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Chaos / Volume 22 / Issue 4 / REGULAR ARTICLES

Chaos 22, 043146 (2012); http://dx.doi.org/10.1063/1.4772967 (5 pages)

Traffic-driven epidemic outbreak on complex networks: How long does it take?

Han-Xin Yang1, Wen-Xu Wang2, and Ying-Cheng Lai3

1Department of Physics, Fuzhou University, Fuzhou 350108, China
2Department of Systems Science, School of Management and Center for Complexity Research, Beijing Normal University, Beijing 100875, China
3School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA

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(Received 13 August 2012; accepted 4 December 2012; published online 28 December 2012)

Recent studies have suggested the necessity to incorporate traffic dynamics into the process of epidemic spreading on complex networks, as the former provides support for the latter in many real-world situations. While there are results on the asymptotic scope of the spreading dynamics, the issue of how fast an epidemic outbreak can occur remains outstanding. We observe numerically that the density of the infected nodes exhibits an exponential increase with time initially, rendering definable a characteristic time for the outbreak. We then derive a formula for scale-free networks, which relates this time to parameters characterizing the traffic dynamics and the network structure such as packet-generation rate and betweenness distribution. The validity of the formula is tested numerically. Our study indicates that increasing the average degree and/or inducing traffic congestion can slow down the spreading process significantly.

© 2012 American Institute of Physics

Lead Paragraph

A spreading process cannot occur on a complex network without a backbone traffic dynamics that transports certain physical quantity across the network. For example, a computer virus may become widespread through email exchanges, and a disease/virus can spread globally through air transportation. A complete understanding of epidemic spreading dynamics thus requires incorporation of some kind of traffic dynamics into the spreading process. This has been a topic of several recent studies, where the focus has been on the asymptotic extent of the spreading dynamics, e.g., the fraction of infected nodes after the process terminates. The aim of our work is to address the issue of timing associated with traffic-driven epidemic spreading dynamics. In particular, we incorporate a shortest-path-based type of traffic dynamics into the standard two-state epidemic spreading model. We find that the density of the infected nodes exhibits an exponential increase with time in the early stage, rendering definable a characteristic time for the spreading process. Numerical results and theoretical reasoning indicate that this time depends on various parameters characterizing the traffic dynamics and network structure, such as packet-generation rate, spreading rate, and betweenness distribution of the network. For example, large-scale outbreaks occur more quickly as the packet-generation rate and the spreading rate increase. A somewhat counterintuitive finding is that an increase in the average connectivity tends to slow down the spreading process. Our study of how epidemic propagates among nodes of different degrees indicates that large-degree nodes are infected first, followed by a progressive spreading over small-degree nodes. In addition, we find that the emergence of traffic congestion can slow down the epidemic outbreak significantly, providing insights into developing effective strategies to prevent large-scale epidemic outbreaks on complex networks.

Article Outline

  1. INTRODUCTION
  2. MODEL
  3. RESULTS
  4. CONCLUSION

RELATED DATABASES

Web of Science

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KEYWORDS and PACS

Keywords

complex networks, epidemics

PACS

  • 87.10.-e

    General theory and mathematical aspects

  • 05.60.-k

    Transport processes

  • 89.75.Hc

    Networks and genealogical trees

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.4772967

PUBLICATION DATA

ISSN

1054-1500 (print)  
1089-7682 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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    References

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Figures (5)

Figures (click on thumbnails to view enlargements)

FIG.1
Density of infected nodes i(t) as a function of time t for (a) different values of the spreading rate β, (b) different values of the packet-generation rate λ, and (c) different values of the average connectivity 〈k〉. For (a), λ = 2 and 〈k〉 = 10; For (b), β = 0.005 and 〈k〉 = 10; For (c), β = 0.005 and λ = 2. The insets show the evolution of i(t) in the early times.

FIG.1 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.2
Comparison between numerical and theoretical values of the characteristic time τ: (a) τ versus the spreading probability β for λ = 2 and 〈k〉 = 10, (b) τ versus the packet-generation rate λ for β = 0.005 and 〈k〉 = 10, and (c) τ versus the average connectivity 〈k〉 for β = 0.005 and λ = 2. The theoretical values are calculated from Eq. ( 6 ).

FIG.2 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.3
Average degree 〈kinf(t)〉 of the newly infected nodes as a function of time for different values of the average connectivity 〈k〉. The spreading and packet-generation rates are β = 0.02 and λ = 2, respectively.

FIG.3 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.4
Density of infected nodes i(t) as a function of time t for C = 10 and C→∞. The average connectivity is 〈k〉 = 10, the spreading probability is β = 0.01 and the packet-generation rate is λ = 1. For C = 10, the critical packet-generating rate is λc ≃ 0.05. The inset shows the evolution of i(t) at the initial stage of epidemic spreading.

FIG.4 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.5
For the two cases where the capacity parameter C is infinite and finite (C = 10), the characteristic time τ as a function of the packet-generation rate λ. Other parameters are 〈k〉 = 10 and β = 0.01. For C = 10, the critical packet-generating rate is λc ≃ 0.05. Line represents the theoretical prediction [Eq. ( 6 )].

FIG.5 Download High Resolution Image (.zip file) | Export Figure to PowerPoint



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