Abstract
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.
Highlights
Many real-world complex systems that can be represented as networks [1,2] require weighted representations in which connections between nodes are characterized by positive weights [3]
We introduce the weighted hypersoft configuration model (WHSCM) which is a hypercanonical ensemble of networks with a fixed joint distribution of degrees and strengths
The Hypersoft configuration model (HSCM) can be viewed as a hyperparametrization of the Soft configuration model (SCM), in that the Lagrange multipliers ν are not fixed by any degree sequence as solutions of (8), but random, sampled independently for each node i from a fixed distribution ρ(ν): νi ← ρ(ν )
Summary
Many real-world complex systems that can be represented as networks [1,2] require weighted representations in which connections between nodes are characterized by positive weights [3]. The soft configuration model (SCM) [6,8,10,35,36] is a grand-canonical ensemble of random graphs with soft constraints on the degree sequence This means that the expected degree sequence in the ensemble is equal to a given degree sequence. We introduce the weighted hypersoft configuration model (WHSCM) which is a hypercanonical ensemble of networks with a fixed joint distribution of degrees and strengths. Besides introducing the WHSCM in general, the main focus of this paper is a much more involved task, which is to identify the joint distribution of latent parameters that reproduces several features of degree and strength distributions observed in many real weighted networks [3,44,45,46,47,48].
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