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A Game Theoretic Model for Network Virus Protection Iyed Khammassi, Rachid Elazouzi, Majed Haddad and Issam Mabrouki University of Avignon, 84 Avignon, FRANCE

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A Game Theoretic Model for Network Virus Protection Iyed Khammassi, Rachid Elazouzi, Majed Haddad and Issam Mabrouki University of Avignon, 84 Avignon, FRANCE University of Manouba Manouba, Tunisia arxiv: v1 [cs.gt] 14 Oct 214 Abstract The network virus propagation is influenced by various factors, and some of them are neglected in most of the existed models in the literature. In this paper, we study the network virus propagation based on the the epidemiological viewpoint. We assume that nodes can be equipped with protection against virus and the security of a node depends not only on his protection strategy but also by those chosen by other nodes in the network. A crucial aspect is whether owners of device, e.g., either smartphones, machines or tablets, are willing to be equipped to protect themselves or to take the risk to be contaminated in order to avoid the payment for a new antivirus. We model the interaction between nodes as a non-cooperative games where the node has two strategies: either to update the antivirus or not. To this aim, we provide a full characterization of the equilibria of the game and we investigate the impact of the price of protection on the equilibrium as well as the efficiency of the protection at equilibrium. Further we consider more realistic scenarios in which the dynamic of sources that disseminate the virus, evolves as function of the popularity of virus. In this work, the interest in the virus by sources evolves under the Influence Linear Threshold (HILT) model. I. INTRODUCTION The Internet continues to grow exponentially and many applications continue to multiply on the Internet, with immediate benefits to end users. However, these network-based applications and services can pose security risks to devises. Recently many attacks have been launched against business, users and governments, that are attributed to some decentralized online communities acting anonymously in a coordinated manner. However, despite the important efforts spent by the many security companies, researchers, and government institutes, information systems security is still a great concern [1], [2]. One of important security risks is the propagation of some sophistical virus in the internet in which each infected node becomes a new source of infection. The problem of virus propagation has been studied through huge papers focusing on mainly on epidemic thresholds for real and realistic network and immunization policies [3], [4], [5]. Many researchers have taken help of the biological system to study the behavior of spread of virus in a computer network and how to immune the computer system [6]. These epidemic models were very useful in network security modeling and immunization strategy. To manage the network security, lots of efforts have been devoted to study virus propagation and characteristics [7]. To protect from the spread of virus in the network, nodes, e.g., either smartphones, machines or tablets, can use some antivirus software with curing tools. In existing research, there are many perspectives to protect against network infections [8]. An important issue of protection is the frequency of the update in order to provide a protection against new virus. But many of the anti-virus applications are of server-client in which the system may provide the last update especially when there is an intensive epidemic of new virus. But an important fundamental source of difficulty in developing efficient protection is the difficulty to fully observe and control the network. As a consequence, full control and observability is impossible, leading to systems that are vulnerable to local as well as remote attacks. Other factor has to be considered when evaluating the security risk from virus, is the decentralized decision on the protection. Indeed, the protection against virus is typically autonomous nature of decision making in the network and the performance of such security is usually made under the assumption that nodes are willing to use an anti-virus for protection. But, a crucial aspect is whether owners of devices are willing to be equipped to protect them self or to take the risk to be contaminated in order to avoid the payment for a antivirus. Any successful security solution should consider those factors. To date, the problem that attached the most attention from the research community is how efficiency the protection against the virus when the decisions are taken autonomously by nodes according to the cost of antivirus as well as the risk to be contaminated. In fact, the bigger the number of nodes equipped with protection, the lower the infection probability for a node without protection. We model the problem as non-cooperative games in order to establish the behavior of nodes against virus infection. Furthermore, we consider a source S which propagates the virus through the network. We are concerned with spread of sources that disseminate the virus. In particular, we associate the dynamic of sources with the popularity of virus which is measured by the number of infected nodes at least one time. We model this influence process using Homogeneous Influence Linear Threshold (HILT) model [9]. The HILT model focuses on the threshold behavior in influence propagation, which we can frequently relate to when enough of our friends bought a product, we may be influenced and converted to follow the same action. In our context, when a virus reaches some level of popularity, other sources may participate in dissemination of this virus. 