Are we there yet? When to stop a Markov chain while generating random graphs
Abstract
Markov chains are a convenient means of generating realizations of networks, since they require little more than a procedure for rewiring edges. If a rewiring procedure exists for generating new graphs with specified statistical properties, then a Markov chain sampler can generate an ensemble of graphs with prescribed characteristics. However, successive graphs in a Markov chain cannot be used when one desires independent draws from the distribution of graphs; the realizations are correlated. Consequently, one runs a Markov chain for N iterations before accepting the realization as an independent sample. In this work, we devise two methods for calculating N. They are both based on the binary "time-series" denoting the occurrence/non-occurrence of edge (u, v) between vertices u and v in the Markov chain of graphs generated by the sampler. They differ in their underlying assumptions. We test them on the generation of graphs with a prescribed joint degree distribution. We find the N proportional |E|, where |E| is the number of edges in the graph. The two methods are compared by sampling on real, sparse graphs with 10^3 - 10^4 vertices.
Cite
@article{arxiv.1202.3473,
title = {Are we there yet? When to stop a Markov chain while generating random graphs},
author = {Jaideep Ray and Ali Pinar and C. Seshadhri},
journal= {arXiv preprint arXiv:1202.3473},
year = {2012}
}
Comments
12 pages, 4 figures, 1 table. Submitted to 9th Workshop on Algorithms and Models for the Web Graph, Dalhousie University in Halifax, Nova Scotia, Canada, June 22-23, 2012, http://www.mathstat.dal.ca/~mominis/WAW2012/