Generating constrained random graphs using multiple edge switches

The generation of random graphs using edge swaps provides a reliable method to draw uniformly random samples of sets of graphs respecting some simple constraints, e.g. degree distributions. However, in general, it is not necessarily possible to access all graphs obeying some given con- straints through a classical switching procedure calling on pairs of edges. We therefore propose to get round this issue by generalizing this classical approach through the use of higher-order edge switches. This method, which we denote by "k-edge switching", makes it possible to progres- sively improve the covered portion of a set of constrained graphs, thereby providing an increasing, asymptotically certain confidence on the statistical representativeness of the obtained sample.Comment: 15 page

Marathon: An open source software library for the analysis of Markov-Chain Monte Carlo algorithms

In this paper, we consider the Markov-Chain Monte Carlo (MCMC) approach for random sampling of combinatorial objects. The running time of such an algorithm depends on the total mixing time of the underlying Markov chain and is unknown in general. For some Markov chains, upper bounds on this total mixing time exist but are too large to be applicable in practice. We try to answer the question, whether the total mixing time is close to its upper bounds, or if there is a significant gap between them. In doing so, we present the software library marathon which is designed to support the analysis of MCMC based sampling algorithms. The main application of this library is to compute properties of so-called state graphs which represent the structure of Markov chains. We use marathon to investigate the quality of several bounding methods on four well-known Markov chains for sampling perfect matchings and bipartite graph realizations. In a set of experiments, we compute the total mixing time and several of its bounds for a large number of input instances. We find that the upper bound gained by the famous canonical path method is several magnitudes larger than the total mixing time and deteriorates with growing input size. In contrast, the spectral bound is found to be a precise approximation of the total mixing time