Malicious viruses spreading on complex networks with heterogeneous recovery rate

Accepted Manuscript Malicious viruses spreading on complex networks with heterogeneous recovery rate Linbo Long, Kan Zhong, Wei Wang

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S0378-4371(18)30701-5 https://doi.org/10.1016/j.physa.2018.05.149 PHYSA 19689

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Physica A

Received date : 5 April 2018 Revised date : 18 May 2018 Please cite this article as: L. Long, K. Zhong, W. Wang, Malicious viruses spreading on complex networks with heterogeneous recovery rate, Physica A (2018), https://doi.org/10.1016/j.physa.2018.05.149 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Highlights (for review)

Highlights 1. Propose a malicious viruses spreading model which considers the effects healthy

neighbors, 2. There is an optimal strategy of resource contribution. 3. The strategy of resource contribution can alter the phase transition in homogeneous networks. 4. There is always continuous phase transition in heterogeneous network.

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Malicious viruses spreading on complex networks with heterogeneous recovery rate Linbo Long1 , Kan Zhong2 and Wei Wang3 1. College of Computer Science and Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 2. College of Computer Science, Chongqing University, Chongqing 400044, China 3. Cybersecurity Research Institute, Sichuan University, Chengdu 610065, China

Abstract We propose a malicious viruses spreading model on complex networks that considers the effects of the resource contribution of healthy neighbors. Due to the difference in resource amount received from the healthy neighbors, the recovery rate of infected nodes is heterogeneous. Through rigorous theoretical analysis and extensive numerical simulations, we find that the virus spreading can be optimally suppressed if each heathy node contributes equal resource to the infected nodes. There is a maximum outbreak threshold and minimum fraction of infected nodes with the optimal strategy of resource contribution. In addition, we find that in a homogeneous network, the strategy of resource contribution can alter the phase transition. If each healthy node contributes relatively evenly, the phase transition is continuous. Whereas, if the recovery resources of infected nodes is mainly relies on nodes with large or small degrees, there is discontinuous phase transition. In heterogeneous network, there is always continuous phase transition. Keywords: Complex networks, malicious viruses spreading, heterogeneous recovery rate PACS: 89.75.Hc, 87.19.X-, 87.23.Ge

∗

[email protected]

Preprint submitted to Physica A

May 18, 2018

1. Introduction The spreading of malicious viruses, infectious disease, innovation, and healthy behavior in reality can be modelled as spreading dynamics on complex networks, which attracted much attention in recent years [1, 2, 3, 4, 5, 6, 7, 8, 9]. To uncover the underlaying evolution mechanisms of the spreading dynamics, many famous models have been proposed, such as the susceptible-infected-susceptible (SIS) and susceptible-infected-recovered (SIR). Pastor-Satorras and Vespignani developed a seminal work in this field, and found that the threshold vanishes for scale-free networks [4]. After this work, many researches focused on the effects of network topology on spreading dynamics [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Moreno et al found that the critical exponent does not be changed by the heterogeneous degree distribution [26]. Wang et al developed an edge-weight-based compartmental approach for virus spreading dynamics on networks with general degree and weight distributions, and a remarkable agreement with numerics is obtained [27]. They found that the heterogeneity of weight distribution suppressed the epidemic spreading when the average weight is fixed. The role of dynamical parameters on virus spreading dynamics is also widely studied [15, 28, 29, 30, 31, 32, 33]. Miller assumed that nodes have heterogeneous infectivity and susceptibility in the SIR model [15]. Through percolation theory, he revealed that the global outbreak of virus is more likely occurs for nodes with homogeneous infectivity than that for nodes with strong heterogeneous infectivity. For the SIS model, Smilkov found that heterogeneous susceptibility induces small threshold when the correlation between a node’s degree and susceptibility are positive, otherwise, the opposite situation happens [30]. Shu et al studied the effects of the recovery rate on the SIR model, and found that the outbreak threshold does not be changed when we adopted the asynchronous update method to renew the states of nodes [34]. However, the outbreak threshold increases with recovery rate for synchronous update method. The above researches found that the second-order phase transition does not be changed by the heterogeneous susceptibility and infectivity. In recent years researchers focused on the effects of social factors, such as public resource investment [35, 36] or local resource support [37], on dynamics of virus spreading dynamics. Chen et al investigated how the social support affects spreading dynamics by using the SIS model in social-contact multiplex networks [38]. They considered that the recovery rate of nodes are dependent on the resource support from neighborhood in social layer. Both the degree heterogeneity of the network and edge overlap can influence the phase transition of fraction of 2

