Virus-Information Coevolution Spreading Dynamics on Multiplex Networks

Wang, Jian; Qin, Xiaolin; Fang, Hongying

doi:https://doi.org/10.1155/2021/6624612

Complexity

On this page

Abstract Introduction Discussion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Special Issue

Coevolving Spreading Dynamics of Complex Networks

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 6624612 | https://doi.org/10.1155/2021/6624612

Virus-Information Coevolution Spreading Dynamics on Multiplex Networks

Jian Wang,^1,2,3Xiaolin Qin,²and Hongying Fang⁴

Academic Editor: Ning Cai

Received10 Dec 2020

Revised22 Dec 2020

Accepted10 Feb 2021

Published05 Mar 2021

Abstract

Virus and information spreading dynamics widely exist in complex systems. However, systematic study still lacks for the interacting spreading dynamics between the two types of dynamics. This paper proposes a mathematical model on multiplex networks, which considers the heterogeneous susceptibility and infectivity in two subnetworks. By using a heterogeneous mean-field theory, we studied the dynamic process and outbreak threshold of the system. Through extensive numerical simulations on artificial networks, we find that the virus’s spreading dynamics can be suppressed by increasing the information spreading probability, decreasing the protection power, or decreasing the susceptibility and infectivity.

1. Introduction

Coevolution spreading dynamics, ranging from cyberspace security to epidemic contagions, widely exist in the natural systems, in which there are at least two spreading dynamics evolving and interacting simultaneously [1, 2]. For instance, the virus’s information is always spreading on the social network when a computer virus spreads on the Internet. The users whose computers are not infected by the virus will install antivirus software and patches to protect their computers from being infected by the virus [3–6]. In this way, computer viruses can be prevented from spreading widely. Another example is the spreading of the epidemic in society. When a global pandemic was spreading, various kinds of information about the pandemic, such as protecting healthy individuals from infection, spreading on social networks will suppress the pandemic [7–9].

Researchers from the field of computer science and network science have developed some successful mathematical models to model such coevolution spreading dynamics. The state of the art in this field is reviewed in a recent paper by Wang et al. [1]. Historically, Newman [10] studied two viruses spreading on the same computer network in succession, where the two viruses follow the susceptible-infected-recovered model, and the second virus can only infect the remaining susceptible nodes. Using a bond percolation theory, he revealed that a global outbreak of the second virus is possible only if the susceptible nodes form a large cluster of connections and the outbreak threshold of the second virus is much higher than the threshold of the first. Newman and Ferrario [11] further discussed a different situation, i.e., the second virus can only spread on those infected nodes by the first virus. They found that the second virus’s outbreak size can be suppressed by decreasing the spreading probability of the first virus. In reality, the spreading dynamics are always simultaneous. Karrer and Newman [12] proposed a model to include this factor and studied the phase transition by using a competing percolation theory.

The two spreading dynamics always evolve on different networks; that is to say, we should use multiplex or multilayer networks to describe the network topology of the coevolution spreading [13–18]. Previous studies have revealed that the topology of multilayer networks markedly affect the dynamics, such as cascading failures [19–21], virus spreading [22, 23], controllability [24], and synchronization [25–27]. For virus-information spreading on multiplex networks, Granell and her colleagues [8] used an unaware-aware-unaware-susceptible-infected-susceptible (UAU-SIS) model. They revealed a metacritical critical point, above which the global virus will break out by using a generalized Markovian approach. Based on the research framework in Reference [8], researchers studied the global information [8], network topology [28, 29], and different interacting mechanisms [30, 31] on the virus spreading. Wang et al. revealed the asymmetric interaction between the virus and information spreading dynamics. They used a susceptible-informed-recovered-susceptible-infected-recovered-vaccination (SIR-SIRV) model. They found that the information can suppress the virus spreading greatly, especially when there is a positive correlation between layers.

