Proof of uniform sampling of binary matrices with fixed row sums and column sums for the fast Curveball algorithm

Abstract

Randomization of binary matrices has become one of the most important quantitative tools in modern computational biology. The equivalent problem of generating random directed networks with fixed degree sequences has also attracted a lot of attention. However, it is very challenging to generate truly unbiased random matrices with fixed row and column sums. Strona et al. [Nat. Commun. 5, 4114 (2014)] introduce the innovative Curveball algorithm and give numerical support for the proposition that it generates truly random matrices. In this paper, we present a rigorous proof of convergence to the uniform distribution. Furthermore, we show the Curveball algorithm must include certain failed trades to ensure uniform sampling.

A unifying framework for fast randomization of ecological networks with fixed (node) degrees

“…e corresponding graphs or lists for each problem are called realisations or degree sequences, respectively. e Curveball algorithm, as introduced in [35,11], is a Markov chain approach to the uniform random sampling of a realisation with bipartite xed degree sequence. Given one such realization, the Curveball algorithm nds others by repeatedly making small changes to the adjacency list representation of the bipartite graph.…”

“…In this paper we focus on Markov chain approaches to this problem, where a graph is randomised by repeatedly making small changes to it. Even though several Markov chains have been shown to converge to the uniform distribution on their state space [32,4,38,11], the main question for both theoreticians and practitioners remains unanswered: that is, in all but some special cases it is unknown how many changes need to be made, i.e. how many steps the Markov chains needs to take, in order to sample from a distribution that is close to uniform.…”

Grand canonical validation of the bipartite international trade network

“…Remarkably, the method is ergodic, i.e. it explores the phases space uniformly [29] (note that the ergodicity of BiCM is automatically obtained by construction). Although the algorithm is relatively fast, the fact that it is micro canonical does not permit to calculate the expectation values of different quantities, thus preventing the possibility of writing an expression like eq.…”

The connectivity of graphs of graphs with self-loops and a given degree sequence

“…For many different types of graphs, one of the most popular sampling techniques is Markov chain Monte Carlo sampling via 'double edge swaps'. Guarantees about the uniformity of double edge swap MCMC sampling (along with a related family MCMC techniques [4,6]) are founded on several properties, the most difficult of which is whether an MCMC sampler can sample every possible graph, or equivalently, whether the associated Markov chain is irreducible (equivalently, the associated graph of the Markov chain is strongly connected). For any degree sequence, the following spaces are connected and thus can be sampled using MCMC techniques: simple graphs [21,2,1,9,18], simple connected graphs [18,2], multigraphs [11] and multigraphs with selfloops [8].…”