The Evolution of Hyperedge Cardinalities and Bose-Einstein Condensation in Hypernetworks

To depict the complex relationship among nodes and the evolving process of a complex system, a Bose-Einstein hypernetwork is proposed in this paper. Based on two basic evolutionary mechanisms, growth and preference jumping, the distribution of hyperedge cardinalities is studied. The Poisson process theory is used to describe the arrival process of new node batches. And, by using the Poisson process theory and a continuity technique, the hypernetwork is analyzed and the characteristic equation of hyperedge cardinalities is obtained. Additionally, an analytical expression for the stationary average hyperedge cardinality distribution is derived by employing the characteristic equation, from which Bose-Einstein condensation in the hypernetwork is obtained. The theoretical analyses in this paper agree with the conducted numerical simulations. This is the first study on the hyperedge cardinality in hypernetworks, where Bose-Einstein condensation can be regarded as a special case of hypernetworks. Moreover, a condensation degree is also discussed with which Bose-Einstein condensation can be classified.

attachment mechanism as mentioned in the afore-mentioned literatures, Guo and Suo 17 proposed another model and introduced two other factors, such as node's own fitness and competitiveness, to underlie the mechanism of preferential attachment.
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. 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. 18 studied the nonlocal properties of quantum hypergraph states. Bianconi et al. 19 proposed the concept of quantum geometric networks, which has many properties common to those of complex networks. 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. 20 studied single-particle quantum transport on parametrized complex networks. Bianconi and Barabási 21 tried to map an equilibrium Bose gas into a complex network and found the emergence of Bose-Einstein condensation in such evolving networks, which is indeed a pioneering work on the subject matter. Other studies include bosonic networks 22 , fermionic networks 23,24 , and Bose-Einstein condensation in the Apollonian complex network 25 . 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. 21 each same energy level was regarded as a node, which resulted in such a consequence as that all particles at different energy levels shrink into a node. In ref. 26 the nodes in each layer were the same. 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 Related Concepts
The generalization of the concept of complex networks can be categorized into those of network-based supernetworks and hypergraph-based hypernetworks.
A supernetwork is a 'network of networks' . This concept was first proposed by Denning in 1985, while it was clearly defined by Nagurney 27 . In supernetworks, there are large scale and complex connections, resulting in many networks mingled with each other. Such networks are also called 'multilayer networks' 28,29 or 'multilevel networks' 30 . Usually they possess 'multi-stranded' relationships and are formed by layers. 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 6,28 .
Another concept is that of hypergraph-based hypernetworks. Each edge in a hypergraph, known as hyperdeges, contains arbitrary number of nodes 31 . 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 32 . 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. Formally, we define a hypergraph as a binary H = (V, E h ), which is also denoted as (V, ) are the sets of nodes and hyperedges, respectively. 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. 1. 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 V and E h are finite.
h . An example of four-uniform hypergraph is depicted in Fig. 1(a). With the definitions above, we now establish the concept of hypernetworks. Assuming that is a finite hypergraph for any given t ≥ 0. 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.

Model Description
Hypernetwork model. In the real-life world, hyperedge growth and hyperedge preferential attachment are the bases of the evolution mechanism. The generation algorithm in ref. 13 can be described as follows. (1) The hypernetwork starts with m 0 nodes and a hyperedge which concludes all these m 0 nodes. 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. (2) The probability ∏ d i ( ( )) H that an old node i is selected by the new hyperedge depends on the hyperdegree d H (i) of node i: Bose-Einstein hypernetwork. Most of the afore-mentioned models are k-uniform hypernetworks. Results of theoretical analysis show that hyperdegree distributions exhibit the scale-free property. Different from the models described previously, we will construct a non-uniform hypernetwork. In particular, the evolving mechanism reflects the common feature of competitions among hyperedges, resulting in the evolution of the cardinality of hyperedges. Here we show that our model can be mapped into an equilibrium Bose gas for treating particles as nodes while considering different energy levels as hyperedges.
For the growth mechanism, it is often assumed that the nodes are added to the system at equal-lengthtime intervals, and the arrival of nodes follows a uniform distribution. The continuum assumption of discrete problems is the precondition for analyzing node degrees by employing differential equations 2 . Here we assume that the process of node arrivals follows a Poisson process to better describe the arrival patterns in realistic systems, and hence this assumption allows us a more rigorous analysis of the model.
For the preference mechanism, how can we reflect the state of competition among hyperedges? It is clear that a hyperedge with bigger cardinality has more probability to be selected. Another dimension is the energy level of a hyperedge, which closely relates to the hyperedge's competitiveness. These two parameters jointly underlie the evolution process. Our hypernetwork model satisfies the following two steps.
Growth. The arrival process of new node batches is a Poisson process N t ( ) with a constant rate λ. At time t, when a batch of new nodes arrives at the network, a positive integer ς N t ( ) is chosen from a given distribution ζ g ( ), accounting for the number of new nodes. The new ς N t ( ) nodes are encircled by a new hyperedge E N(t) , while the energy ε N(t) > 0 of hyperedge is drawn from a given distribution ρ(ε). And each new node is assigned to a state.
Preferential Jump. At time t, a hyperedge is randomly chosen from the hypernetwork. And a randomly chosen node that belongs to this hyperedge jumps into another hyperedge. The probability W that the chosen node jumping into hyperedge j depends on the cardinality h j of the hyperedge j and on the energy level ε j of the hyperedge j such that is the expected value of the initial cardinality of all hyperedges.
In some cases, the fitness η j of hyperedge j is determined by its energy level. The relationship between the fitness and energy level of the hyperedge j is given as follows: Here we interpret how the hypernetwork model above corresponds to a realistic system. Take online shopping as an example; its features can be depicted by the model above. In electronic commerce, with new products and new customers joining into the network, the sales network is in a constant state of growth. One obvious characteristic of online purchasing behaviors is that customers tend to purchase products associated with good quality and higher sales. This reflects the mechanism of preferential purchase. Customers have preferences for higher quality items. However, it is hard for them to differentiate products in terms of their qualities. In many cases, historical sales data is an effective tool for customers to make their decisions based on the assumption that better products are accompanied by higher sales. Treating customers as nodes while considering products as hyperedges, the fitness and the cardinality of hyperedges represent respectively the competitiveness and the sales history of products. At the beginning, purchasing behaviors are scattered. With the evolution of the network, products with higher competitiveness and sales will attract more customers, thus resulting in preferential jumpings.
A schematic illustration of the dynamical rules for building a Bose-Einstein hypernetwork is shown in Fig. 2.

