Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity

Tao Zhou, Jian-Guo Liu, Wen-Jie Bai, Guanrong Chen, and Bing-Hong Wang

Phys. Rev. E 74, 056109 – Published 13 November 2006

Abstract

In this paper, we propose a susceptible-infected model with identical infectivity, in which, at every time step, each node can only contact a constant number of neighbors. We implemented this model on scale-free networks, and found that the infected population grows in an exponential form with the time scale proportional to the spreading rate. Furthermore, by numerical simulation, we demonstrated that the targeted immunization of the present model is much less efficient than that of the standard susceptible-infected model. Finally, we investigate a fast spreading strategy when only local information is available. Different from the extensively studied path-finding strategy, the strategy preferring small-degree nodes is more efficient than that preferring large-degree nodes. Our results indicate the existence of an essential relationship between network traffic and network epidemic on scale-free networks.

Received 11 April 2006

DOI:https://doi.org/10.1103/PhysRevE.74.056109

Authors & Affiliations

Tao Zhou^1,2,*, Jian-Guo Liu³, Wen-Jie Bai⁴, Guanrong Chen², and Bing-Hong Wang^1,†

¹Department of Modern Physics and Nonlinear Science Center, University of Science and Technology of China, Anhui Hefei 230026, People’s Republic of China
²Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, People’s Republic of China
³Institute of System Engineering, Dalian University of Technology, Dalian 116023, People’s Republic of China
⁴Department of Chemistry, University of Science and Technology of China, Anhui Hefei 230026, People’s Republic of China

^*Electronic address: zhutou@ustc.edu
^†Electronic address: bhwang@ustc.edu.cn

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Vol. 74, Iss. 5 — November 2006

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Images

Figure 1
(Color online) The infected density $i (t)$ vs time, where $i (t) = I (t) ∕ N$ . The black and red curves result from the standard SI network model and the present model. The numerical simulations are implemented based on the BA network [2] of size $N = 5000$ and with average degree $⟨ k ⟩ = 6$ . The spreading rate is given as $λ = 0.01$ , and the data are averaged over 5000 independent runs.Reuse & Permissions
Figure 2
(Color online) The infected density $i (t)$ vs time in normal (a) and single-log (b) plots. The black solid, red dot, green dash, and blue dash-dot curves correspond to $λ = 0.01$ , 0.001, 0.0005, and 0.0001, respectively. In single-log plot (b), the early behavior of $i (t)$ can be well fitted by a straight line, indicating the exponential growth of the infected population. The inset shows the rescaled curves $i (λ t)$ . The four curves for different $λ$ collapse to one curve in the new scale $λ t$ . The numerical simulations are implemented based on a BA network of size $N = 5000$ and with average degree $⟨ k ⟩ = 6$ , and the data are averaged over 5000 independent runs.Reuse & Permissions
Figure 3
(Color online) The infected density $i (t)$ vs time for different $γ$ . The black squares, red circles, blue up-triangles, green down-triangles, and pink diamonds (from up to down) denote the cases of $γ = 2.0$ , 2.5, 3.0, 3.5, and 4.0, respectively. The numerical simulations are implemented based on the scale-free configuration network model. The networks are of size $N = 1000$ and with average degree $⟨ k ⟩ = 6$ , the spreading rate is given as $λ = 0.01$ , and the data are averaged over 10 000 independent runs.Reuse & Permissions
Figure 4
(Color online) The infected density $i (t)$ vs time with different vaccinating ranges. (a) and (b) show the results of targeted immunization for the standard SI process in normal and single-log plots, respectively. Correspondingly, (c) and (d) display the results for the present model. In all the four panels, the black solid, red dash, blue dot, and green dash-dot curves represent the cases of $f = 0$ , 0.001, 0.005, and 0.01, respectively. The numerical simulations are implemented based on a BA network of size $N = 5000$ and with average degree $⟨ k ⟩ = 6$ , the spreading rate is given as $λ = 0.01$ , and the data are averaged over 5000 independent runs. For comparison, the infectivity of the present model is set as $A = ⟨ k ⟩ = 6$ .Reuse & Permissions
Figure 5
(Color online) The infected density $i (t)$ vs time for different $β$ . In (a), the black solid, blue dot, magenta dash-dot, red dash, and green dash-dot-dot curves correspond to $β = 0$ , $- 1$ , $- 2$ , 1, and 2, respectively. In (b), the black solid, red dash, blue dot, green dash-dot, magenta dash-dot-dot, and cyan short-dash curves, from up to down, correspond to $β = 0$ , 0.1, 0.2, 0.3, 0.4, and 0.5, respectively. In (c), the black solid, red dash, blue dot, green dash-dot, magenta dash-dot-dot, and cyan short-dash curves correspond to $β = 0$ , $- 0.1$ , $- 0.2$ , $- 0.3$ , $- 0.4$ , and $- 0.5$ , respectively. The numerical simulations are implemented based on the BA network of size $N = 5000$ and with average degree $⟨ k ⟩ = 6$ , the spreading rate is given as $λ = 0.01$ , and the data are averaged over 5000 independent runs.Reuse & Permissions

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