infected nodes. A hybrid phase transition displaying characteristics of both discontinuous and continuous transitions appears when there is a small degree of edge overlap. We consider that in reality the healthy populations are source of resources, as they can be human resources needed to provide health services, and can also product medicines etc. In addition, the recovery of the infected individuals needs resources that comes from the contributions of their direct healthy neighbors. However, due to individual differences (e.g., economic capacity), the amount of resources that each healthy individual can provide is inconsistent. Thus we are interested in the questions: should the nodes with larger degrees or smaller degrees in complex networks contributed more resources to the infected nodes, and how could it influence the spreading dynamics? To solve the problems, we propose a novel virus spreading model, in which the recovery rate of infected nodes depends on the resource amount provided by healthy neighbors. To adjust the resource quantity of the nodes with different degrees, a controlling parameter is proposed. Through extensive theoretical analysis and numerical simulations, we find that in both homogeneous and heterogeneous networks, each healthy node should contribute equally in order to optimally constrain the virus spreading. Besides, we find that both the contribution strategy and degree heterogeneity can alter the phase transition. Specially, in ER networks, the phase transition changes from continuous transition to discontinuous transition when the resource supply is largely relies on large degree nodes or small degree nodes. Whereas, in scale-free networks, there is always continuous phase transition. 2. Model descriptions We adopt the susceptible-infected-susceptible (SIS) model to describe the virus spreading dynamics on networks. Every node can be one of the susceptible and infected state. Node in the susceptible state means that it has not infected by the virus. An infected node represents that it is infected by the virus and will transmit the infection to every susceptible neighbors. Assuming the malicious virus spreading on complex networks with degree distribution P (k) and N nodes. In simulations, we adopt the synchronous update method to renew the states of nodes [39]. Initially a fraction of ρ0 nodes are randomly selected as seeds (infected), and the remaining nodes are susceptible. At every time step, every infected node i will try to transmit the infection to every susceptible neighbor with rate β, and then recoveries with rate µi . Note that the recovery rate µi of node i is dependent on the surroundings of i. The recovery 3

process of an infected node always needs consuming some resources, which are always provided from susceptible neighbors, we therefore assume the recovery rate µi of node i is µi = 1 − (1 − µ0 )ωri , (1)

where µ0 is the basic recovery rate, ω is the utilization factor of the resources, ri is the totally provided resources from neighbors as ri =

N ∑ j=1

Aij sj

kjα α α } max{kmin , kmax

,

(2)

where A is the adjacent matrix of the complex networks, if nodes i and j are connected by a link, Aij = 1, otherwise Aij = 0. The parameters kmin and kmax are the minimum and maximum degrees, sj = 1 if node j is susceptible, otherwise sj = 0. The parameter α is an adjustable value, which is used to describe the recourse providing strategy. For the case of α = 0, every susceptible neighbor provide unity resource. For the case of α > 0, nodes with large degrees provide more resources to neighbors; otherwise, the opposite situations happens. 3. Theoretical analysis 3.1. Discrete Markov chains approach To describe the viruses spreading dynamics, we develop a generalized discrete Markov chains approach [40]. Denoting si (t) as the probability that node i is in the susceptible state at time t, and ρi (t) = 1 − si (t) is the probability that node i is in the infected state at time t. For the evolution of ρi (t), we should consider two situations: (1) the infected node i recovers with probability µi ρi (t), and (2) the susceptible node i is infected by neighbors with probability si (t)θi (t), where θi (t) is the probability that node i is infected by neighbors. The expression of θi (t) is θi (t) = 1 − Πj∈Ω(i) [1 − βρj (t)],

