Abstract

To restrain escalating computer viruses, new virus patches must be constantly injected into networks. In this scenario, the patch-developing cost should be balanced against the negative impact of virus. This article focuses on seeking best-balanced patch-injecting strategies. First, based on a novel virus-patch interactive model, the original problem is reduced to an optimal control problem, in which (a) each admissible control stands for a feasible patch-injecting strategy and (b) the objective functional measures the balance of a feasible patch-injecting strategy. Second, the solvability of the optimal control problem is proved, and the optimality system for solving the problem is derived. Next, a few best-balanced patch-injecting strategies are presented by solving the corresponding optimality systems. Finally, the effects of some factors on the best balance of a patch-injecting strategy are examined. Our results will be helpful in defending against virus attacks in a cost-effective way.

1. Introduction

Computer networks bring huge convenience to our work and life [1, 2]. Meanwhile, digital viruses can propagate rapidly through computer networks, posing a severe threat to human society. For example, Wanna Decryptor, the notorious ransomware, has recently swept across the globe, leading to massive computer paralysis [3]. Consequently, the problem of how to mitigate the negative impact of computer virus in a cost-effective way has long been a hotspot of research in the field of cyber security [4].

To restrain evolving computer viruses, new virus patches must be constantly injected into networks. In this scenario, there is an obvious conflict between the patch-developing cost and the impact of virus; reducing the former would increase the latter, whereas mitigating the latter would enhance the former. Therefore, the patch-developing cost should be balanced against the impact of virus. We refer to a dynamic patch-injecting strategy that achieves the best balance between the two aspects as a best-balanced patch-injecting strategy, and we refer to the problem of seeking best-balanced patch-injecting strategies as the virus-patch tradeoff (VPT) problem. Solving the VPT problem would be helpful in defending against virus attacks in a cost-effective way.

This article addresses the VPT problem. First, based on a novel virus-patch interactive model, the original problem is reduced to an optimal control problem which we refer to as the VPT control problem, in which (a) each admissible control stands for a feasible patch-injecting strategy and (b) the objective functional measures the balance of a feasible patch-injecting strategy. Second, the solvability of the VPT control problem is shown, and the optimality system for solving the VPT control problem is derived. Next, a few best-balanced patch-injecting strategies are given by solving the corresponding optimality systems. Finally, the effects of some factors on the best balance of a patch-injecting strategy are examined.

The remaining materials are organized in this fashion: Section 2 reviews the related work. Sections 3 and 4 establish and solve the VPT control problem, respectively. Section 5 illustrates how to solve the VPT control problem, and Section 6 examines the effects of some factors on the best balance. This work is summarized by Section 7.

2. Related Work

In order to solve the VPT problem, the expected total loss of all network users resulting from a patch-injecting strategy must be estimated [5, 6]. As this quantity relies on the expected network states at all times, we need to characterize the evolutionary process of the expected network state. The resulting evolutionary model is essentially a propagation model that captures the interactive propagation of viruses and patches [7, 8]. In the available literature, propagation models of this kind are referred to as Susceptible-Infected-Patched-Susceptible (SIPS) models.

Compartmental propagation models are propagation models in which all nodes of the same state are classified as a class, with the goal of understanding the evolutionary trend of the size or fraction of each class [9]. Compartmental models are suited to capturing propagation phenomena occurring on homogeneously mixed networks but fail to characterize propagation phenomena occurring on highly heterogeneous networks. The compartmental SIPS models proposed in [10–13] take patch forwarding into account but leave patch injection out of consideration. Very recently, [14] proposed a compartmental SIPS model with static patch-injecting mechanism and thereby assessed the effectiveness of patch injection.

Node-level propagation models are propagation models in which each node is regarded as a separate class, with the goal of gaining insight into the evolutionary trend of the expected network state [15, 16]. One striking advantage of node-level propagation models is that they can accurately characterize propagation phenomena occurring on arbitrary networks. With the progress of wireless and mobile communication technologies, most existing computer networks admit an irregular topology [17–19]. As a result, a number of node-level computer virus propagation models have been advised [20–25]. In particular, [26] introduced a node-level SIPS model with no patch injection. Recently, [27] proposed a node-level SIPS model with dynamic patch-injecting mechanism and thereby addressed a problem that is something like the VPT problem through differential game approach. In our opinion, this work has two weaknesses: (i) It is assumed that the network defender is aware of the total attack budget of all relevant cyber malefactors. However, in practice the budget is usually unknown to the defender. (ii) It is assumed that new patches can be injected into any network node. Due to the limited network bandwidth, in practice new patches are typically injected into a small subset of nodes and then forwarded to the unpatched nodes [28].

