Abstract
The interactions between the components of complex networks are often directed. Proper modeling of such systems frequently requires the construction of ensembles of digraphs with a given sequence of in- and out-degrees. As the number of simple labeled graphs with a given degree sequence is typically very large even for short sequences, sampling methods are needed for statistical studies. Currently, there are two main classes of methods that generate samples. One of the existing methods first generates a restricted class of graphs and then uses a Markov chain Monte-Carlo algorithm based on edge swaps to generate other realizations. As the mixing time of this process is still unknown, the independence of the samples is not well controlled. The other class of methods is based on the configuration model that may lead to unacceptably many sample rejections due to self-loops and multiple edges. Here we present an algorithm that can directly construct all possible realizations of a given bi-degree sequence by simple digraphs. Our method is rejection-free, guarantees the independence of the constructed samples and provides their weight. The weights can then be used to compute statistical averages of network observables as if they were obtained from uniformly distributed sampling or from any other chosen distribution.
Highlights
Introduction and definitionsIn network modeling problems [1, 2, 3, 4, 5, 6, 7], one often needs to generate ensembles of graphs obeying a given constraint
There are two main classes of algorithms that are used today to achieve the construction of graphs with given degree sequences
We have developed a graph construction and sampling algorithm to construct simple directed graphs realizing a given sequence of in- and out-degrees. Such constructions are needed in practical modeling situations, ranging from epidemics and sociology through food-webs to transcriptional regulatory networks, where we are interested in learning about the statistical properties of the network observables as determined only by the bi-degree sequence and nothing else
Summary
Introduction and definitionsIn network modeling problems [1, 2, 3, 4, 5, 6, 7], one often needs to generate ensembles of graphs obeying a given constraint. The interest lies in the study of network observables, as determined by the given sequence of degrees, and unbiased by anything else These can be graph theoretical measures, or properties of processes happening on the network (e.g., spreading processes, such as of opinion or disease). The other class consists of direct construction methods, which perform pairwise matchings of the half-edges emanating from randomly chosen nodes until all edges are realized. This method can generate multiple edges and self-loops, i.e., edges starting and ending on the same node, after which the sample must be rejected in order to avoid biases [35]. For a comparison of the two classes of methods see Ref. [23]
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