Many real-world data analyses proved that the spreading dynamics on the network are heterogeneous. There are three aspects. On the one hand, the network topology is heterogeneous, e.g., heterogeneous degree distribution. Scholars revealed that heterogeneous degree distribution could decrease the virus’s outbreak threshold spreading [32–34]. An important result is that Pastor-Satorras and Vespignani [32] used a heterogeneous mean-field theory to describe the computer virus spreading on the Internet and revealed that the existence of some hubs may make the outbreak threshold vanish. On the other hand, infectivity and susceptibility are heterogeneous since different computers have distinct circumstances. Miller [35] revealed that the global virus is more likely to break out for homogeneous infectivity when the average transmissibility is fixed. In addition, he found that the attack rate was highest when the susceptibility was homogeneous and lowest when the variance was maximum. Lastly, the virus and information always transmit through different networks. Generally, the virus spreads on the computer network and the information transmits on the social network. Therefore, the virus-information dynamics spreading on two-layered multiplex networks are more realistic. Previous paragraphs have stated the state-of-the-art progresses for virus-information spreading on multiplex networks. To our best knowledge, systematic study still lacks for the interacting spreading dynamics including the above three aspects. In the paper, we first describe the mathematical model in Section 2. In Section 3, we develop a heterogeneous mean-field theory to describe the spreading dynamics. In Section 4, we perform extensive numerical simulations. Finally, we conclude the paper in Section 5.

2. Model Descriptions

In this section, we propose the virus-information coevolution spreading model on computer-social network . We first introduce the computer-social network and then present the virus-information spreading model.

2.1. Computer-Social Network

We denote the two subnetworks as and , respectively. The computer virus spreads on subnetwork , and the information spreads on subnetwork . In subnetwork , nodes represent the computers (users), and the edges stand for the relationships among computers (users). To build the two-layered complex networks, we use the following steps: (i) assigning the subnetwork sizes ; (ii) building subnetworks and by using the uncorrelated configuration model [36] (the degree distributions of subnetworks and are and , respectively); and (iii) randomly matching nodes in two subnetworks. Specifically, we build an interlayer connection for node and , which means that the user uses computer . By using the above methods, there are node inter- and intradegree correlations. As shown in Figure 1, we illustrate the computer-social network.

Mathematically, the computer-social network can be represented by a adjacency matrix , where and , respectively, stand for the adjacency matrixes of subnetworks and . An element of subnetwork means that nodes and are connected. The matrixes and are the adjacency matrixes of interlayer network, where means that node uses computer . Note that for any values of and . The average degrees of the two subnetworks can be denoted as and , respectively.

2.2. Virus-Information Spreading Model

We here adopt a susceptible-infected-recovered (SIR) model to describe the virus spreading on subnetwork . A node in the susceptible state means that it does not get infected by the computer virus. An infected node represents that it is infected by the virus and can transmit it to one of its neighbors. A node in the recovered state means that it has recovered and does not change its state.

For the information spread on subnetwork , we consider using the irreversible susceptible-informed-recovered (SIR) model [37]. A susceptible node means that it does not obtain information about the virus. An informed node indicates that it knows information about the virus and is willing to share it with its neighbors. A node in the recovered state means that it loses interest in the information and will not transmit it to its neighbors. In this paper, we denote the virus-information coevolution spreading as a SIR-SIR model.

The virus-information coevolution spreading dynamic evolves as follows. Initially, we randomly select a node in subnetwork , that is to say, the computer virus infects node . The corresponding node , i.e., the user of computer , is also set to be the infected state, since the user can release the information of his infection to his neighbors. At every time step , each infected node in subnetwork tries to transmit the computer virus to one of its neighbors , since every infected node usually communicates with one neighbor at a short time interval. In reality, the infection transmission depends on the “source” and “target” nodes [35]. That is to say, the infectivity and susceptibility of the system are distinct for different nodes. To include this factor, we assume that nodes’ infectivity and susceptibility depend on the degree of nodes. More specifically, the infectivity of node with degree iswhere is the unit infectivity for a node with degree 1. Similarly, the susceptibility of node with degree iswhere is the unit susceptibility for node with degree 1. Varying the values of and , we get different infectivities and susceptibilities of the system. If node is susceptible, two different situations should be considered. (i) If the user of computer is in the susceptible state, the computer infects the virus with probability . Meanwhile, node obtains the information. Otherwise, is in the infected or recovered state, and nothing happens. (ii) If the user has already obtained the information before, the computer is infected by the virus with probability , where . We here use the parameter to describe the degree of protection when a user knows the virus’s spreading. The smaller the value of , the stronger the protection against computer viruses. Every infected node recovers with probability .