(3)

where Ω(i) is the neighbor set of node i. The evolution of ρi (t) can be expressed as ρi (t + 1) = (1 − µi )ρi (t) + si (t)θi (t). (4) In the steady state, t → ∞, the value of ρi (t) does not change, we denote ρi (∞) as ρi for convenience. We know that Eq. (4) always has a trivial solution ρi = 0 for i = 1, · · · , N . In this situation, there is no global virus outbreak. If ρi > 0, a 4

global outbreak becomes possible. The order parameter is given by the expected infection density N 1 ∑ ρ∞ = ρi . (5) N i=1

Now we determine the critical point βc , only above which the global virus becomes possible, i.e., ρ∞ > 0. Otherwise, there is no global virus that is ρ∞ = 0. Near the critical point βc , we know ρi → 0, and thus the second order terms in ρi can be neglected. Rewriting Eq. (3) as θi = β

N ∑

(6)

Aij ρj .

j=1

Inserting Eq. (6) into Eq. (4), and neglecting the second order of ρi , we obtain ρi = (1 − µi )ρi + β

N ∑

Aij ρj .

(7)

j=1

For small values of µ0 , we obtain µi = µ0 ωri . Combining Eq. (7) and only consider the first order of ρi , we get N ∑ j=1

[

1 1 α α max{kmin , kmax }Aij − δji ]ρj = 0, α µ0 ωkj β

(8)

where δji represents the Kronecker delta. The global virus occurs if and only if 1/β is an eigenvalue of the matrix B, where the element of B is Bij = 1 α α max{kmin , kmax }Aij . Thus, the critical point is µ0 ωkα j

βc =

1 , Λc (B)

(9)

where Λc (B) is largest eigenvalue of matrix B. Note that βc is correlated with A and α. 3.2. Heterogeneous mean-field approach To have a better understanding the interplay between the spreading dynamics and network structures directly, especially to get an analysis results of the optimal value of α, we develop a heterogeneous mean-field approach. In this approach, we 5

assume that nodes with the same degree are the statistical same. Denoting Ik (t) as the probability that nodes i with degree k is in the infected state at time t. The evolution of Ik (t) can be expressed as ∑ (k ′ − 1)p(k ′ ) dIk (t) = βk[1 − Ik (t)] Ik′ (t) − µk Ik (t), dt ⟨k⟩ ′ k

(10)

where ⟨k⟩ is the average degree, p(k) is the degree distribution, µk is the recovery rate of node with degree k. The first item of Eq. (10) is the probability that nodes with degree k are infected by neighbors. The second item of Eq. (10) is the probability of nodes with degree k recover to susceptible. The parameter µk = 1 − (1 − µ0 )ωrk , where rk = k

∑ (k ′ − 1)p(k ′ ) ⟨k⟩

k′

[1 − Ik′ (t)]

k ′α . α α } max{kmin , kmax

(11)

For small value of µ0 , we obtain µk ≈ µ0 ωrk . In the steady state, t → ∞, we get dIk (t)/dt = 0, and further obtain ∑ (k ′ − 1)p(k ′ )

∑ (k ′ − 1)p(k ′ )

k ′α , α α } ⟨k⟩ ⟨k⟩ max{kmin , kmax k′ k′ (12) where Ik = Ik (∞). Near the critical point βc , neglecting the second order terms of Ik in Eq. (12), we get β(1−Ik )

β

Ik′ = µ0 ωIk

∑ (k ′ − 1)p(k ′ ) ⟨k⟩

k′

where ∆α = µ0 ω

Ik′ = Ik ∆α ,

∑ (k ′ − 1)p(k ′ ) k′

⟨k⟩

(1−Ik′ )

k ′α . α α } max{kmin , kmax

(13)

(14)

At the critical point, the following condition should fulfill Λc (M) = 0,

(15)

where the element of matrix M is Mkk′ =

1 β(k ′ − 1)p(k ′ ) − ∆α δkk′ . ⟨k⟩ 6

(16)

