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October  2017, 14(5&6): 1215-1232. doi: 10.3934/mbe.2017062

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

 a. Department of Mathematics, Shanghai University, Shanghai 200444, China b. Complex Systems Research Center, Shanxi University, Taiyuan 030051, Shanxi, China

* Corresponding authorr: Zhen Jin

Received  April 07, 2016 Revised  November 11, 2016 Published  May 2017

Fund Project: This work is supported by the National Natural Science Foundation of China (11331009,11171314)

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.

Citation: Junbo Jia, Zhen Jin, Lili Chang, Xinchu Fu. Structural calculations and propagation modeling of growing networks based on continuous degree. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1215-1232. doi: 10.3934/mbe.2017062
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show all references

##### References:
  R. Albert and A. L. Barabási, Statistical mechanics of complex networks, Reviews of Modern Physics, 74 (2002), 47-97.  doi: 10.1103/RevModPhys.74.47.  Google Scholar  A. L. Barabási, R. Albert and H. Jeong, Mean-field theory for scale-free random networks, Physica A: Statistical Mechanics and its Applications, 272 (1999), 173-187.   Google Scholar  A. L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512.  doi: 10.1126/science.286.5439.509.  Google Scholar  S. N. Dorogovtsev, J. F. F. Mendes and A. N. Samukhin, Structure of growing networks with preferential linking, Physical Review Letters, 85 (2000), 4633. doi: 10.1103/PhysRevLett.85.4633. Google Scholar  S. N. Dorogovtsev and J. F. F. Mendes, Scaling properties of scale-free evolving networks: Continuous approach, Physical Review E, 63 (2001), 056125. doi: 10.1103/PhysRevE.63.056125. Google Scholar  S. N. Dorogovtsev and J. F. F. Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW, Oxford University Press, New York, 2013. Google Scholar  P. Erdős and A. Rényi, On the strength of connectedness of a random graph, Acta Mathematica Hungarica, 12 (1961), 261-267.  doi: 10.1007/BF02066689.  Google Scholar  M. Faloutsos, P. Faloutsos and C. Faloutsos, On power-law relationships of the internet topology, ACM SIGCOMM Computer Communication Review, 29 (1999), 251-262.  doi: 10.1145/316188.316229. Google Scholar  M. J. Gagen and J. S. Mattick, Accelerating, hyperaccelerating, and decelerating networks, Physical Review E, 72 (2005), 016123. doi: 10.1103/PhysRevE.72.016123. Google Scholar  T. House and M. J. Keeling, Insights from unifying modern approximations to infections on networks, Journal of The Royal Society Interface, 8 (2011), 67-73.  doi: 10.1098/rsif.2010.0179. Google Scholar  M. J. Keeling, The effects of local spatial structure on epidemiological invasions, Proceedings of the Royal Society of London. Series B: Biological Sciences, 266 (1999), 859-867.  doi: 10.1098/rspb.1999.0716. Google Scholar  K. T. D. Ken and M. J. Keeling, Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases, Proceedings of the National Academy of Sciences, 99 (2002), 13330-13335.   Google Scholar  P. L. Krapivsky, S. Redner and F. Leyvraz, Connectivity of growing random networks, Physical Review Letters, 85 (2000), 4629. Google Scholar  P. L. Krapivsky and S. Redner, Organization of growing random networks, Physical Review E, 63 (2001), 066123. doi: 10.1103/PhysRevE.63.066123. Google Scholar  J. Lindquist, J. Ma, P. van den Driessche and F. H. Willeboordse, Effective degree network disease models, Journal of Mathematical Biology, 62 (2011), 143-164.  doi: 10.1007/s00285-010-0331-2.  Google Scholar  C. Liu, J. Xie, H. Chen, H. Zhang and M. Tang, Interplay between the local information based behavioral responses and the epidemic spreading in complex networks, Chaos: An Interdisciplinary Journal of Nonlinear Science, 25 (2015), 103111, 7 pp. doi: 10.1063/1.4931032.  