2 Fig. 1. Transition condition of activation state. Fig. 2. Flow chart of states transitions. The remainder of this paper is organized as follows. The epidemic model and the network game is introduced in section II. In section III, we study the dynamics and the different equilibrium properties of the proposed security game. Numerical illustrations of the system behaviors and the equilibrium characteristics are given in Section IV. In order to enhance the readability of the paper, the mathematical proofs are given in Appendices. II. NETWORK MODEL Consider a sparse network that consists of a large population of N computer systems (CS) or nodes. We assume that the network is a complete graph where all computers can communicate with each others. A set of sources generates a virus with a rate µ s and nodes in the network becomes susceptible to be infected by this virus. Each node i chooses or not to be equipped in order to protect himself from the virus by paying a relative price of protection. Mixed strategies, i.e, probability distribution over the actions, are also possible. Obviously, all nodes have an incentive to protect themselves until the virus extinction. However, every antivirus update costs a price U c. Hence, the strategy adopted by a node corresponds to a certain utility it receives and this utility depends on actions performed by N nodes. Nodes with outdated antivirus are vulnerable to the virus spread process, and lose an infection cost I c if they were infected. An infected node can recover after a curing time using various tools (e.g., through a clean-up software). Under this setting, nodes shall immunise themselves during the period of the virus spread while minimizing the antivirus update cost. A. Modeling active sources evolution Assume that there are a set of sources that generate the virus. We associate the dynamic of active sources, which still disseminate the virus, with the dynamic of the number of infected nodes. In particular, we associate the dynamic of the number of active sources with the popularity of virus which is measured by the number of infected nodes at least one time. We model this influence process using Homogeneous Influence Linear Threshold (HILT) model [9]. The HILT model focuses on the threshold behavior in influence propagation. The evolution of the number of active sources is modeled following process evolving in continuous time. Each source j chooses a random threshold θ j from an arbitrary threshold distribution with cumulative density function (c.d.f) F. Hence a source becomes active if the popularity, which measured by the number of infected nodes, exceeds its threshold. An active source disseminate the virus during a random time T m, which follows the exponential distribution with rate δ S. Under the HILT model, the following proposition describes the dynamic of the number of actives sources at time t. Proposition 1. The dynamic of the number of active sources that disseminate the virus, is given by f( X(t)) Ṡ(t) = δ S S(t) + λ (N S(t)), 1 F ( X(t)) where X(t) is the number of infected nodes till time t. B. Modeling Infection Dynamic To model the spreading process under the influence of a curing process, we choose the SIS (Susceptible Infected Susceptible) model, which is one of the most studied epidemic models [1], [11]. A node at time t can be in one out of two states: infected or susceptible. We assume that the curing process is a Poisson process with rate δ, after which the node becomes healthy, but susceptible again to the virus. Let X(t) denote the number of infected nodes and each infected node can infect healthy nodes with a contact rate β. Note that the total infection rate of a susceptible node is β times the size of infected nodes. C. The Security Game We consider a security game, in which nodes choose individually whether or not to invest in the protection by updating their antivirus versions. Indeed, nodes prefer to not invest in antivirus update if the network is enough protected, i.e., there is enough nodes in the network that have chosen to update. Each node has two strategies: either to invest in the antivirus protection, i.e., pure strategy update (U), or not to invest, i.e., pure strategy not update (N U). Each strategy corresponds to certain playoff for the node. Notice that, a node