Double-layer hypernetwork model with bimodal peak characteristics
Authors: Lu Wen , Zhao Hai-Xing , Meng Lei , Hu Feng ,   School of Computer Science, Shaanxi Normal University, Xi’an 710119, China,   College of Computer, Qinghai Normal University, Xining 810008, China,   Key Laboratory of Tibetan Information Processing and Machine Translation of Qinghai Province, Xining 810008, China,   Key Laboratory of Tibetan Information Processing, Ministry of Education, Xining 810008, China
Published: December 2020

With the rapid development of social economy, the relationship between social members and groups has shown more complex and diverse characteristics. As a network depicting complex relation and multi-layer, hyper network has been widely used in different fields. Random network that obeys Poisson distribution is one of the pioneering models studying complex networks. In the existing hyper network researches, the hyper network based on ER random graph is still a blank. In this paper, we first propose an ER random hyper network model which is based on the hypergraph structure and it adopts the ER random graph theory. Furthermore, using this model, the node hyper degree distribution of this hyper network model is analyzed theoretically, and the node hyper degree distribution is simulated under different hyper edge probabilities: <inline-formula><tex-math id="M1">\begin{document}$ p=0.004$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20201065_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20201065_M1.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ p=0.006$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20201065_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20201065_M2.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ p=0.008$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20201065_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20201065_M3.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ p=0.01$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20201065_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20201065_M4.png"/></alternatives></inline-formula>. The results show that the node hyper degree distribution of this hyper network model complies to the Poisson distribution <inline-formula><tex-math id="M5">\begin{document}$p(k)\approx \dfrac{{{\left\langle \lambda \right\rangle }^{k}}}{k!}{{e}^{-\left\langle \lambda \right\rangle }}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20201065_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20201065_M5.png"/></alternatives></inline-formula>, which conforms with the characteristics of random networks and is consistent with the theoretical derivation. Further, in order to more accurately and effectively describe the multiple heterogeneous relationship in real life, in this paper we construct three different kinds of double-layer hyper network models with node hyper degree distribution with bimodal peak characteristics. The three kinds respectively are ER-ER, BA-BA and BA-ER, where ER represents the ER random hyper network, and BA denotes the scale-free hyper network, and the layers are connected by a random manner. The analytical expressions of node hyper degree distribution of the three kinds of double-layer hyper network models are obtained by theoretical analysis, and the average node hyper degrees of the three double-layer hyper networks are closely related to the inter-layer hyper edge probability. As the inter-layer hyper edge probability increases, the average node hyper degree increases. The results of simulation experiments show that the node hyper degree distributions of three kinds of double-layer hyper network models proposed in this paper possess the characteristics of bimodal peaks. The ER random hyper network model and the double-layer hyper network model proposed in this paper provide the theories for further studying the hyper network entropy, hyper network dynamics, hyper network representation learning, hyper network link prediction, and traffic hyper network optimization of such hyper networks in the future, and also it has certain reference significance for studying the evolution of multilayer hyper networks.