walk on a directed graph with a synchronizing road coloring, ensuring the existence of appropriate random mappings for Propp–Wilson's coupling from the past. The key contribution is a necessary and sufficient condition for approximate entropy preservation in such realizations, derived using the road coloring theorem and entropy analysis.

ABSTRACT

A mixing Markov chain is proved to be realized as a random walk in a directed graph subject to a synchronizing road coloring. The result ensures existence of appropriate random mappings in Propp--Wilson's coupling from the past. The proof is based on the road coloring theorem. A necessary and sufficient condition for approximate preservation of entropies is also given.

Motivation & Objective

• To establish a realization of finite-state mixing Markov chains as random walks on directed graphs with synchronizing road coloring.
• To ensure the existence of suitable random mappings for Propp–Wilson's coupling from the past algorithm.
• To derive a necessary and sufficient condition for approximate preservation of entropy in the random walk representation.

Proposed method

• Construct a mapping law μ on the set of endomorphisms of the state space V such that the transition probabilities of the Markov chain are preserved.
• Use the road coloring theorem to ensure the existence of a synchronizing set of mappings (i.e., a set of road colors) that can synchronize the system to a single state.
• Define a μ-random walk (X, N) where X_k = N_k X_{k-1}, with N_i i.i.d. under μ, to model the Markov chain dynamics.
• Characterize the support of μ as synchronizing if there exists a finite sequence of mappings whose composition collapses all states to a single state.
• Analyze entropy via the joint law of the random walk and the driving noise, comparing the entropy of the noise h(N) to the entropy of the chain h(Y).
• Derive a condition under which h(N) approximates h(Y) arbitrarily closely, using a perturbation of the mapping law μ with a synchronizing subset Σ₁.

Experimental results

Research questions

• RQ1 Can every finite-state mixing Markov chain be represented as a random walk on a directed graph with a synchronizing road coloring?
• RQ2 What conditions ensure that the entropy of the driving noise in the random walk approximates the entropy of the original Markov chain?
• RQ3 Is there a systematic way to construct a mapping law μ with synchronizing support for any given mixing Markov chain?
• RQ4 Under what conditions does the entropy of the noise process h(N) converge to the entropy of the chain h(Y) as the perturbation parameter ε → 0+?
• RQ5 When is the mapping law μ such that the induced random walk preserves the entropy of the original chain up to arbitrary precision?

Key findings

• Every finite-state mixing Markov chain admits a mapping law μ with synchronizing support, enabling realization as a random walk under a synchronizing road coloring.
• A necessary and sufficient condition for approximate entropy preservation is that the Markov chain is p-uniform, meaning the transition probabilities satisfy q_{x,y} = q_{x_0, au_x(y)} for all x,y and some permutation τ_x.
• For the two-state example with p ≠ 1/2, h(N^{(ε)}) > h(Y) for all ε > 0, but h(N^{(ε)}) → h(Y) as ε → 0+, showing asymptotic entropy preservation.
• When p = 1/2, h(N^{(1/2)}) = h(Y), indicating exact entropy preservation under symmetric transition probabilities.
• The perturbation construction μ^{(n)}(σ) = ∑_{i:σ^{(i)}=σ} (ν(x_i) - 1/(nd)) + 1/(n|Σ₁|)1_{σ∈Σ₁} ensures both synchronizing support and convergence of h(N^{(n)}) to h(Y) as n → ∞.
• The entropy of the noise process h(N) equals the entropy of the chain h(Y) if and only if the chain is p-uniform, as shown via equality in the entropy inequality chain.