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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.

No MeSH data available.


Schematic illustration of the process at each time step of a Bose-Einstein hypernetwork.(a) At time step t, there are four energy hyperedges e1~e4 described by closed curves, which contain different number of nodes respectively. Each node is assigned a state. And nodes with the same state are encircled by a red ellipse representing a state hyperedge. (b) At time step t + 1, a batch of new nodes encircled by a new energy hyperedge e5 arrives at the network. (c) Select an energy hyperedge randomly (e2 is selected and shown as a red hollow ellipse) from the existing network. Then select a node randomly in e2 (shown as the red one encircled by a yellow hollow ellipse). (d) The selected yellow node jumps from the original energy hyperedge e2 to another energy hyperedge e4 according to the preferential mechanism. The quantum state of the node remains unchanged while jumping.
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f2: Schematic illustration of the process at each time step of a Bose-Einstein hypernetwork.(a) At time step t, there are four energy hyperedges e1~e4 described by closed curves, which contain different number of nodes respectively. Each node is assigned a state. And nodes with the same state are encircled by a red ellipse representing a state hyperedge. (b) At time step t + 1, a batch of new nodes encircled by a new energy hyperedge e5 arrives at the network. (c) Select an energy hyperedge randomly (e2 is selected and shown as a red hollow ellipse) from the existing network. Then select a node randomly in e2 (shown as the red one encircled by a yellow hollow ellipse). (d) The selected yellow node jumps from the original energy hyperedge e2 to another energy hyperedge e4 according to the preferential mechanism. The quantum state of the node remains unchanged while jumping.

Mentions: A schematic illustration of the dynamical rules for building a Bose-Einstein hypernetwork is shown in Fig. 2.


The Evolution of Hyperedge Cardinalities and Bose-Einstein Condensation in Hypernetworks
Schematic illustration of the process at each time step of a Bose-Einstein hypernetwork.(a) At time step t, there are four energy hyperedges e1~e4 described by closed curves, which contain different number of nodes respectively. Each node is assigned a state. And nodes with the same state are encircled by a red ellipse representing a state hyperedge. (b) At time step t + 1, a batch of new nodes encircled by a new energy hyperedge e5 arrives at the network. (c) Select an energy hyperedge randomly (e2 is selected and shown as a red hollow ellipse) from the existing network. Then select a node randomly in e2 (shown as the red one encircled by a yellow hollow ellipse). (d) The selected yellow node jumps from the original energy hyperedge e2 to another energy hyperedge e4 according to the preferential mechanism. The quantum state of the node remains unchanged while jumping.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f2: Schematic illustration of the process at each time step of a Bose-Einstein hypernetwork.(a) At time step t, there are four energy hyperedges e1~e4 described by closed curves, which contain different number of nodes respectively. Each node is assigned a state. And nodes with the same state are encircled by a red ellipse representing a state hyperedge. (b) At time step t + 1, a batch of new nodes encircled by a new energy hyperedge e5 arrives at the network. (c) Select an energy hyperedge randomly (e2 is selected and shown as a red hollow ellipse) from the existing network. Then select a node randomly in e2 (shown as the red one encircled by a yellow hollow ellipse). (d) The selected yellow node jumps from the original energy hyperedge e2 to another energy hyperedge e4 according to the preferential mechanism. The quantum state of the node remains unchanged while jumping.
Mentions: A schematic illustration of the dynamical rules for building a Bose-Einstein hypernetwork is shown in Fig. 2.

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.