Using the theory of matrix analysis, the outbreak threshold is βc =

⟨k⟩∆α . ⟨k⟩ − 1

(17)

Combining Eqs. (14) and (17), we know that the outbreak threshold is correlated the average degree ⟨k⟩ and α. In addition, we find that βc non-monotonously varies with α. Specifically, βc increases with α, and then decreases with α. To locate the maximal value of βc , differentiating Eq. (17) with respect α, and set dβc /dα = 0, we get the optimal value of α as αopt = 0,

(18)

λopt = µ0 ω. c

(19)

and the outbreak threshold is From Eqs. (18)-(19), we know the optimal resource providing strategy is independent on the network topology. 4. Numerical Validation We perform extensive numerical simulations on artificial networks to validate the theoretical analysis above. Each simulation runs long enough to ensure there is either no nodes are infected or the number of infected nodes fluctuates within a small range. In the simulations, the basic recovery rate is set at µ0 = 0.1, and the utilization factor is set at ω = 0.6. To determine the outbreak threshold through simulations, we adopt the susceptibility measure [41] ⟨ρ2 ⟩ − ⟨ρ∞ ⟩2 χ=N ∞ . (20) ⟨ρ∞ ⟩ where ⟨· · · ⟩ denotes the ensemble average, and the value of χ shows peak at the critical point.

4.1. Results on ER networks First of all, we perform simulations on ER network of size N = 5000, the average degree of the network is set at ⟨k⟩ = 8. The initial fraction of infected nodes is set at ρ0 = 0.01. The resutls for three tyipcal values of α, α = −1.0 (yellow triangles), α = 0.0 (red circles) and α = 1.0 (green squares) are shown in Fig. 1. We find that when α = −1.0, the healthy nodes with smaller degrees 7

1 (a)

250

(b)

α=−1.0 α=0.0 α=1.0

0.8 200 ER

0.4

0.02

0.04

β

0.06

150 100

α=0.0 α=−1.0 α=1.0

0.2 0 0

χ

ρ

∞

0.6

50 0 0

0.08

0.02

0.04

β

0.06

0.08

Figure 1: (Color online) Influence of resource contribution strategy on virus spreading. (a) Fraction of infectd nodes ρ∞ at the steady state as a function of transmission rate β at α = −1.0 (yellow triangles), α = 0.0 (red cirles) and α = 1.0 (green squares). Lines represent the theoretical prediction of ρ∞ . (b) The susceptibility measure χ as a function of β for the corresponding values of α respectively. Peaks indicate the outbreak thresholds. Each point is abtained by averaging over 500 independent simulations.

contribute more resources to their infected neighbors than the larger degree nodes, there is an abrupt jump of ρ∞ at the threshold βc , and the threshold is smallest among the three values of α. When, α = 0.0, each healthy nodes allocates one unit resource to its infected neighbors, the phase transition of ρ∞ is continuous, and the threshold is maximum among the three values. When, α = 1.0, the nodes with larger degrees contribute more resources, then we find that both the values of βc and ρ∞ is between the cases of α = −1.0 and α = 0.0. The phase transition of ρ∞ for α = 1.0 is continuous. The peaks in Fig. 1 indicate the thresholds of the phase transition of ρ∞ for the three values of α. When β is below the threshold βc , the virus dies out quickly, and can not spread globally. Otherwise, i.e., β > βc , the global virus outbreaks. Our theoretical predictions agree well with numerical simulations. The deviations between the theoretical and numerical predictions are induced by the strong dynamical correlations among the states of neighbors and finite-size network effect. From above analysis, it can be obtained that both α and β can affect ρ∞ and phase transition. We find that the plane is divided into two regions, i.e., virus free and global virus regions, by the threshold βc . When β ≤ βc , there is no virus in 8