The information about the virus spread on the subnetwork is as follows. At each time step , every informed node transmits the information to one of its neighbor in subnetwork depending on the infectivity of and the susceptibility of . The infectivity of with degree iswhere is the unit infectivity. The susceptibility of with degree iswhere is the unit susceptibility. The infection probability is . Finally, every informed node loses interest in transmitting the information with probability . The spreading ends when there are no nodes in the infected or informed state. In Table 1, we present the definitions of parameters and abbreviations.

3. Heterogeneous Mean-Field Theory

In this section, we develop a heterogeneous mean-field approach to describe the evolution of the virus-information spreading dynamics. In theory, we assume that nodes with same degrees have the same infection probability in statistical [32, 33, 38, 39]. In other words, the probability of nodes with the degree is the same as each other.

We use the following parameters to describe the coevolution process. Denote , , and as the probability that a node with degree is in the susceptible, infected, and recovered states at time in subnetwork , respectively. Similarly, we denote , , and as the probability of node with degree in the susceptible, informed, and recovered states at time in subnetwork , respectively. Considering the degree distributions of subnetworks and , we know the fraction of nodes in each state. For instance, the fraction of nodes in the susceptible state at time is . In the final state, i.e., , the fraction of nodes in the susceptible state is .

Now we derive the expressions of the probability of nodes in each state. We know that decreases with time when nodes are infected by the virus. A susceptible computer with degree infected by the virus has two situations. (1) The corresponding node (i.e., the user) of node is in the susceptible state. In this situation, the infection transmitted to node should fulfill two necessary conditions.

(i) An infected neighbor of node contacts node . In uncorrelated complex networks, the probability of node connecting to an infected neighbor with degree is is , where is the degree of node . Considering the degree distribution of subnetwork , the average probability that a node connects to an infected neighbor through an edge is (ii) The infection is transmitted successfully with probability . According to the description of the model, for a node in the susceptible state, its corresponding node must also be in the susceptible state. However, the opposite situation does not always exist. Combining conditions (i) and (ii), we know situation (1) happens with probability . Situation (2) indicates that node of node is in the informed state. Using a similar discussion with situation (1), we obtain the probability of situation (2) as follows: , where is the probability that the corresponding node of is informed by neighbors in subnetwork at time . will be defined later. The rate equation of is

The evolution of iswhere is the fraction of nodes recovered at time . Finally, the evolution of is

According to equations (6)–(8), we obtain the evolution of computer virus spreading on subnetwork .

Now, we study the rate equations of the information about the virus spreading on social network . There are two different situations for the reduction of . For the first situation, the susceptible node with degree is infected by its informed neighbor . The infection probability is , where denotes the probability of an edge connecting to an informed neighbor in subnetwork . The expression of can be expressed as

The second situation of node obtaining the information is that the corresponding node of is infected by the computer virus through an edge of with probability . Since and are randomly coupled, the averaged infection probability of is , where . Combining the two situations, we obtain the rate equation of as

With the similar discussion about the virus spreading on subnetwork , we have

With the above equations, we know the fraction of nodes in each state at subnetworks and .

In the following, we study the global outbreak conditions of the computer virus and information spreading. For global information outbreak condition, we can linearize equations (7) and (11) around the initial conditions, i.e., , . We know that , , , and are trivial solutions. Denote a vector , where , , and and represent the maximal degrees of subnetworks and , respectively. We perform a Taylor expansion for equations (7) and (11) at , , , and and neglect the high order of . We havewhere is the Jacobian matrix. The expression of is

The Jacobian matrix can be further expressed as block matrix aswhere dimensions of , , , and are , , , and , respectively. When the global information on subnetwork breaks out, the largest eigenvalue of is larger than zero. The global information outbreak condition iswhere is the largest eigenvalue of . For the virus outbreak condition, we cannot solve it directly. When the network is extensive, we can use the competing percolation theory [40]. That is to say, we can process the information spreading on subnetwork first and then the virus spreading on subnetwork .

