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

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.


The number of nodes is equal to 100000, the number of new nodes is equal to 15, β = 1, the energy level follows a uniform distribution on [0, 1].O denotes the simulation result, and + the theoretical prediction.
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f4: The number of nodes is equal to 100000, the number of new nodes is equal to 15, β = 1, the energy level follows a uniform distribution on [0, 1].O denotes the simulation result, and + the theoretical prediction.

Mentions: Numerical simulations for the distributions of hyperedge cardinalities are given. The simulations are performed with the scale of N = 100000 (the total number of nodes), and each simulation result is obtained by averaging over 30 independent runs. The simulation results are shown in Figs 3 and 4 in double-logarithmic axis. 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.


The Evolution of Hyperedge Cardinalities and Bose-Einstein Condensation in Hypernetworks
The number of nodes is equal to 100000, the number of new nodes is equal to 15, β = 1, the energy level follows a uniform distribution on [0, 1].O denotes the simulation result, and + the theoretical prediction.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5037361&req=5

f4: The number of nodes is equal to 100000, the number of new nodes is equal to 15, β = 1, the energy level follows a uniform distribution on [0, 1].O denotes the simulation result, and + the theoretical prediction.
Mentions: Numerical simulations for the distributions of hyperedge cardinalities are given. The simulations are performed with the scale of N = 100000 (the total number of nodes), and each simulation result is obtained by averaging over 30 independent runs. The simulation results are shown in Figs 3 and 4 in double-logarithmic axis. 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.

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.