Figure 2: (Color online) Dependece of ρ∞ on parameters α and β on ER network. Colors represent the values of ρ∞ . Dotted line indicates the optimal value of αopt , and white dots denote the outbreak threshold βc obtained from simulations. Green line stands for the theoretical prediction of βc . Initial fraction of infected nodes is set at ρ∞ = 0.01. the system, i.e., virus free region. The global virus outbreaks when β > βc . Thus, to gain further insights into the effects of parameters α and β on the spreading dynamics, the dependence of ρ∞ on α and β is shown in Fig. 2. Colors in Fig. 2 represent the value of ρ∞ . From the figure, we can see that there is an optimal value αopt = 0, where the βc reaches maximum (see dotted line), and ρ∞ stays lowest. The phase transition of ρ∞ is continuous in the vicinity of α = 0.0, and then changes to discontinuous [42] with the decrease (increase) of α. In addition, we find that there is a lower threshold βc when α < 0 than α > 0. Namely, when nodes with large degrees contribute more resources, the virus can be suppressed better. Since in a homogeneous network, the degree of nodes are concentrated around the mean degree ⟨k⟩ according to the degree distribution of ER network [43]. Thus the vast majority of of nodes in the network have relatively large degrees, i.e., P (ki ≥ ⟨k⟩) is large, the fraction of small degree nodes decay exponentially, i.e., P (ki < ⟨k⟩). Thus when the recovery of infected nodes are dependent on the small degree nodes, there would be not enough resources contributed by these nodes to control the outbreaks of virus. Consequently, there is a smaller value of βc when α < 0 than α > 0. White dots in Fig. 2 denote the 9

thresholds at various α abtained from the peaks of χ. Green line stands for the theoretical prediction of βc obtained from Eq. (15), which matches well with the simulations. We can learn from above results that in ER networks, there is an optimal strategy of resource contribution (αopt = 0.0), where the virus can be suppressed to the most extend as predicted by theoretical analysis. In addition, the strategy of resource contribution can alter the phase transition of ρ∞ from continuous transition to discontinuous transition. 4.2. Effect of degree heterogeneity We next explore the effect of degree heterogeneity on the spreading dynamics. To build networks with heterogeneous distributions P (k) = ck −γ , where γ ∑ degree is the degree exponent and c = 1/ k k −γ , we adopt the uncorrelated configuration model (UCM) [44]. In simulations, network size is set at N = 5000, average degree ⟨k⟩ = 8 and γ = 2.4. 1 (a)

80 (b)

α=−1.0 α=0.0 α=1.0

60 SF

0.4

χ

0.6

ρ

∞

0.8

40

α=0.0 α=−1.0

0.2

20

α=1.0

0 0

0.02

0.04

β

0.06

0 0

0.08

0.02

β

0.04

0.06

Figure 3: (Color online) Influence of resource contribution strategy on virus spreading on scale-free network. (a) Infected fraction ρ∞ at the steady state versus β for α = −1.0 (squres), α = 0.0 (circles) and α = 1.0 (triangles). Lines represent the theoretical prediction of ρ∞ . (b) Susceptibility measure χ as a function of β. Peaks indicate the outbreak thresholds. Each point is abtained by averaging over 500 independent simulations.

Figure 3 exhibites the results for α = −1.0 (squares), α = 0.0 (circles) and α = 1.0. We find that when α = 0.0 (triangles), the value of βc is maximux, and ρ∞ is minimum at each fixed transmission rate β. While, when the large 10

degree nodes contribute more resources, i.e., α = 1.0, virus is most likely to breakout. The threshold βc is minimum, and ρ∞ is maximum. When α = 1.0, there is a relatively larger βc and smaller spreading size ρ∞ compared to α = −1.0. In addition, we find that the phase transitions of ρ for the three values of α are continuous. Lines in Fig. 3 stand for the theoretical results abtained from iteration of Eq. (7), which match well with the simulation results. Peaks in Fig. 3 (b) show the outbreak thresholds for the three values of α respectively.

Figure 4: (Color online) Phase diagram on parameter space (α, β) on scale-free network. Colors represent the value of ρ∞ . The dotted line indicates αopt , and the white dots denote the outbreak threshold obtained from simulations. Green line represents the outbreak threshold predicted by theoretical analysis. Initial infected fraction is set at ρ0 = 0.01.

