# Non-Markovian Monte Carlo on Directed Graphs

## TL;DR

This paper introduces Non-Markovian Monte Carlo (NMMC), a novel framework for sampling and inference on directed graphs, overcoming limitations of traditional MCMC methods that require undirected or reciprocal edges.

## Contribution

The paper develops the first NMMC framework for directed graphs, enabling arbitrary distribution sampling and eigenvector centrality estimation through history-dependent random walks.

## Key findings

- Successfully achieves any target distribution on directed graphs.
- Estimates eigenvector centrality in a distributed manner.
- Numerical experiments confirm theoretical results.

## Abstract

Markov Chain Monte Carlo (MCMC) has been the de facto technique for sampling and inference of large graphs such as online social networks. At the heart of MCMC lies the ability to construct an ergodic Markov chain that attains any given stationary distribution $\boldsymbol{\pi}$, often in the form of random walks or crawling agents on the graph. Most of the works around MCMC, however, presume that the graph is undirected or has reciprocal edges, and become inapplicable when the graph is directed and non-reciprocal. Here we develop a similar framework for directed graphs, which we call Non-Markovian Monte Carlo (NMMC), by establishing a mapping to convert $\boldsymbol{\pi}$ into the quasi-stationary distribution of a carefully constructed transient Markov chain on an extended state space. As applications, we demonstrate how to achieve any given distribution $\boldsymbol{\pi}$ on a directed graph and estimate the eigenvector centrality using a set of non-Markovian, history-dependent random walks on the same graph in a distributed manner. We also provide numerical results on various real-world directed graphs to confirm our theoretical findings, and present several practical enhancements to make our NMMC method ready for practical use in most directed graphs. To the best of our knowledge, the proposed NMMC framework for directed graphs is the first of its kind, unlocking all the limitations set by the standard MCMC methods for undirected graphs.