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This relates to system time measured by the formation of simplices as system events. Multilevel hypernetworks are classes of sets of relational simplices that represent the system backcloth and the traffic of systems activity it supports.


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Hypernetworks provide a significant generalisation of network theory, enabling the integration of relational structure, logic, and topological and analytic dynamics. They provide structures that are likely to be necessary if not sufficient for a science of complex multilevel socio-technical systems.

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Advertisement Hide. Recently, the concept of hypernetworks 6 has attracted much attention in the scientific community. Hyperlink prediction in hypernetworks using latent social features was studied 7.

Kim et al. Komarov and Pikovsky reported on finite-sized-induced transitions to synchrony in a population of phase oscillators coupled via nonlinear mean field, which is microscopically equivalent to a hypernetwork organization of interactions 9. However, few studies in the literature focus on dynamic evolution models of hypernetworks.

Although the content of Wikipedia was described with a hypernetwork model 10 , it represents just a special case of the general evolution model of complex networks in ref.

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In addition, Hu et al. Wang et al. Hu et al. Although several hyperedges were added at each time step in ref. Consequently, Guo and Zhu 16 developed a hypernetwork evolution model that unified the hypernetwork models introduced in refs 13 , 14 , 15 and the BA model. The common feature of all the hypernetwork models reviewed above is that node degree distribution is an extension of the concept of degree distributions in complex networks.

Nevertheless, the cardinality of a hyperedge is also an important parameter.

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ISBN 13: 9781860949722

In the study of hypernetworks, one of the tasks is to depict the topological characteristics with the hyperedge cardinality. In addition to the above researches, models of network dynamics based on quantum statistics have also been well studied. Gachechiladze et al. Bianconi et al. Quantum geometric networks can be distinguished from Fermi-Dirac networks and Bose-Einstein networks that obey respectively the Fermi-Dirac and Bose-Einstein statistics.

Kulvelis et al.

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Other studies include bosonic networks 22 , fermionic networks 23 , 24 , and Bose-Einstein condensation in the Apollonian complex network Recently, Bianconi 26 constructed a multiplex network which is described by coupled Bose-Einstein and Fermi-Dirac quantum statistics. They extended the definition of entanglement entropy of multiplex structures. However, in ref. In ref. The structural properties, including the degree distribution in different layers and different types of correlations have been obtained. Contrary to these works, the originality of our present work is to regard particles as nodes and energy levels as hyperedges, on which an evolving hypernetwork model is developed and its essential properties are studied.

On one hand, nodes in different hyperedges can be different from each other. On the other hand, the cardinalities of hyperedges are dynamic changing as time goes on. One purpose of the paper is to obtain the stationary average hyperedge cardinality distribution. Furthermore, our model is able to capture the phenomenon of Bose-Einstein condensation in the evolution of hypernetworks.

The rest of the paper is organized as follows. In next section, we introduce respectively the concepts of hypergraphs and hypernetworks. After that, we propose Bose-Einstein hypernetwork model. That is followed by theoretical analysis and numerical simulations of our model. At the end, our presentation is concluded with some conclusion remarks. The generalization of the concept of complex networks can be categorized into those of network-based supernetworks and hypergraph-based hypernetworks.

This concept was first proposed by Denning in , while it was clearly defined by Nagurney In supernetworks, there are large scale and complex connections, resulting in many networks mingled with each other. Each layer can be seen as a graph, and interconnections are existed between nodes of different layers. Such networks constitute a natural environment to describe systems interconnected through different categories of connections. For further information, please consult with review articles 6 , Another concept is that of hypergraph-based hypernetworks.


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Each edge in a hypergraph, known as hyperdeges, contains arbitrary number of nodes The extension from edge to hyperededge, make it possible to relate groups of more than two nodes. Meanwhile, the network structure is simple and clear. A complex system represented by a hypergraph will be referred to as a hypernetwork The following is the mathematical definition of hypergraphs and hypernetworks. The concept of hypergraphs generalizes that of graphs by allowing for edges of higher cardinality.

While for graphs edges connect only two nodes, each hyperedge can connect more than two nodes; to this end, examples of hypergraphs are depicted in Fig. Two nodes are said to be adjoined if they belong to the same hyperedge. Two hyperedges are said to be adjoined if their intersection set is not empty. H is said to be a finite hypergraph if both and are finite.

An example of four-uniform hypergraph is depicted in Fig. When each edge in a hypergraph contains only two nodes, the hypergraph degenerates into an ordinary graph. With the definitions above, we now establish the concept of hypernetworks. Here the indicator t is often interpreted as time. A hypernetwork is a set of hypergraphs. The degree or hyperdegree of node v i is defined as the number of hyperedges containing the node. The cardinality of a hyperedge E i is defined as the number of nodes contained in the hyperedge.

In the real-life world, hyperedge growth and hyperedge preferential attachment are the bases of the evolution mechanism. The generation algorithm in ref. At each time step, m 1 nodes are added to the system, and a new hyperedge is constructed by connecting these new nodes and an existing old nodes.