To systematically study the effect of parameters α and β on the spreading dynamics, we plot the phase diagram of ρ∞ on parameter space (α, β). As shown in Fig. 4, colors represent the values of ρ∞ . When α = 0.0, there is an optimal value of βc and ρ∞ , as predicted by theoretical analysis. The plane is divided into virus free and global virus regions by the threshold βc . When β ≤ βc , there is no virus in the system, i.e., virus free region. The global virus outbreaks when β > βc . The value of βc reaches maximum, and the infected fraction ρ∞ reaches minimum at α = 0.0. Interestingly, α does not alter the phase transition type of ρ∞ on networks with heterogeneity degree distribution. The infected fraction ρ∞ always increase continuously with β in the whole parameter space, which is in 11

contrast to spreading dynamics on ER networks (see Fig. 2). In addition, we find that virus breaks out more easily when α > 0 than when α < 0. When α > 0, the threshold βc is smaller than that of α < 0, and the infected fraction ρ∞ is larger. Similar to the explanation in Sec. 4.1, we can explain the phenomenon as follow. Since in scale-free networks, the majority of nodes are small degree nodes according to the degree distribution in scale-free networks [45]. Thus, if resources are mainly contributed by the small degree nodes, there would be much more resources available in the network to cure the infected nodes. On the contrary, there would be not enough resources available if resources are contributed by the large degree nodes. Consequently, the virus breaks out more easily when α > 0 than α < 0, which is opposite to the case of ER networks. Note that, the white dots denote the threshold βc obtained from the susceptibility measure, and green line represents the threshold predicted by Eq. (15) that matches well with the simulation result. 5. Discussions In this paper, we studied how heterogeneous recovery rate induced by heterogeneous resources supply of healthy nodes affects the spreading dynamics of virus on complex networks. To describe the spreading dynamics, an virus model based on the classical SIS model is proposed. Specially, we considered that the recovery of the infected individuals need resources which are contributed by local healthy (susceptible) neighbors. Thus, depending on the neighborhood of each infected individuals, the recovery rate is diverse from each other. We analyzed the spreading dynamics theoretically by adopting two different approaches. Namely the discrete Markov chains approach and the heterogeneous mean-field approach. From extensive theoretical analysis and experimental simulations we find that on both homogeneous and heterogeneous networks, when heathy individuals contributes equally to their infected neighbors (αopt = 0), the virus could be suppressed optimally, there is maximum threshold βc and minimum fraction of infected nodes ρ∞ . In addition, we found that in homogeneous networks (ER), the heterogeneous recovery rate change the phase transition from continuous transition to discontinuous phase transition when there is a largely bias of α. Whereas, in heterogeneous network, there is always continuous phase transition. Thus we found that the heterogeneity of networks can also alter the phase transition type. Our results could provide some insight into controlling the malicious viruses spreading in the Internet. The theoretical method can be applied to other dynamics on networks, such as information spreading and infectious diseases spreading with an SIR model. In 12

this work, we studied the recovery rate of nodes following a nonlinear function, other expression of the recovery rate should be addressed in the future works, such as a linear function. 6. Acknowledgements This work was supported by the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No.KJ1704085), and Chongqing High-Tech Research Program (Grant No.cstc2017jcyjAX0164). [1] R. Pastor-Satorras, C. Castellano, P. Van Mieghem, A. Vespignani, Epidemic processes in complex networks, Reviews of Modern Physics 87 (3) (2015) 925. [2] M. E. Newman, Spread of epidemic disease on networks, Physical Review E 66 (1) (2002) 016128. [3] R. Zhang, D. Zeng, S. Zhong, Y. Yu, Event-triggered sampling control for stability and stabilization of memristive neural networks with communication delays, Applied Mathematics & Computation 310 (C) (2017) 57–74. [4] R. Pastor-Satorras, A. Vespignani, Epidemic dynamics and endemic states in complex networks, Physical Review E 63 (6) (2001) 066117. [5] W. Wang, M. Tang, H. E. Stanley, L. A. Braunstein, Unification of theoretical approaches for epidemic spreading on complex networks, Reports on Progress in Physics 80 (2017) 036603. [6] L. Wang, J. T. Wu, Characterizing the dynamics underlying global spread of epidemics, Nature Communications 9 (1) (2018) 218. [7] Z.-K. Zhang, C. Liu, X.-X. Zhan, X. Lu, C.-X. Zhang, Y.-C. Zhang, Dynamics of information diffusion and its applications on complex networks, Physics Reports 651 (2016) 1–34. [8] W. Wang, M. Tang, H. Yang, Y. Do, Y.-C. Lai, G. Lee, Asymmetrically interacting spreading dynamics on complex layered networks, Scientific Reports 4 (2014) 5097. [9] H. Chen, G. Li, H. Zhang, Z. Hou, Optimal allocation of resources for suppressing epidemic spreading on networks, Physical Review E 96 (1) (2017) 012321. 13

