Fastest Mixing Markov Chain on Symmetric K-Partite Network

TL;DR

This paper analytically solves the fastest mixing Markov chain problem for specific symmetric K-partite networks using stratification and semidefinite programming, revealing key insights into network mixing rates and optimal transition probabilities.

Contribution

It provides an analytical solution for the fastest mixing Markov chain on four types of K-partite networks, utilizing convexity and polynomial comparison methods.

Findings

Symmetric and semi-symmetric K-PPDR networks share the same SLEM.

Symmetric K-PPDR networks have faster initial mixing rates.

Optimal transition probabilities outperform Metropolis-Hasting in mixing time.

Abstract

Solving fastest mixing Markov chain problem (i.e. finding transition probabilities on the edges to minimize the second largest eigenvalue modulus of the transition probability matrix) over networks with different topologies is one of the primary areas of research in the context of computer science and one of the well known networks in this issue is K-partite network. Here in this work we present analytical solution for the problem of fastest mixing Markov chain by means of stratification and semidefinite programming, for four particular types of K-partite networks, namely Symmetric K-PPDR, Semi Symmetric K-PPDR, Cycle K-PPDR and Semi Cycle K-PPDR networks. Our method in this paper is based on convexity of fastest mixing Markov chain problem, and inductive comparing of the characteristic polynomials initiated by slackness conditions in order to find the optimal transition probabilities. The presented results shows that a Symmetric K-PPDR network and its equivalent Semi Symmetric K-PPDR network have the same SLEM despite the fact that Semi symmetric K-PPDR network has less edges than its equivalent symmetric K-PPDR network and at the same time symmetric K-PPDR network has better mixing rate per step than its equivalent semi symmetric K-PPDR network at first few iterations. The same results are true for Cycle K-PPDR and Semi Cycle K-PPDR networks. Also the obtained optimal transition probabilities have been compared with the transition probabilities obtained from Metropolis-Hasting method by comparing mixing time improvements numerically.