Structural calculations and propagation modeling of growing networks based on continuous degree

Junbo Jia; Zhen Jin; Lili Chang; Xinchu Fu; Junbo Jia; Zhen Jin; Lili Chang; Xinchu Fu

doi:10.3934/mbe.2017062

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 5&6: 1215-1232. doi: 10.3934/mbe.2017062

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Structural calculations and propagation modeling of growing networks based on continuous degree

. Department of Mathematics, Shanghai University, Shanghai 200444, China

Received: 07 April 2016 Revised: 11 November 2016 Published: 01 October 2017
MSC : Primary: 35R02; Secondary: 37N25

When a network reaches a certain size, its node degree can be considered as a continuous variable, which we will call continuous degree. Using continuous degree method (CDM), we analytically calculate certain structure of the network and study the spread of epidemics on a growing network. Firstly, using CDM we calculate the degree distributions of three different growing models, which are the BA growing model, the preferential attachment accelerating growing model and the random attachment growing model. We obtain the evolution equation for the cumulative distribution function $F(k,t)$, and then obtain analytical results about $F(k,t)$ and the degree distribution $p(k,t)$. Secondly, we calculate the joint degree distribution $p(k_1, k_2, t)$ of the BA model by using the same method, thereby obtain the conditional degree distribution $p (k_1|k_2) $. We find that the BA model has no degree correlations. Finally, we consider the different states, susceptible and infected, according to the node health status. We establish the continuous degree SIS model on a static network and a growing network, respectively. We find that, in the case of growth, the new added health nodes can slightly reduce the ratio of infected nodes, but the final infected ratio will gradually tend to the final infected ratio of SIS model on static networks.
- Epidemic models,
- propagation modeling,
- complex networks,
- growing networks,
- partial differential equation
Citation: Junbo Jia, Zhen Jin, Lili Chang, Xinchu Fu. Structural calculations and propagation modeling of growing networks based on continuous degree[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1215-1232. doi: 10.3934/mbe.2017062

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Abstract

When a network reaches a certain size, its node degree can be considered as a continuous variable, which we will call continuous degree. Using continuous degree method (CDM), we analytically calculate certain structure of the network and study the spread of epidemics on a growing network. Firstly, using CDM we calculate the degree distributions of three different growing models, which are the BA growing model, the preferential attachment accelerating growing model and the random attachment growing model. We obtain the evolution equation for the cumulative distribution function $F(k,t)$, and then obtain analytical results about $F(k,t)$ and the degree distribution $p(k,t)$. Secondly, we calculate the joint degree distribution $p(k_1, k_2, t)$ of the BA model by using the same method, thereby obtain the conditional degree distribution $p (k_1|k_2) $. We find that the BA model has no degree correlations. Finally, we consider the different states, susceptible and infected, according to the node health status. We establish the continuous degree SIS model on a static network and a growing network, respectively. We find that, in the case of growth, the new added health nodes can slightly reduce the ratio of infected nodes, but the final infected ratio will gradually tend to the final infected ratio of SIS model on static networks.

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