Abstract
Firstly, this paper proposes a non-uniform evolving hypergraph model with nonlinear preferential attachment and an attractiveness. This model allows nodes to arrive in batches according to a Poisson process and to form hyperedges with existing batches of nodes. Both the number of arriving nodes and that of chosen existing nodes are random variables so that the size of each hyperedge is non-uniform. This paper establishes the characteristic equation of hyperdegrees, calculates changes in the hyperdegree of each node, and obtains the stationary average hyperdegree distribution of the model by employing the Poisson process theory and the characteristic equation. Secondly, this paper constructs a model for weighted evolving hypergraphs that couples the establishment of new hyperedges, nodes and the dynamical evolution of the weights. Furthermore, what is obtained are respectively the stationary average hyperdegree and hyperstrength distributions by using the hyperdegree distribution of the established unweighted model above so that the weighted evolving hypergraph exhibits a scale-free behavior for both hyperdegree and hyperstrength distributions.
Highlights
Complex networks can be used to describe and understand a variety of real-life systems, be they complex interacting systems or the microscopic nature of space-time
Some real-life systems have been represented by bipartite graphs or tripartite graphs, their properties may be different when depicted by hypergraphs
We will extend these concepts for complex networks that are represented by hypergraphs
Summary
A non-uniform model with an attractiveness is defined as follows: (i) The network starts from an initial seed of m0 nodes and a hyperedge containing m0 nodes. The probability that a new node will connect to the jth node of the ith batch, is proportional to a sublinear function of the hyperdegree h j (t, ti) and attractiveness a such that. The solution of this equation, with the initial condition that a node in the ith batch at its birth time satisfies h j (ti, ti) = m, is h j (t, ti). Where x is a common positive solution of Eq (5) This result shows that the hyperdegree distribution depends on the exponent α of the nonlinear preferential attachment, and relates with the distribution of the number of chosen existing nodes. The theoretical prediction of the hyperdegree distribution is in good agreement with the simulation results
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