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Double-layer hypernetwork model with bimodal peak characteristics

## Double-layer hypernetwork model with bimodal peak characteristics

Lu Wen, Zhao Hai-Xing, Meng Lei, Hu Feng
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• #### Abstract

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: $p=0.004$, $p=0.006$, $p=0.008$ and $p=0.01$. The results show that the node hyper degree distribution of this hyper network model complies to the Poisson distribution $p(k)\approx \dfrac{{{\left\langle \lambda \right\rangle }^{k}}}{k!}{{e}^{-\left\langle \lambda \right\rangle }}$, 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.

#### References

 [1] Wuchty S 2001 Mol. Biol. Evol. 18 1694 [2] Wasserman S, Faust K 1994 Social Network Analysis (Cambridge: Cambridge University Press) pp1−66 [3] 汪小帆, 李翔, 陈关荣 2012 网络科学导论 (北京: 高等教育出版社) 第194 −226页 Wang X F, Li X, Chen G R 2012 Network Science: An Introduction (Beijing: Higher Education Press) pp194−226 (in Chinese) [4] Wang P, Xu B W, Wu Y R, Zhou X Y 2015 Sci. Chin. Inf. 58 011101 [5] Lü L Y, Zhou T 2011 Phys. A 390 1150 [6] Liben-Nowell D, Kleinberg J 2007 J. Am. Soc. Inf. Sci. Technol. 58 1019 [7] Newman M E J 2001 Proc. Natl. Acad. Sci. 98 404 [8] Zhou T, Wang B H, Jin Y D, He D R, Zhang P P, He Y, Su B B, Chen K, Zhang Z Z, Liu J G 2007 Int. J. Mod. Phys. C 18 297 [9] Berge C 1973 Graphs and Hypergraphs (New York: American Elsevier Publishing Company, Inc.) pp389−425 [10] Camarinha-Matos L M, Afsarmanesh H 2003 Comput. Ind. 51 139 [11] Wu Z Y, Duan J Q, Fu X C 2014 Appl. Math. Modell. 38 2961 [12] 胡枫, 赵海兴, 何佳倍, 李发旭, 李淑玲, 张子柯 2013 物理学报 62 198901 Hu F, Zhao H X, He J B, Li F X, Li S L, Zhang Z K 2013 Acta Phys. Sin. 62 198901 [13] Estrada E, Rodríguez-Velázquez J A 2006 Phys. A 364 581 [14] Ghoshal G, Zlatić V, Caldarelli G, Newman M E J 2009 Phys. Rev. E 79 066118 [15] Zlatić V, Ghoshal G, Caldarelli G 2009 Phys. Rev. E 80 036118 [16] Zhang Z K, Liu C 2010 J. Stat. Mech. 10 1742 [17] Wang J W, Rong L L, Deng Q H 2010 Eur. Phys. J. B 77 493 [18] 胡枫, 赵海兴, 马秀娟 2013 中国科学: 物理学 力学 天文学 43 16 Hu F, Zhao H X, Ma X J 2013 Sci. China, Ser. G 43 16 [19] 郭进利, 祝昕昀 2014 物理学报 63 090207 Guo J L, Zhu X J 2014 Acta Phys. Sin. 63 090207 [20] 索琪, 郭进利 2017 系统工程理论与实践 37 720 Suo Q, Guo J L 2017 System Eng. Theor. Prac. 37 720 [21] Zhou Z, Jin Z, Jin J 2019 J. Phys. A 123 765 [22] 李甍娜, 郭进利, 卞闻, 常宁戈, 肖潇, 陆睿敏 2017 复杂系统与复杂性科学 4 66 Li M N, Guo J L, Bian W, Chang N G, Xiao X, Lu R M 2017 Complex Systems and Complexity Science 4 66 [23] 胡枫, 刘猛, 赵静 2018 复杂系统与复杂性科学 4 31 Hu F, Liu M, Zhao J 2018 Complex Systems and Complexity Science 4 31 [24] Fang J Q, Liu Q H, Tang M 2016 JAAC 6 12 [25] 刘强, 方锦清, 李永 2015 复杂系统与复杂性科学 12 64 Liu Q, Fang J Q, Li Y 2015 Complex Systems and Complexity Science 12 64 [26] Boccaletti S, Bianconi G, Criado R 2014 Phys. Rep. 544 1 [27] 蒋文君, 刘润然, 范天龙, 刘霜霜, 吕琳媛 2020 物理学报 69 088904 Jiang W J, Liu R R, Fan T L, Liu S S, Lü L Y 2020 Acta Phys. Sin. 69 088904 [28] 杨喜艳, 吴亚豪, 张家军 2019 电子科技大学学报 10 12178 Yang X Y, Wu Y H, Zhang J J 2019 J. Elec. Sci. Tech. Univ. 10 12178 [29] Erdös P, Rényi A 1960 Publ. Math. Inst. Hung. Acad. Sci. 5 17 [30] Xu X P, Liu F 2008 Phys. Lett. A 372 6727 [31] Xue X F 2017 Phys. A 486 434 [32] Lima F W S, Sousa A O, Sumuor M A 2008 Phys. A 387 3503 [33] Zehmakan A N 2020 Discrete. Appl. Math. 277 280 [34] 李炎, 唐刚, 宋丽建, 寻之朋, 夏辉, 郝大鹏 2013 物理学报 62 046401 Li Y, Tang G, Song L J, Xu Z P, Xia H, Hao D P 2013 Acta Phys. Sin. 62 046401

