Security and Communication Networks

Volume 2018, Article ID 2153195, 9 pages

https://doi.org/10.1155/2018/2153195

## Global Behavior of a Computer Virus Propagation Model on Multilayer Networks

School of Information Engineering, Guangdong Medical University, Dongguan 523808, China

Correspondence should be addressed to Chunming Zhang; moc.361@2002iefnuhc

Received 10 October 2017; Revised 2 February 2018; Accepted 4 March 2018; Published 12 April 2018

Academic Editor: Vasileios A. Karyotis

Copyright © 2018 Chunming Zhang. 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.

#### Abstract

This paper presents a new linear computer viruses propagation model on multilayer networks to explore the mechanism of computer virus propagation. Theoretical analysis demonstrates that the maximum eigenvalue of the sum of all the subnetworks is a vital factor in determining the viral prevalence. And then, a new sufficient condition for the global stability of virus-free equilibrium has been obtained. The persistence of computer virus propagation system has also been proved. Eventually, some numerical simulation results verify the main conclusions of the theoretical analysis.

#### 1. Introduction

In 1987, the first computer viruses propagation model was proposed by Cohen [1]. Since then, numbers of typical computer viruses propagation models had been proposed, for instance, susceptible–infected–susceptible (SIS) models [2], susceptible–infected–removed (SIR) models [3, 4], susceptible–infected–recovered–susceptible (SIRS) models [5], susceptible–exposed–infected–removed–susceptible (SEIRS) models [6], susceptible–infected–patched–susceptible (SIPS) models [7], and susceptible–infected–external–susceptible (SIES) [8]. However, some of these models simply ignore the fact that the dominating majority of computer viruses have a quite long propagation period before breaking out which can vividly express the node with high infectious ability, while some other models assume that, during its latency, an infected computer just infected recently that has low infectious ability compared with breaking out nodes, also known as an* E* computer, has no infectivity. This, however, is inconsistent with the fact that, in general, an infected computer does possess infectivity [9]. To overcome these deficiencies, Yang et al. proposed a novel computer virus propagation model, called Susceptible–Latent–Breaking–Susceptible (SLBS) model, in which all the computers connected to the Internet are divided into three groups: virus-free computers, known as susceptive computer (*S* computer), infected computers that are latent (*L* computer), and infected computers in which the viruses are breaking out that means the computer with a high infectious level (*B* computer). One remarkable distinction between the SLBS model and the classical SEIS model is that a latent computer possesses infecting capability [9].

While the mechanism of computer virus propagation on networks is an important research area, lots of network-based computer viruses propagation models ranging from susceptible–infected (SI) models [2, 10] and SIS models [3–5, 8, 11–14] to SIR models [5, 6, 13, 15, 16] and SLBS models [14] and SIPS models [7] have also been inspected. However, these studies mainly focus on single layer networks [11–13, 15, 17–24]. In reality, computer viruses can spread not only through single layer networks but also through multilayer networks; for example, mobile phone viruses (a type of computer virus) can utilize 3G network, 4G network, Wi-Fi network, and even Bluetooth network as their communication network.

On the other hand, from the point of view of research methods, Markov chain method, which is proposed by Van Mieghem et al. [10, 25], can exactly describe computer viruses propagation process with constant transition rates between compartments on any networks. Nevertheless, this method is complex in mathematical analysis. For the purpose of overcoming this deficiency, several approximate methods of researching computer viruses propagation on networks are also proposed in recent years. For example, based on the assumption that the dynamic state of every node is statistically independent of the state of its nearest neighbors, Wang et al. [26], Youssef and Scoglio [27], and Yang et al. [14] proposed the so-called Individual-based mean-field theory (IBMF); based on the assumption that all nodes of degree are statistically equivalent, Pastor-Satorras and Vespignani [28, 29] and Barthélemy et al. [30] proposed the so-called degree-based mean-field theory (DBMF), and so on.

For the purpose of more accurately understanding the propagation mechanism of computer viruses on multilayer networks, in this paper, we propose a novel SLBS computer virus propagation model on multilayer network. Highlights of this paper are as follows.

Based on the assumption of multilayer network, by applying the IBMF to the existing SLBS model, a high-dimensional computer virus propagation dynamic model, which is known as the individual-based SLBS model, is formulated. This model forms the foundation of this work. To our knowledge, there is no report about the spread of computer viruses on multilayer networks.

To find out the influence of multilayer networks topology on computer virus spreading, by means of mathematical analysis, I find out that the propagation threshold is the maximum eigenvalue of the sum of all the subnetworks on multilayer networks. Then, the global stability of the virus-free equilibrium has been analyzed. The persistence of system has been proved. Extensive experiments confirmed the conclusions of the mathematical analysis.

The subsequent materials of this paper are organized as follows. In Section 2, we present the multilayer networks and computer viruses propagation model in detail; and then in Section 3 the model is analyzed comprehensively; numerical simulation results are given in Section 4; eventually, in Section 5, we outline this work.

#### 2. Assumptions and Modeling

For the purpose of describing the model in detail, the following notations are proposed:(i): the multilayer network, which consists of subnetworks(ii) (): the th layer subnetwork; each subnetwork has nodes(iii): the set of nodes contained in (iv): the set of edges contained in (v): the link from node to node in , (vi): the corresponding parameterized adjacency matrix of graph .

In addition, a dynamic switching network also must satisfy the following conditions:(I).(II) and for all .

Condition (I) presents that nodes in all subnetworks are identical. Condition (II) shows that the edges of any two subnetworks are different.

A simple example of a multilayer network is shown in Figure 1.