Non-uniform Evolving Hypergraphs and Weighted Evolving Hypergraphs

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.

Hypernetwork science via high-order hypergraph walks

“…The suite of generative models we considered are effective as structural null models, but do not explicitly posit a process or mechanism through which hypernetworks grow. In contrast, other researchers have put forth and studied hypergraph evolution mechanisms, such as a preferential attachment inspired model for non-uniform hypergraphs [78,79]. An analysis of these, or the development of entirely new, temporal hypergraph models might shed insight into how high-order structural properties put forth here emerge in network topology.…”

Networks beyond pairwise interactions: structure and dynamics

“…In Wu et al [296], choices are not made preferentially but instead proportionally to the "joint degree" of nodes, i.e., the number of hyperedges they share with the hyperedges that are already in the set. Finally, yet another model uses a complicated choice function to decide which nodes should be involved in new hyperedges [297], namely…”

The stability of persistent homology of hypergraphs

“…An important question is that as time goes on, how does a network evolve (cf. [1,12]). In this section, by using the stability of the persistent homology for hypergraphs in Theorem 3.5 and the stability of the persistent homomorphisms between the persistent homology in Theorem 6.5, we give a potential homological approach for the evolution problem of hypergraph-modeled networks.…”