#### Cited By

• 图 1  “航空-铁路”双层超网络模型

Figure 1.  Airline-Railway double-layer hyper network.

图 2  500个节点的随机3均匀超网络在不同连接概率p值时的节点超度分布　(a) $p = 0.004$; (b) $p = 0.006$; (c) $p = 0.008$; (d) $p = 0.01$

Figure 2.  The hyper degree distribution of 3-uniform random hyper networks under different p: (a) $p = 0.004$; (b) $p = 0.006$; (c) $p = 0.008$; (d) $p = 0.01$.

图 3  双层3均匀EE超网络在不同层间超边连接概率时的节点超度分布　(a)${p_{12}} = 0.001$; (b)${p_{12}} = 0.01$

Figure 3.  The EE hyper degree distribution of double-layer 3-uniform hyper network under different ${p_{12}}$: (a)${p_{12}} = 0.001$; (b)${p_{12}} = 0.01$.

图 4  双层3均匀BB超网络节点超度分布　(a)${p_{12}} = 0.001$; (b)${p_{12}} = 0.01$

Figure 4.  The BB hyper degree distribution of double-layer 3-uniform hyper network under different ${p_{12}}$: (a)${p_{12}} = 0.001$; (b)${p_{12}} = 0.01$.

图 5  双层3均匀BE超网络模型节点超度分布　(a)${p_{12}} = 0.001$; (b)${p_{12}} = 0.01$

Figure 5.  The BE hyper degree distribution of double-layer 3-uniform hyper network under different ${p_{12}}$: (a)${p_{12}} = 0.001$; (b)${p_{12}} = 0.01$.