Google Scholar  S. Milgram, The small world problem, Psychology Today, 2 (1967), 60-67.   Google Scholar  J. C. Miller, A. C. Slim and E. M. Volz, Edge-based compartmental modelling for infectious disease spread, Journal of the Royal Society Interface, 9 (2012), 890-906.   Google Scholar  J. C. Miller and I. Z. Kiss, Epidemic spread in networks: Existing methods and current challenges, Mathematical Modelling of Natural Phenomena, 9 (2014), 4-42.  doi: 10.1051/mmnp/20149202.  Google Scholar  Y. Moreno, R. Pastor-Satorras and A. Vespignani, Epidemic outbreaks in complex heterogeneous networks, The European Physical Journal B-Condensed Matter and Complex Systems, 26 (2002), 521-529.  doi: 10.1140/epjb/e20020122. Google Scholar  M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45 (2003), 167-256.  doi: 10.1137/S003614450342480.  Google Scholar  R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Physical Review Letters, 86 (2001), 3200. doi: 10.1103/PhysRevLett.86.3200. Google Scholar  D. Shi, Q. Chen and L. Liu, Markov chain-based numerical method for degree distributions of growing networks, Physical Review E, 71 (2005), 036140. Google Scholar  E. Volz, SIR dynamics in random networks with heterogeneous connectivity, Journal of Mathematical Biology, 56 (2008), 293-310.  doi: 10.1007/s00285-007-0116-4.  Google Scholar  D. J. Watts and S. H. Strogatz, Collective dynamics of "small-world" networks, Nature, 393 (1998), 440-442.   Google Scholar  H. Zhang, J. Xie, M. Tang and Y. Lai, Suppression of epidemic spreading in complex networks by local information based behavioral responses, Chaos: An Interdisciplinary Journal of Nonlinear Science, 24 (2014), 043106, 7 pp. doi: 10.1063/1.4896333.  Google Scholar The degree distribution of the BA growing model. The marked points are the simulation results, and the corresponding solid lines are the analytical results. (a): $t=20000$ with $m_0=m=2, 4$ and 7. (b): $m_0=m=4$ with $t=200,2000,10000$ and 20000 The degree distribution of the preferential attachment accelerating growing model with $m$-varying. The marked points are the simulation results, and the corresponding solid lines are the analytical results. (a): $t=5000$, $\alpha=0.2$ with $m=2$, $4$ and $7$. (b): $t=5000$, $m=4$ with $\alpha=0.1$, $0.2$ and $0.3$. (c): $m=4$, $\alpha=0.2$ with $t=200,500,2000$ and 5000 The degree distribution of the random attachment growing model. The marked points are the simulation results, and the corresponding solid lines are the analytical results. (a): $t=20000$ with $m_0=m=2, 4$ and 7. (b): $m_0=m=4$ with $t=200,2000,10000$ and 20000 The schematic diagram of added directed edges $\Delta L_1$ and $\Delta L_2$ The ratio of infected nodes over time $t$ for the SIS model on static BA network with size $N=5000$. The marked lines are the average of 500 runs of stochastic simulations, and the corresponding solid lines are the results of numerical simulation. The initial ratio of infected nodes is set to 5% The relative ratio of infected nodes with degree $k$ at time $t$ for the SIS model on static BA network. The time $t=1000$ and the initial ratio of infected nodes is 5% The relative ratio of infected nodes with degree $k$ at time $t$ for the SIS model on BA growing network. The time $t=6000$, the initial time $t_0=5000$ and the initial ratio of infected nodes is 5% The ratio of infected nodes over time $t$ for the SIS model on BA growing network with different epidemic occurrence time $t_0$. The time step $t$ is recorded from $t0$. The marked lines are the average of 200 runs of stochastic simulations, and the corresponding solid lines are the results of numerical simulation. (a): $t_0=150$; (b): $t_0=200$; (c): $t_0=1000$