Optimal control theory [29, 30] provides a powerful tool for studying the problem of how to contain the prevalence of computer virus in a cost-effective way [31–35]. In view of the defects of the research approach used in [27], in this paper we deal with the VPT problem through optimal control approach. For this purpose, we propose a novel node-level SIPS model with dynamic patch-injecting mechanism, where new patches can be injected into only a small subset of nodes. Thereby, we accurately estimate the expected total loss of all network users. On this basis, we reduce the VPT problem to an optimal control problem and then solve the problem by means of optimal control theory. Our optimal control model is promising, because, by collecting and analyzing the relevant actual data, the model parameters can be estimated quite accurately.

3. The Modeling of the VPT Problem

This section is devoted to the modeling of the VPT problem. First, we introduce basic terms and notations. Second, we establish a node-level SIPS model. Finally, we model the VPT problem as an optimal control problem.

3.1. Terms and Notations

Consider a computer network with nodes labeled through . Let denote the topology of the network, i.e., , and each edge stands for a communication link between the two endpoints. Let denote the adjacency matrix of , i.e, or 0 according as or not. Suppose new computer viruses can be injected into any node of the network and can propagate over the network, and suppose new virus patches can be injected into only the node subset of the network and can be forwarded to other nodes through the network.

Consider the finite time horizon . Assume each and every node of the network is in one of three possible states: susceptible, infected, and patched. Susceptible nodes are nodes that are not infected with any virus but have not received the newest patch. This implies these nodes are vulnerable to new viruses. Infected nodes are nodes that are infected with some virus. Patched nodes are nodes that are not infected with any virus and have received the newest patch. This implies that these nodes possess temporary immunity to new viruses. Let , 1, and 2 denote that the node is susceptible, infected, and patched at time , respectively. Then the state of the network at time can be characterized by the vectorLet , , and denote the probabilities of the node being susceptible, infected, and patched at time , respectively.Since , the expected state of the network at time can be characterized by the vector

3.2. A Virus-Patch Interactive Model

In order to establish a virus-patch interactive model, we introduce a set of assumptions as follows.(A₁)Due to virus injection, each susceptible node gets infected at the average rate which we refer to as the virus injection rate.(A₂)Due to virus propagation, the susceptible node gets infected at time at the average rate , where is a constant which we refer to as the virus propagation rate.(A₃)Due to patch injection, each unpatched node in gets patched at time at the average rate which we refer to as the patch injection rate at time .(A₄)Due to patch forwarding, the unpatched node gets patched at time at the average rate , where is a constant which we refer to as the patch forwarding rate.(A₅)Due to appearance of new viruses, each patched node becomes susceptible at the average rate which we refer to as the patch failure rate.

Remark 1. The virus injection rate, the virus propagation rate, the patch forwarding rate, and the patch failure rate can be estimated accurately by collecting and analyzing the relevant historical data. All patch injection rates are under control of the network defender.

Figure 1 shows the above assumptions schematically.

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Figure 1 

Diagram of the assumptions (A₁)-(A₅).

Based on the above assumptions, the expected network state evolves over time according to the following differential dynamical system: This is a novel SIPS model, which can be written in matrix-vector notation as

3.3. The Modeling of the VPT Problem

The function defined by , , is under control of the network defender. We refer to the function as a patch-injecting strategy. Let denote the set of all Lebesgue integrable functions defined on the interval [36]. Henceforth, we assume the set of all allowable patch-injecting strategies is We refer to as the minimum allowable patch injection rate, as the maximum allowable patch injection rate.

Let denote the cost per unit time at time for patch developing. Obviously, is increasing with . In this paper we simply assume that is linearly proportional to . That is, , where is a constant which we refer to as the cost coefficient. As a result, the total patch-developing cost is units.

Remark 2. In practice, may be dependent on in a more complex way. For example, may be proportional to the square of . That is, , where is a constant. If this is the case, the total patch-developing cost would be units. The exact form in which depends on is yet to be determined through analysis of massive actual data. Nevertheless, our research approach can easily be applied to any other dependence relationship.

On the other hand, we assume that the average loss per unit time caused by the infected node is unit. Then, the expected total loss of all network users is units. Let . Therefore, we get a measure of the balance of a patch-injecting strategy as follows.