•  [1] Wuchty S 2001 Mol. Biol. Evol. 18 1694 [2] Wasserman S, Faust K 1994 Social Network Analysis (Cambridge: Cambridge University Press) pp1−66 [3] 汪小帆, 李翔, 陈关荣 2012 网络科学导论 (北京: 高等教育出版社) 第194 −226页 Wang X F, Li X, Chen G R 2012 Network Science: An Introduction (Beijing: Higher Education Press) pp194−226 (in Chinese) [4] Wang P, Xu B W, Wu Y R, Zhou X Y 2015 Sci. Chin. Inf. 58 011101 [5] Lü L Y, Zhou T 2011 Phys. A 390 1150 [6] Liben-Nowell D, Kleinberg J 2007 J. Am. Soc. Inf. Sci. Technol. 58 1019 [7] Newman M E J 2001 Proc. Natl. Acad. Sci. 98 404 [8] Zhou T, Wang B H, Jin Y D, He D R, Zhang P P, He Y, Su B B, Chen K, Zhang Z Z, Liu J G 2007 Int. J. Mod. Phys. C 18 297 [9] Berge C 1973 Graphs and Hypergraphs (New York: American Elsevier Publishing Company, Inc.) pp389−425 [10] Camarinha-Matos L M, Afsarmanesh H 2003 Comput. Ind. 51 139 [11] Wu Z Y, Duan J Q, Fu X C 2014 Appl. Math. Modell. 38 2961 [12] 胡枫, 赵海兴, 何佳倍, 李发旭, 李淑玲, 张子柯 2013 物理学报 62 198901 Hu F, Zhao H X, He J B, Li F X, Li S L, Zhang Z K 2013 Acta Phys. Sin. 62 198901 [13] Estrada E, Rodríguez-Velázquez J A 2006 Phys. A 364 581 [14] Ghoshal G, Zlatić V, Caldarelli G, Newman M E J 2009 Phys. Rev. E 79 066118 [15] Zlatić V, Ghoshal G, Caldarelli G 2009 Phys. Rev. E 80 036118 [16] Zhang Z K, Liu C 2010 J. Stat. Mech. 10 1742 [17] Wang J W, Rong L L, Deng Q H 2010 Eur. Phys. J. B 77 493 [18] 胡枫, 赵海兴, 马秀娟 2013 中国科学: 物理学 力学 天文学 43 16 Hu F, Zhao H X, Ma X J 2013 Sci. China, Ser. G 43 16 [19] 郭进利, 祝昕昀 2014 物理学报 63 090207 Guo J L, Zhu X J 2014 Acta Phys. Sin. 63 090207 [20] 索琪, 郭进利 2017 系统工程理论与实践 37 720 Suo Q, Guo J L 2017 System Eng. Theor. Prac. 37 720 [21] Zhou Z, Jin Z, Jin J 2019 J. Phys. A 123 765 [22] 李甍娜, 郭进利, 卞闻, 常宁戈, 肖潇, 陆睿敏 2017 复杂系统与复杂性科学 4 66 Li M N, Guo J L, Bian W, Chang N G, Xiao X, Lu R M 2017 Complex Systems and Complexity Science 4 66 [23] 胡枫, 刘猛, 赵静 2018 复杂系统与复杂性科学 4 31 Hu F, Liu M, Zhao J 2018 Complex Systems and Complexity Science 4 31 [24] Fang J Q, Liu Q H, Tang M 2016 JAAC 6 12 [25] 刘强, 方锦清, 李永 2015 复杂系统与复杂性科学 12 64 Liu Q, Fang J Q, Li Y 2015 Complex Systems and Complexity Science 12 64 [26] Boccaletti S, Bianconi G, Criado R 2014 Phys. Rep. 544 1 [27] 蒋文君, 刘润然, 范天龙, 刘霜霜, 吕琳媛 2020 物理学报 69 088904 Jiang W J, Liu R R, Fan T L, Liu S S, Lü L Y 2020 Acta Phys. Sin. 69 088904 [28] 杨喜艳, 吴亚豪, 张家军 2019 电子科技大学学报 10 12178 Yang X Y, Wu Y H, Zhang J J 2019 J. Elec. Sci. Tech. Univ. 10 12178 [29] Erdös P, Rényi A 1960 Publ. Math. Inst. Hung. Acad. Sci. 5 17 [30] Xu X P, Liu F 2008 Phys. Lett. A 372 6727 [31] Xue X F 2017 Phys. A 486 434 [32] Lima F W S, Sousa A O, Sumuor M A 2008 Phys. A 387 3503 [33] Zehmakan A N 2020 Discrete. Appl. Math. 277 280 [34] 李炎, 唐刚, 宋丽建, 寻之朋, 夏辉, 郝大鹏 2013 物理学报 62 046401 Li Y, Tang G, Song L J, Xu Z P, Xia H, Hao D P 2013 Acta Phys. Sin. 62 046401
•  Citation:
##### Metrics
• Abstract views:  658
• Cited By: 0
##### Publishing process
• Received Date:  04 July 2020
• Accepted Date:  31 August 2020
• Available Online:  22 December 2020
• Published Online:  05 January 2021

## Double-layer hypernetwork model with bimodal peak characteristics

###### Corresponding author: Hu Feng, qhhuf@163.com
• 1. School of Computer Science, Shaanxi Normal University, Xi’an 710119, China
• 2. College of Computer, Qinghai Normal University, Xining 810008, China
• 3. Key Laboratory of Tibetan Information Processing and Machine Translation of Qinghai Province, Xining 810008, China
• 4. Key Laboratory of Tibetan Information Processing, Ministry of Education, Xining 810008, China

Abstract: 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: $p=0.004$, $p=0.006$, $p=0.008$ and $p=0.01$. The results show that the node hyper degree distribution of this hyper network model complies to the Poisson distribution $p(k)\approx \dfrac{{{\left\langle \lambda \right\rangle }^{k}}}{k!}{{e}^{-\left\langle \lambda \right\rangle }}$, 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.

Reference (34)

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