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

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ABSTRACT

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.

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Examples of hypergraphs.(a) A four-uniform hypergraph. (b) An ordinary graph. When each edge in a hypergraph contains only two nodes, the hypergraph degenerates into an ordinary graph. (c) A non-uniform hypergraph.
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f1: Examples of hypergraphs.(a) A four-uniform hypergraph. (b) An ordinary graph. When each edge in a hypergraph contains only two nodes, the hypergraph degenerates into an ordinary graph. (c) A non-uniform hypergraph.

Mentions: Another concept is that of hypergraph-based hypernetworks. Each edge in a hypergraph, known as hyperdeges, contains arbitrary number of nodes31. 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 hypernetwork32. 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, Eh), which is also denoted as (V, Eh) or H, where ( denotes the cardinality of the set V) and ( denotes the cardinality of the set Eh, ) 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 and are finite. A hypergraph H = (V, Eh) is a k-uniform hypergraph if , for . An example of four-uniform hypergraph is depicted in Fig. 1(a).


The Evolution of Hyperedge Cardinalities and Bose-Einstein Condensation in Hypernetworks
Examples of hypergraphs.(a) A four-uniform hypergraph. (b) An ordinary graph. When each edge in a hypergraph contains only two nodes, the hypergraph degenerates into an ordinary graph. (c) A non-uniform hypergraph.
© Copyright Policy - open-access
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC5037361&req=5

f1: Examples of hypergraphs.(a) A four-uniform hypergraph. (b) An ordinary graph. When each edge in a hypergraph contains only two nodes, the hypergraph degenerates into an ordinary graph. (c) A non-uniform hypergraph.
Mentions: Another concept is that of hypergraph-based hypernetworks. Each edge in a hypergraph, known as hyperdeges, contains arbitrary number of nodes31. 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 hypernetwork32. 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, Eh), which is also denoted as (V, Eh) or H, where ( denotes the cardinality of the set V) and ( denotes the cardinality of the set Eh, ) 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 and are finite. A hypergraph H = (V, Eh) is a k-uniform hypergraph if , for . An example of four-uniform hypergraph is depicted in Fig. 1(a).

View Article: PubMed Central - PubMed

ABSTRACT

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.

No MeSH data available.


Related in: MedlinePlus