4. Simulation Results

In this section, we perform numerical simulations to study the virus-information spreading dynamics on computer-social network. To build the computer-social network, we use the uncorrelated configuration model [36]. We set the degree distributions of subnetworks and as and , respectively, where and , respectively, represent the degree exponents. There is no inter- and intralayer degree-degree correlations. In numerical simulations, we set the average degrees of the two subnetworks as , the network sizes as , and the degree exponent as . We set and . Initially, we randomly select 5 seeds in subnetwork . All results presented in this paper are averaged over at least 2000 times.

We first study the virus and information spreading sizes, respectively, denoted as and , versus virus transmission probability as shown in Figure 2. We find that increases with , i.e., the virus is more likely to spread when the infectivity and susceptibility are large. Specifically, we note that the virus cannot spread for any values of when are small, e.g., and 0.2. When are large enough, enlarging their values cannot promote the virus spreading. When comparing the effects of information spreading on virus spreading, i.e., increasing , the virus spreading can be suppressed, as shown in Figures 2(a)–2(d). That is to say, to contain the virus spreading, we can transmit more information about the virus on the social network. For the information spreading on subnetwork , i.e., the social network, we find that increases with , , and , since the users have more chances to obtain the information.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

In Figure 3, we further investigate the effects of protection power on the virus-information spreading for different values of and . Generally speaking, we find similar results with that discussed in Figure 2. When the protection power is large, we find that the virus spreading size is relatively smaller, i.e., , since the susceptible nodes are less likely to be infected by neighbors. We also note that since the promotion of virus spreading on information spreading is decreased.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

In Figure 4, we study the effects of susceptibility and infectivity in detail. We find that decreases with the increase of susceptibility and infectivity of . That is to say, the virus spreading can be suppressed by increasing the susceptibility and infectivity. We can explain the results as follows. Increasing susceptibility and infectivity, the information will be widely spread on social network (see Figures 4(e)–4(g)), and more susceptible nodes in subnetwork will take measures to protect themselves from being infected. As a result, decreases with .

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Finally, we studied the virus-information spreading as a function of when the protection power is lower with in Figure 5. We reveal similar phenomena as shown in Figure 4. We note that and since the protection power is decreased.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

5. Discussion

In this paper, we studied the virus-information spreading dynamics on computer-social multiplex networks. We first proposed a mathematical model to describe the coevolution spreading dynamics. In this model, we assumed that nodes’ susceptibility and infectivity are heterogeneous and positively correlated with the node’s degree. To describe the spreading dynamics, we adopt a generalized heterogeneous mean-field approach. Using extensive numerical simulations, we revealed that the virus spreading dynamics can be significantly suppressed by promoting the information spreading on the computer network or decreasing the susceptibility and infectivity of nodes. Our results provide some insight into containing the virus spreading.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This study was partially supported by the Sichuan Science and Technology Program (2020YFG0010).

