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

  • 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.

    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

    Related Papers:

  • 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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    [1] [ R. Albert,A. L. Barabási, Statistical mechanics of complex networks, Reviews of Modern Physics, 74 (2002): 47-97.
    [2] [ A. L. Barabási,R. Albert,H. Jeong, Mean-field theory for scale-free random networks, Physica A: Statistical Mechanics and its Applications, 272 (1999): 173-187.
    [3] [ A. L. Barabási,R. Albert, Emergence of scaling in random networks, Science, 286 (1999): 509-512.
    [4] [ S. N. Dorogovtsev, J. F. F. Mendes and A. N. Samukhin, Structure of growing networks with preferential linking, Physical Review Letters, 85 (2000), 4633.
    [5] [ S. N. Dorogovtsev and J. F. F. Mendes, Scaling properties of scale-free evolving networks: Continuous approach, Physical Review E, 63 (2001), 056125.
    [6] [ 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.
    [7] [ P. Erdős,A. Rényi, On the strength of connectedness of a random graph, Acta Mathematica Hungarica, 12 (1961): 261-267.
    [8] [ M. Faloutsos,P. Faloutsos,C. Faloutsos, On power-law relationships of the internet topology, ACM SIGCOMM Computer Communication Review, 29 (1999): 251-262.
    [9] [ M. J. Gagen and J. S. Mattick, Accelerating, hyperaccelerating, and decelerating networks, Physical Review E, 72 (2005), 016123.
    [10] [ T. House,M. J. Keeling, Insights from unifying modern approximations to infections on networks, Journal of The Royal Society Interface, 8 (2011): 67-73.
    [11] [ 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.
    [12] [ K. T. D. Ken,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.
    [13] [ P. L. Krapivsky, S. Redner and F. Leyvraz, Connectivity of growing random networks, Physical Review Letters, 85 (2000), 4629.
    [14] [ P. L. Krapivsky and S. Redner, Organization of growing random networks, Physical Review E, 63 (2001), 066123.
    [15] [ J. Lindquist,J. Ma,P. van den Driessche,F. H. Willeboordse, Effective degree network disease models, Journal of Mathematical Biology, 62 (2011): 143-164.
    [16] [ 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.
    [17] [ S. Milgram, The small world problem, Psychology Today, 2 (1967): 60-67.
    [18] [ J. C. Miller,A. C. Slim,E. M. Volz, Edge-based compartmental modelling for infectious disease spread, Journal of the Royal Society Interface, 9 (2012): 890-906.
    [19] [ J. C. Miller,I. Z. Kiss, Epidemic spread in networks: Existing methods and current challenges, Mathematical Modelling of Natural Phenomena, 9 (2014): 4-42.
    [20] [ Y. Moreno,R. Pastor-Satorras,A. Vespignani, Epidemic outbreaks in complex heterogeneous networks, The European Physical Journal B-Condensed Matter and Complex Systems, 26 (2002): 521-529.
    [21] [ M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45 (2003): 167-256.
    [22] [ R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Physical Review Letters, 86 (2001), 3200.
    [23] [ D. Shi, Q. Chen and L. Liu, Markov chain-based numerical method for degree distributions of growing networks, Physical Review E, 71 (2005), 036140.
    [24] [ E. Volz, SIR dynamics in random networks with heterogeneous connectivity, Journal of Mathematical Biology, 56 (2008): 293-310.
    [25] [ D. J. Watts,S. H. Strogatz, Collective dynamics of "small-world" networks, Nature, 393 (1998): 440-442.
    [26] [ 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.
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