The definition of main notations
 Notation Definition $m_0$ The total number of nodes in the initial network. $l_0$ The total number of edges in the initial network. $\Pi_i$ The probability for the node $i$ connect to the new added node. $N(t)$ The total number of nodes at time $t$. $\hat{N}(k, t)$ The total number of nodes which degree not more than $k$ at time $t$ (note there has a hat $\text{ha}{{\text{t}}^{\text{ }\!\!\hat{\ }\!\!\text{ }}}$ on letter $N$). $F(k, t)$ The cumulative distribution function of node degree, or the proportion of nodes which degree not more than $k$, at time $t$. $p(k, t)$ The degree distribution, or the probability density of node which degree equal to $k$, at time $t$, $p(k, t)=\frac{\partial F(k, t)}{\partial k}$. $L(t)$ The total number of directed edges at time $t$. $\hat{L}(k_1, k_2, t)$ The total number of directed edges which degree sequentially not larger than $k_1$ and $k_2$ at time $t$ (also note the hat). $F(k_1, k_2, t)$ The joint cumulative distribution function at time $t$. $p(k_1, k_2, t)$ The joint degree distribution at time $t$, $p(k_1, k_2, t)=\frac{\partial^2 F(k_1, k_2, t)}{\partial k_1 \partial k_2}$. $p(k_2|k_1, t)$ The conditional degree distribution at time $t$. $q(k, t)$ The marginal distribution at time $t$. $F_S(k, t)$ The cumulative distribution function of susceptible nodes at time $t$. $F_I(k, t)$ The cumulative distribution function of infected nodes at time $t$. $p_S(k, t)$ The probability density of susceptible nodes which degree equal to $k$ at time $t$. $p_I(k, t)$ The probability density of infected nodes which degree equal to $k$ at time $t$. $\Theta_k$ The probability that a edge emitted by degree $k$ node points to an infected node. $\Theta$ The probability that a edge points to an infected node in degree unrelated network.
 Notation Definition $m_0$ The total number of nodes in the initial network. $l_0$ The total number of edges in the initial network. $\Pi_i$ The probability for the node $i$ connect to the new added node. $N(t)$ The total number of nodes at time $t$. $\hat{N}(k, t)$ The total number of nodes which degree not more than $k$ at time $t$ (note there has a hat $\text{ha}{{\text{t}}^{\text{ }\!\!\hat{\ }\!\!\text{ }}}$ on letter $N$). $F(k, t)$ The cumulative distribution function of node degree, or the proportion of nodes which degree not more than $k$, at time $t$. $p(k, t)$ The degree distribution, or the probability density of node which degree equal to $k$, at time $t$, $p(k, t)=\frac{\partial F(k, t)}{\partial k}$. $L(t)$ The total number of directed edges at time $t$. $\hat{L}(k_1, k_2, t)$ The total number of directed edges which degree sequentially not larger than $k_1$ and $k_2$ at time $t$ (also note the hat). $F(k_1, k_2, t)$ The joint cumulative distribution function at time $t$. $p(k_1, k_2, t)$ The joint degree distribution at time $t$, $p(k_1, k_2, t)=\frac{\partial^2 F(k_1, k_2, t)}{\partial k_1 \partial k_2}$. $p(k_2|k_1, t)$ The conditional degree distribution at time $t$. $q(k, t)$ The marginal distribution at time $t$. $F_S(k, t)$ The cumulative distribution function of susceptible nodes at time $t$. $F_I(k, t)$ The cumulative distribution function of infected nodes at time $t$. $p_S(k, t)$ The probability density of susceptible nodes which degree equal to $k$ at time $t$. $p_I(k, t)$ The probability density of infected nodes which degree equal to $k$ at time $t$. $\Theta_k$ The probability that a edge emitted by degree $k$ node points to an infected node. $\Theta$ The probability that a edge points to an infected node in degree unrelated network.
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