References

W. Wang, Q.-H. Liu, J. Liang, Y. Hu, and T. Zhou, “Coevolution spreading in complex networks,” Physics Reports, vol. 820, p. 1, 2019.
View at: Google Scholar
S. Lehmann and Y.-Y. Ahn, Complex Spreading Phenomena in Social Systems, Springer, Berlin, Germany, 2018.
P. Szor, The Art of Computer Virus Research and Defense: Art Comp Virus Res Defense _p1, Pearson Education, London, UK, 2005.
L.-X. Yang, X. Yang, L. Wen, and J. Liu, “A novel computer virus propagation model and its dynamics,” International Journal of Computer Mathematics, vol. 89, no. 17, p. 2307, 2012.
View at: Publisher Site | Google Scholar
F. D. Sahneh and C. Scoglio, “Competitive epidemic spreading over arbitrary multilayer networks,” Physical Review E, vol. 89, Article ID 062817, 2014.
View at: Publisher Site | Google Scholar
L.-X. Yang, X. Yang, and Y. Y. Tang, “An effective rumor-containing strategy,” IEEE Transactions on Network Science and Engineering, vol. 5, p. 2, 2017.
View at: Google Scholar
C. Granell, S. Gómez, and A. Arenas, “Dynamical interplay between awareness and epidemic spreading in multiplex networks,” Physical Review Letters, vol. 111, Article ID 128701, 2013.
View at: Publisher Site | Google Scholar
C. Granell, S. Gómez, and A. Arenas, “Competing spreading processes on multiplex networks: awareness and epidemics,” Physical Review E, vol. 90, Article ID 012808, 2014.
View at: Publisher Site | Google Scholar
P. C. Ventura, Y. Moreno, and F. A. Rodrigues, “The role of time scale in the spreading of asymmetrically interacting diseases,” 2020, http://arxiv.org/abs/2007.02774.
View at: Google Scholar
M. E. Newman, “Threshold effects for two pathogens spreading on a network,” Physical Review Letters, vol. 95, Article ID 108701, 2005.
View at: Publisher Site | Google Scholar
M. E. Newman and C. R. Ferrario, “Interacting epidemics and coinfection on contact networks,” PLoS One, vol. 8, Article ID e71321, 2013.
View at: Publisher Site | Google Scholar
B. Karrer and M. E. Newman, “Competing epidemics on complex networks,” Physical Review E, vol. 84, Article ID 036106, 2011.
View at: Publisher Site | Google Scholar
M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, and M. A. Porter, “Multilayer networks,” Journal of Complex Networks, vol. 2, no. 3, p. 203, 2014.
View at: Publisher Site | Google Scholar
M. De Domenico, A. Solé-Ribalta, E. Cozzo et al., “Mathematical formulation of multilayer networks,” Physical Review X, vol. 3, Article ID 041022, 2013.
View at: Publisher Site | Google Scholar
M. De Domenico, V. Nicosia, A. Arenas, and V. Latora, “Structural reducibility of multilayer networks,” Nature Communications, vol. 6, p. 1, 2015.
View at: Publisher Site | Google Scholar
G. Bianconi, Multilayer Networks: Structure and Function, Oxford University Press, Oxford, UK, 2018.
Z. Wang, L. Wang, A. Szolnoki, and M. Perc, “Evolutionary games on multilayer networks: a colloquium,” The European Physical Journal B, vol. 88, p. 1, 2015.
View at: Publisher Site | Google Scholar
X. Zhang, S. Boccaletti, S. Guan, and Z. Liu, “Explosive synchronization in adaptive and multilayer networks,” Physical Review Letters, vol. 114, Article ID 038701, 2015.
View at: Publisher Site | Google Scholar
J. Gao, S. V. Buldyrev, H. E. Stanley, and S. Havlin, “Networks formed from interdependent networks,” Nature Physics, vol. 8, no. 1, p. 40, 2012.
View at: Publisher Site | Google Scholar
D. Zhou, J. Gao, H. E. Stanley, and S. Havlin, “Percolation of partially interdependent scale-free networks,” Physical Review E, vol. 87, Article ID 052812, 2013.
View at: Publisher Site | Google Scholar
F. Radicchi and A. Arenas, “Abrupt transition in the structural formation of interconnected networks,” Nature Physics, vol. 9, no. 11, p. 717, 2013.
View at: Publisher Site | Google Scholar
S. Gomez, A. Diaz-Guilera, J. Gomez-Gardenes, C. J. Perez-Vicente, Y. Moreno, and A. Arenas, “Diffusion dynamics on multiplex networks,” Physical Review Letters, vol. 110, Article ID 028701, 2013.