may also be protected by the other nodes update: the risk to be infected decreases with the number of antivirus activated throughout the network. Accordingly, the payoff of a node depends on the actions performed by the N 1 nodes. We denote by V j (k U ) the long term fitness of a computer, given that it plays the strategy j {U, NU}, and that k U is 3 the number of updated antivirus. The fitness is given by { Uc j U V j (k U ) = P i I c j / U where P i is the probability to be infected until the virus extinction. III. CHARACTERIZATION OF EQUILIBRIUM The nodes which invest in the antivirus protection are directly immune to the virus. Therefore, they can not infect the other nodes or be infected. The dynamic of sources S(t) depends on the the virus popularity. The activation process describes how the infectious nodes cause the inactive sources to become active. A source j is influenced by the accumulative number of infected nodes X and it will be active when (1) X(t) θ (2) We assume that a node contacts an active source with a rate γ. Before evaluating the dynamics of the the infected nodes X N (t), we study the dynamic of sources S N (t) under the activation process. A source S is active when the number of infected nodes X N (t) reaches the target value θ. Let X(t) be the dynamic of infected nodes disregarding the curing process. The dynamics of X(t) is given by X(t) = (βx(t) + γs(t))(n k U X(t)) Recall that S N (t) is the set of active sources which participate in the infection process by time t, and δ S S N (t) is the set of the sources which are no longer interested to the virus and move from the active state to the susceptible state. A source is influenced by the accumulative infection process with a rate λ. By applying Condition (2), we can write the sources dynamics as follows f( X(t)) Ṡ(t) = δ S S(t) + λ (N S(t)) 1 F ( X(t)) We know that the hazard function [12] for the c.d.f F (.) is given by h F (x) = f(x) 1 F (x) where f(x) is the corresponding density function. Hence, the ordinary differential equation (ODE) becomes Ṡ(t) = δ S S(t) + λh F ( X(t))(N S(t)) The sources contact (N k U X N (t)) susceptible nodes with a rate γ.therefore, we can write the dynamics of X(t) as following Ẋ(t) = δx(t) + (βx(t) + γs(t))(n k U X(t)) This growth equation gives the dynamics of infected nodes under the sources activation process. All nodes aim to be enough protected during the lifetime of the virus. A node i can be infected by a source or by another infected node. The infection probability P i is expressed as follows [ τc ] P i (t) = 1 E e (βx(t)+γs(t)) dt (3) 1) Pure Nash Equilibrium: Definition 1. At a Nash equilibrium (NE), no player can improve its fitness by unilaterally deviating from the equilibrium. For the proposed game a NE in pure strategies exists if and only if the following two conditions are satisfied { Vj (NU, k 1 j N = U 1) V j (U, k U ) (4) V j (NU, k U ) V j (U, k U + 1) We are interested in the existence and uniqueness of the pure NE which is characterized by the number Ψ of players investing in the antivirus. A unique pure NE exists for the proposed security game when V j (NU, ψ) = V j (U, ψ). 2) Mixed Nash Equilibrium: Let us now discuss the case when every node maintain a probability distribution over the two actions (U, NU). The advantage of this mixed equilibrium compared to the pure one is that a node can invest in protection only for a fraction of the time and stay susceptible the rest of the time. This kind of equilibrium is more efficient for our case because we study a homogeneous population with fixed update and infection cost. In a mixed strategy game, a node i can decide to invest in protection (playing U) with probability p i or keep protected only by his neighbors (playing NU) with probability (1 p i ). p = (p 1, p 2,..., p N ), ip i, is the mixed strategy profile. For p i / {, 1} we have a fully mixed strategy profile. We note (p i, p i ) if node i uses strategy p i and other use p i = (p 1,..., p i 1, p i+1,..., p N ). We denote by V j (p, p i ) the playoff of a node i which invest in antivirus with probability p. Definition 2. A mixed strategy p i [, 1] is a NE if for each player i (where i = 1,..., N) we have U i (p 1,..., p i 1, p i, p i+1,., p N) U i (p 1,..., p i 1, p i, p i+1,..., p N) (5) for every mixed strategy p i [, 1]. If i, p i / {, 1} then we call p fully mixed NE. Every finite strategic game has a mixed strategy NE [13]. There exists a unique fully mixed NE p for the proposed game and it is solution of N k=1 C N 1 k 1 (p ) k 1 (1 p ) N k V (U, k) = (6) 4 3) Equilibrium with Mixers and Non-Mixers: The mixers are the players that choose a mixed strategy. We suppose that a part of the population chooses to play a pure strategy U or NU and the rest of the players are mixers. We will study the existence of the equilibrium in this case. Let N U {, 1,..., N} be the number of players choosing the pure strategy U, and N NU {, 1,..., N} be the number of players choosing the pure strategy NU. The N N U N NU players use the mixed