[10] Z. Yang, T. Zhou, Epidemic spreading in weighted networks: an edge-based mean-field solution, Physical Review E 85 (5) (2012) 056106. [11] O. Ya˘gan, V. Gligor, Analysis of complex contagions in random multiplex networks, Physical Review E 86 (3) (2012) 036103. [12] M. Bogun´a, R. Pastor-Satorras, A. Vespignani, Absence of epidemic threshold in scale-free networks with degree correlations, Physical Review Letters 90 (2) (2003) 028701. [13] M. Bogu˜na´ , C. Castellano, R. Pastor-Satorras, Nature of the epidemic threshold for the susceptible-infected-susceptible dynamics in networks, Physical Review Letters 111 (6) (2013) 068701. ´ Serrano, M. Bogun´a, Percolation and epidemic thresholds in clustered [14] M. A. networks, Physical Review Letters 97 (8) (2006) 088701. [15] J. C. Miller, Epidemic size and probability in populations with heterogeneous infectivity and susceptibility, Physical Review E 76 (1) (2007) 010101. [16] C. Kamp, M. Moslonka-Lefebvre, S. Alizon, Epidemic spread on weighted networks, PLoS Comput Biol 9 (12) (2013) e1003352. [17] P. Rattana, K. B. Blyuss, K. T. Eames, I. Z. Kiss, A class of pairwise models for epidemic dynamics on weighted networks, Bulletin of Mathematical Biology 75 (3) (2013) 466–490. [18] M. Karsai, N. Perra, A. Vespignani, Time varying networks and the weakness of strong ties, arXiv preprint arXiv:1303.5966. [19] X. Wang, X. Liu, K. She, S. Zhong, Pinning impulsive synchronization of complex dynamical networks with various time-varying delay sizes, Nonlinear Analysis Hybrid Systems 26 (2017) 307–318. [20] N. Perra, B. Gonc¸alves, R. Pastor-Satorras, A. Vespignani, Activity driven modeling of time varying networks, arXiv preprint arXiv:1203.5351. [21] X. Zhang, S. Boccaletti, S. Guan, Z. Liu, Explosive synchronization in adaptive and multilayer networks, Physical review letters 114 (3) (2015) 038701.