View at: Publisher Site | Google Scholar
M. Dickison, S. Havlin, and H. E. Stanley, “Epidemics on interconnected networks,” Physical Review E, vol. 85, Article ID 066109, 2012.
View at: Publisher Site | Google Scholar
M. Pósfai, J. Gao, S. P. Cornelius, A.-L. Barabási, and R. M. D’Souza, “Controllability of multiplex, multi-timescale networks,” Physical Review E, vol. 94, Article ID 032316, 2016.
View at: Publisher Site | Google Scholar
L. V. Gambuzza, M. Frasca, and J. Gómez-Gardeñes, “Intra-layer synchronization in multiplex networks,” EPL (Europhysics Letters), vol. 110, no. 2, Article ID 20010, 2015.
View at: Publisher Site | Google Scholar
L. Tang, X. Wu, J. Lü, J.-a. Lu, and R. M. D’Souza, “Master stability functions for complete, intralayer,” Physical Review E, vol. 99, Article ID 012304, 2019.
View at: Publisher Site | Google Scholar
S. Jalan, V. Rathore, A. D. Kachhvah, and A. Yadav, “Inhibition-induced explosive synchronization in multiplex networks,” Physical Review E, vol. 99, Article ID 062305, 2019.
View at: Publisher Site | Google Scholar
Z. Wang, Q. Guo, S. Sun, and C. Xia, “The impact of awareness diffusion on SIR-like epidemics in multiplex networks,” Applied Mathematics and Computation, vol. 349, p. 134, 2019.
View at: Publisher Site | Google Scholar
C. Zheng, C. Xia, Q. Guo, and M. Dehmer, “Interplay between SIR-based disease spreading and awareness diffusion on multiplex networks,” Journal of Parallel and Distributed Computing, vol. 115, p. 20, 2018.
View at: Publisher Site | Google Scholar
J.-Q. Kan and H.-F. Zhang, “Effects of awareness diffusion and self-initiated awareness behavior on epidemic spreading-an approach based on multiplex networks,” Communications in Nonlinear Science and Numerical Simulation, vol. 44, p. 193, 2017.
View at: Publisher Site | Google Scholar
E. Massaro and F. Bagnoli, “Epidemic spreading and risk perception in multiplex,” Physical Review E, vol. 90, Article ID 052817, 2014.
View at: Publisher Site | Google Scholar
R. Pastor-Satorras and A. Vespignani, “Epidemic dynamics and endemic states in complex networks,” Physical Review E, vol. 63, Article ID 066117, 2001.
View at: Publisher Site | Google Scholar
R. Pastor-Satorras and A. Vespignani, “Epidemic dynamics in finite size scale-free networks,” Physical Review E, vol. 65, Article ID 035108, 2002.
View at: Publisher Site | Google Scholar
R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, “Epidemic processes in complex networks,” Reviews of Modern Physics, vol. 87, no. 3, p. 925, 2015.
View at: Publisher Site | Google Scholar
J. C. Miller, “Natural visions: the power of images in american environmental reform,” Physical Review E, vol. 76, Article ID 010101, 2007.
View at: Publisher Site | Google Scholar
M. Catanzaro, M. Boguñá, and R. Pastor-Satorras, “Diffusion-annihilation processes in complex networks,” Physical Review E, vol. 71, Article ID 027103, 2005.
View at: Publisher Site | Google Scholar
Z.-K. Zhang, C. Liu, X.-X. Zhan, X. Lu, C.-X. Zhang, and Y.-C. Zhang, “Dynamics of information diffusion and its applications on complex networks,” Physics Reports, vol. 651, p. 1, 2016.
View at: Publisher Site | Google Scholar
W. Wang, M. Tang, H. E. Stanley, and L. A. Braunstein, “Unification of theoretical approaches for epidemic spreading on complex networks,” Reports on Progress in Physics, vol. 80, Article ID 036603, 2017.
View at: Publisher Site | Google Scholar
Y. Moreno, R. Pastor-Satorras, and A. Vespignani, “Epidemic outbreaks in complex heterogeneous networks,” The European Physical Journal B, vol. 26, no. 4, p. 521, 2002.
View at: Publisher Site | Google Scholar
B. Karrer and M. E. J. Newman, “Random graphs containing arbitrary distributions of subgraphs,” Physical Review E, vol. 82, Article ID 016101, 2010.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 Jian Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies