Weighted hypersoft configuration model

Maximum entropy null models of networks come in different flavors that depend on the type of constraints under which entropy is maximized. If the constraints are on degree sequences or distributions, we are dealing with configuration models. If the degree sequence is constrained exactly, the corresponding microcanonical ensemble of random graphs with a given degree sequence is the configuration model per se. If the degree sequence is constrained only on average, the corresponding grand-canonical ensemble of random graphs with a given expected degree sequence is the soft configuration model. If the degree sequence is not fixed at all but randomly drawn from a fixed distribution, the corresponding hypercanonical ensemble of random graphs with a given degree distribution is the hypersoft configuration model, a more adequate description of dynamic real-world networks in which degree sequences are never fixed but degree distributions often stay stable. Here, we introduce the hypersoft configuration model of weighted networks. The main contribution is a particular version of the model with power-law degree and strength distributions, and superlinear scaling of strengths with degrees, mimicking the properties of some real-world networks. As a byproduct, we generalize the notions of sparse graphons and their entropy to weighted networks.