14

[22] X. Zhang, X. Hu, J. Kurths, Z. Liu, Explosive synchronization in a general complex network, Physical Review E 88 (1) (2013) 010802. [23] J.-B. Wang, L. Wang, X. Li, Identifying spatial invasion of pandemics on metapopulation networks via anatomizing arrival history, IEEE transactions on cybernetics 46 (12) (2016) 2782–2795. [24] D. He, R. Lui, L. Wang, C. K. Tse, L. Yang, L. Stone, Global spatio-temporal patterns of influenza in the post-pandemic era, Scientific reports 5 (2015) 11013. [25] L. Wang, X. Li, Spatial epidemiology of networked metapopulation: An overview, Chinese Science Bulletin 59 (28) (2014) 3511–3522. [26] Y. Moreno, R. Pastor-Satorras, A. Vespignani, Epidemic outbreaks in complex heterogeneous networks, The European Physical Journal B 26 (4) (2002) 521–529. [27] W. Wang, M. Tang, H.-F. Zhang, H. Gao, Y. Do, Z.-H. Liu, Epidemic spreading on complex networks with general degree and weight distributions, Physical Review E 90 (4) (2014) 042803. [28] F. M. Neri, F. J. P´erez-Reche, S. N. Taraskin, C. A. Gilligan, Heterogeneity in susceptible–infected–removed (sir) epidemics on lattices, Journal of The Royal Society Interface (2010) rsif20100325. [29] F. M. Neri, A. Bates, W. S. F¨uchtbauer, F. J. P´erez-Reche, S. N. Taraskin, W. Otten, D. J. Bailey, C. A. Gilligan, The effect of heterogeneity on invasion in spatial epidemics: from theory to experimental evidence in a model system, PLoS computational biology 7 (9) (2011) e1002174. [30] D. Smilkov, C. A. Hidalgo, L. Kocarev, Beyond network structure: How heterogeneous susceptibility modulates the spread of epidemics, Scientific reports 4 (2014) 4795. [31] J. Cheng, H. Zhu, S. Zhong, F. Zheng, Y. Zeng, Finite-time filtering for switched linear systems with a mode-dependent average dwell time , Nonlinear Analysis Hybrid Systems 15 (2015) 145–156. [32] K. Shi, X. Liu, H. Zhu, S. Zhong, New master-slave synchronization criteria of chaotic lure systems with time-varying-delay feedback control, Applied Mathematics & Computation 282 (C) (2016) 137–154. 15

[33] P. Rodrigues, A. Margheri, C. Rebelo, M. G. M. Gomes, Heterogeneity in susceptibility to infection can explain high reinfection rates, Journal of theoretical biology 259 (2) (2009) 280–290. [34] P. Shu, W. Wang, M. Tang, P. Zhao, Y.-C. Zhang, Recovery rate affects the effective epidemic threshold with synchronous updating, Chaos 26 (2) (2016) 063108. [35] L. B¨ottcher, O. Woolley-Meza, N. A. Ara´ujo, H. J. Herrmann, D. Helbing, Disease-induced resource constraints can trigger explosive epidemics, Scientific Reports 5 (2015) 16571. [36] X. Chen, T. Zhou, L. Feng, J. Liang, F. Liljeros, S. Havlin, Y. Hu, Non-trivial resource amount requirement in the early stage for containing fatal diseases, arXiv preprint arXiv:1611.00212. [37] X. Chen, W. Wang, S. Cai, H. E. Stanley, L. A. Braunstein, Optimal resource diffusion for suppressing disease spreading in multiplex networks, arXiv preprint arXiv:1801.03632. [38] X. Chen, R. Wang, M. Tang, S. Cai, H. E. Stanley, L. A. Braunstein, Suppressing epidemic spreading in multiplex networks with social-support, New Journal of Physics 20 (1) (2018) 013007. [39] B. Sch¨onfisch, A. de Roos, Synchronous and asynchronous updating in cellular automata, BioSystems 51 (3) (1999) 123–143. [40] S. G´omez, A. Arenas, J. Borge-Holthoefer, S. Meloni, Y. Moreno, Discretetime markov chain approach to contact-based disease spreading in complex networks, EPL (Europhysics Letters) 89 (3) (2010) 38009. [41] S. C. Ferreira, C. Castellano, R. Pastor-Satorras, Epidemic thresholds of the susceptible-infected-susceptible model on networks: A comparison of numerical and theoretical results, Physical Review E 86 (4) (2012) 041125. [42] X. Chen, C. Yang, L. Zhong, M. Tang, Crossover phenomena of percolation transition in evolution networks with hybrid attachment, Chaos: An Interdisciplinary Journal of Nonlinear Science 26 (8) (2016) 083114. [43] P. Erd¨os, A. R´enyi, On random graphs i, Publicationes Mathematicae 4 (1959) 3286–3291. 16

[44] M. Catanzaro, M. Bogu˜na´ , R. Pastor-Satorras, Generation of uncorrelated random scale-free networks, Physical Review E 71 (2) (2005) 027103. [45] A. L. Barabasi, Scale-free network, Scientific American 288.

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