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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 \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 called Non- Markovian Monte Carlo (NMMC) by establishing a mapping to convert \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 \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 inmost 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.
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Sequential stratified regeneration: MCMC for large state spaces with an application to subgraph count estimation
This work considers the general task of estimating the sum of a bounded function over the edges of a graph, given neighborhood query access and where access to the entire network is prohibitively expensive. To estimate this sum, prior work proposes Markov chain Monte Carlo (MCMC) methods that use random walks started at some seed vertex and whose equilibrium distribution is the uniform distribution over all edges, eliminating the need to iterate over all edges. Unfortunately, these existing estimators are not scalable to massive real-world graphs. In this paper, we introduce Ripple, an MCMC-based estimator that achieves unprecedented scalability by stratifying the Markov chain state space into ordered strata with a new technique that we denote sequential stratified regenerations. We show that the Ripple estimator is consistent, highly parallelizable, and scales well. We empirically evaluate our method by applying Ripple to the task of estimating connected, induced subgraph counts given some input graph. Therein, we demonstrate that Ripple is accurate and can estimate counts of up to 12-node subgraphs, which is a task at a scale that has been considered unreachable, not only by prior MCMC-based methods but also by other sampling approaches. For instance, in this target application, we present results in which the Markov chain state space is as large as 1043, for which Ripple computes estimates in less than 4 h, on average.
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- Award ID(s):
- 1943364
- PAR ID:
- 10323552
- Date Published:
- Journal Name:
- Data mining and knowledge discovery
- ISSN:
- 1573-756X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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