Model Analyses
Here we focus on the dynamics of hyperedge cardinality. Firstly, we write down the rate equation for the distribution of hyperedge cardinalities. Then the theoretical results are given based on the Poisson process theory and a continuity technique 21   The first term in the right of Eq. (4) corresponds to preferential attachment of a hyperedge that is selected by a node. And the second term corresponds to the randomly selection of a node to jump out of the current hyperedge. Let For sufficiently large t, we have Because of the initial condition that each hyperedge j satisfies ς = h t ( ) j j j , the solution of Eq. (6) is  Equation (8) is called a characteristic equation of the hyperedge cardinality of the Bose-Einstein hypernetwork. According to the Possion process theory, the arrival time t j of node batches obeys Gamma distribution having parameters (i, λ), thus  For simplicity, for given ς = m, the stationary average hyperedge cardinality distribution of the Bose-Einstein hypernetwork is as follows  From the evolution mechanism of the model, we know that the cardinalities and energy levels of hyperedges jointly determine the evolution. Thus the ability for hyperedges to compete for nodes is not the same from one hyperedge to another. Nodes tend to jump to the most attractive hyperedges, and these hyperedges thus acquire more and more nodes over time. And this process results in that a tiny fraction of the hyperedges will acquire respectively good numbers of nodes. As the figures show, the theoretical prediction result which is obtained from Eq. (15) is consistent with the tail of the distributions of hyperedge cardinalities in simulations.
Bose-Einstein condensation. Assume that M ε=0 represents the number of nodes on the energy level where ε = 0, then we have , then the nodes condense on the energy level of ε = 0 where α is called a condensation degree of the hypernetwork. The condition the condensation degree α on the energy level where ε = 0 satisfies is given as follows If α = 1, the hypernetwork almost completely condenses on the energy level of ε = 0. Regarding particles as nodes, Bose-Einstein condensation can be described by the model above. According to the condensation degree, Bose-Einstein condensations can be classified.
The particles of a Bose-Einstein condensation model follow the stationary average cardinality distribution Eq. (10) at each energy level. By introducing the concept of chemical potential μ, we let = βµ , we have μ ≤ 0. That is, the chemical potential is nonpositive. The maximum of I(β, μ) is obtained when μ = 0, for given β, m, and ρ(ε); thus we have The condensation degree is α on the lowest energy level.  From Eq. (17), it follows that Bose-Einstein condensation appears when Eq. (8) has no solution, at which point Eqs (7) and (8) break down. The absence of a solution indicates that almost all hyperedges have only a few of nodes, while some "gel" hyperedges have the rest of the nodes of the hypernetwork. This end seems to be a well-known signature of Bose-Einstein condensation.

Conclusions
By taking into account the fact that the concept of hypernetworks is more general as hypernetwroks allow for the dynamics of hyperedge cardinality and node degree, we propose Bose-Einstein hypernetwork evolution model by combining the growth and preferential jump mechanisms to investigate hyperedge cardinalities of the hypernetwork structure. We obtain the distribution of hyperedge cardinalities by using theoretical analysis and numerical simulations. Our simulation results are in good agreement with theoretical conclusions. Specially, when treating particles as nodes while considering different energy levels as hyperedges, the Bose-Einstein condensation model can be regarded as a special case of our model. Furthermore, we establish the condensation condition of the hypernetwork on the zero-energy level. The solid results established in this paper lead us to believe that the concept of hypernetworks can be used as a new tool for the study of statistical physics.
Presently, the research on the topological characteristics and evolution mechanism of hypernetworks are just started. Although we have obtained some essential theoretical result of the model, the corresponding empirical studies are still absent. Such empirical studies, as a complement to the model, can further enrich the current research. Besides hyperdegrees and cardinalities, there are other important parameters, such as clustering coefficient, assortativity that have been discussed in the study of complex networks. The definitions and analyses of such parameters in hypernetworks need to be explored. Furthermore, there is a real need to understand whether or not hypernetworks pose common topological features in a self-organized way. Followup studies could focus on the evolutionary dynamics of hypernetwork structures and dynamical processes that occur over hypernetworks.