![]() There are two main classes of algorithms that are used today to achieve the construction of graphs with given degree sequences. degree-based graph construction , is a well-known and challenging problem that has attracted considerable interest among researchers . The problem of creating and sampling graphs with a given degree sequence, i.e. spreading processes, such as of opinion or disease). These can be graph theoretical measures or properties of processes happening on the network (e.g. Often, the interest lies in the study of network observables, as determined by the given sequence of degrees, and unbiased by anything else. Full graph connectivity is uniquely determined by the degree sequence only for a special class of sequences (see for the case of undirected graphs). Note that the node degrees by themselves, that is, the degree sequence in general, does not determine a graph uniquely: there can be a very large number of graphs having the same degree sequence . A typical constraint is the case when the only information available is the degrees of the nodes and not the actual connectivity matrix. In network modeling problems , one often needs to generate ensembles of graphs obeying a given constraint. 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. ![]() Our method is rejection-free, guarantees the independence of the constructed samples and provides their weight. Here we present an algorithm that can directly construct all possible realizations of a given bi-degree sequence by simple digraphs. 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. As the mixing time of this process is still unknown, the independence of the samples is not well controlled. 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. Currently, there are two main classes of methods that generate samples. 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. Proper modeling of such systems frequently requires the construction of ensembles of digraphs with a given sequence of in- and out-degrees. The interactions between the components of complex networks are often directed.
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