strategy. Let p (, 1) be the probability with which the mixers choose the strategy U. Moreover, we denote by V U (N U, N NU, p) the fitness of the node who updates its antivirus and V NU (N U, N NU, p) the fitness for the node who does not update its antivirus. A necessary condition for the strategy (N U, N NU, p ) to be a NE (with at list one mixer) is that the mixer is indifferent whether it chooses a pure strategy U or N U. This translates mathematically as follows V U (N U + 1, N NU, p ) = V U (N U, N NU + 1, p ) (7) A unique NE of type (N U, N NU, p ) exist for this case, and is solution of N N U N NU k= C N N U N NU k 1 (p ) k 1 (1 p ) N N U N NU k V (U, N U + k) =. (8) We prove that this NE of type (N U, N NU, p ) exists only for N U ψ and N U + N NU N 2. In this section, we have studied different NE types under the S(t) activation process. We summarize the different Nash equilibrium types as following: Pure Nash Equilibrium: There exists a unique NE when the utility of U is equal to the utility of NU and we must update exactly ψ nodes to get this equilibrium, Mixed Nash Equilibrium: A unique fully mixed NE p exists and it is solution of Equation (6), Mixer and Non-Mixer Nash Equilibrium: We characterize this equilibrium by a necessary condition (7). A unique NE exists and it is solution of Equation (8). Moreover, we have proved that a NE exists only for N U ψ and for N U + N NU N 2. IV. NUMERICAL EVALUATION In this section, we provide a numerical analysis of the performances of the proposed security game. We first evaluate the infection probability at the equilibrium. To do so we solve Equation (6) to get the activation probability at the equilibrium. We show how the activation and the infection process depend on the system parameters, such as the number of nodes N and the update cost Uc. X(t) p * =.1 p * =.1 p * = t Fig. 3. The infection process for different activation probabilities p, where N = 1, N s = 5, β = 1 1 3, γ = 1 1 3, δ = 1 1 1, δ S = 1 1 1, λ = 5 1 6, X() =, S() = 5, Ic = 1 and Uc =.1. A. System characteristics Fig. 3 and Fig. 4 illustrate the behavior of the infected nodes X(t) and the sources S(t) as function of the time for different activation probabilities (.1,.1,.5) and a contact rate β = We further take X() = and S() = 5. As expected, we remark a cause-effect phenomenon between the nodes and sources. The number of infected nodes increases as a result of the virus spread till reaching a given infection rate. Then, when the virus popularity reaches a certain level, the participating in the virus spread increases yielding an increase in the number of sources. In the proposed game model, activation probability is a fundamental parameter and is related to how many nodes install the new anti-virus software. To analyze effects of the activation probability, p is set to three different values (.1,.1,.5). Fig. 3 shows that the number of infected nodes X(t) slightly fluctuates before reaching a stable (absorbing) state. In general, the higher the activation probability is, the faster X(t) decreases. This is due to the fact that increasing the activation probability implies a decrease in the risk of being infected for susceptible nodes. Notice that, depending on the activation probability, the virus may disappear completely or become scars. We will discuss this point later in the paper. Fig. 4 depicts the dynamics of the interested sources in the virus spread for different p. We clearly notice that, for low activation probability values, e.g., p =.1, S(t) decreases until the virus popularity reaches a target value. When the activation probability increases to p =.5, we can see that the number of sources are decreasing gradually to vanish eventually. Fig. 5 illustrates the time evolution of the infection probability for different activation probabilities p. We remark that, from p =.495, the infection probability monotonically decreases till completely vanishing at t = 23. This suggests that using an activation probability higher that.495 is worthless as, from p =.495, the virus is going to disappear in any case. 5 S(t) p * =.1 p * =.1 p * =.5 Activation probability β/δ=3 1 2 β/δ=5 1 2 β/δ=8 1 2 epidemic threshold t Number of nodes Fig. 4. Sources behaviour for different activation probabilities. Fig. 6. The activation probability for increasing number of nodes. Infection probability 9 x p=.4 p=.42 p=.45 p=.47 p= t Fig. 5. The infection probability for different activation probabilities, where N = 1, β = 1 1 3, γ = 1 1 3, δ = 1 1 1, δ S = 1 1 1, λ = 1 1 4, X() =, S() = 5, Ic = 1 and Uc =.1. Unless otherwise stated, we will use the following parameters: N = 5, N s = 5, β = 1 1 4, γ = 1 1 3, δ =.1, δ S =.1, λ = 1 1 4, X() =, S() = 1, Ic = 1 and Uc =.1. We notice by t f the time of epidemic extinction corresponding to the time for which we have X(t) =. For this parameters, a virus extinction time t f exists and we can compute the infection probability in [,..., t f ]. B. System Performances at the Equilibr

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