By combining the above discussions, the VPT problem is reduced to the following optimal control problem:We refer to the optimal control problem as the VPT control problem. In this problem, each admissible control stands for a feasible patch-injecting strategy, and the objective functional measures the balance of a feasible patch-injecting strategy. Each instance of the VPT control problem is given by the 11-tuple

4. Theoretical Study of the VPT Control Problem

This section is dedicated to the theoretical study of the VPT control problem. First, we show that the problem is solvable. Second, we present a method for solving this problem.

4.1. Solvability

Let . We have the following lemma [30].

Lemma 3. The VPT game problem (8) admits an optimal control if the following five conditions are met.(C₁) is closed and convex.(C₂)There is such that the differential system is solvable.(C₃) is bounded by a linear function in .(C₄) is concave on .(C₅) for some , and .

The solvability of the VPT control problem is guaranteed by the following theorem.

Theorem 4. The VPT control problem (8) admits an optimal control.

Proof. (a) Let be a limit point of . Then there is a sequence of points in , denoted , that approaches . As is complete [36], we get that . As , we get that . So, is closed. (b) Let , . . As is a real vector space [36], we have . As , we get that . So, is convex. (c) As is continuously differentiable, it follows from Continuation Theorem for Differential Systems [37] that the differential system is solvable. (d) Obviously, for , for , (e) Let , . As we get that is convex with respect to . (f) Obviously, . Hence, the five conditions in Lemma 3 are met. By Lemma 3, the VPT control problem admits an optimal control.
This theorem implies that the VPT problem admits a best-balanced patch-injecting strategy.

4.2. The Optimality System

According to optimal control theory [29], when the solvability of an optimal control problem is guaranteed, we may solve the problem by solving the optimality system associated with the problem. Now, let us derive the optimality system associated with the VPT control problem (8). The associated Hamiltonian iswhere is the adjoint.

The following result is a necessary condition for the optimal control of the VPT control problem.

Theorem 5. Suppose is an optimal control for the VPT control problem (8). Let be the solution to the differential system (5). Then there exists with such that

Proof. According to Pontryagin Minimum Principle [29], there exists such that Thus, the first equations in the system (14) follow by direct calculations. As the terminal cost is unspecified and the final state is free, the transversality condition holds true. Again by Pontryagin Minimum Principle, we have The last equation in the system (14) follows by direct calculations.
The optimality system associated with the VPT control problem (8) consists of the system (5), the system (14), and . In practice, we may apply the well-known Forward-Backward Euler Scheme [38] to solve the optimality system.

5. Examples of Best-Balanced Patch-Injecting Strategy

In this section, we present a few best-balanced patch-injecting strategies by solving the corresponding instances of the VPT control problem. For comparative purpose, for the VPT control problem (8) and the admissible control , we define the cumulative balance function asObviously, . For convenience, let denote an all-one row vector with appropriate number of dimensions.

Small-world networks are networks with small diameter and high clustering coefficient [39]. By invoking Pajek [40], the well-known social network analysis software, we get a synthetic small-world network , which is plotted in Figure 2, where the patch injection subset consists of the red nodes.

Figure 2 

A synthetic small-world network , where consists of the red nodes.

Example 1. Consider the following instance of the VPT control problem: By solving the corresponding optimality system, we get an optimal control, which is depicted in Figure 3(a). Figure 3(b) plots the cumulative balance functions for the optimal control and three static controls, from which it is seen that the optimal control is superior to these static controls in terms of the balance.

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Figure 3 

Results in Example 1: (a) an optimal control; (b) a comparison between the optimal control and three static controls.

Scale-free networks are networks with power-law degree distribution [41]. Again by invoking Pajek, we get a synthetic scale-free network , which is portrayed in Figure 4, where the patch injection subset consists of the red nodes.

Figure 4 

A synthetic scale-free network , where consists of the red nodes.

Example 2. Consider the following instance of the VPT control problem: By solving the corresponding optimality system, we get an optimal control, which is exhibited in Figure 5(a). Figure 5(b) plots the cumulative balance functions for the optimal control and three static controls, from which it is seen that the optimal control outperforms these static controls in terms of the balance.

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Figure 5 

Results in Example 2: (a) an optimal control; (b) a comparison between the optimal control and three static controls.

Figure 6 exhibits a real-world email network , which comes from [42]. Here, the patch injection subset consists of the red nodes.

Figure 6 

An email network , where consists of the red nodes.

Example 3. Consider the following instance of the VPT control problem: By solving the corresponding optimality system, we get an optimal control, which is shown in Figure 7(a). Figure 7(b) plots the cumulative balance functions for the optimal control and three static controls, from which it is seen that the optimal control overmatches these static controls in terms of the balance.

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Figure 7 

Results in Example 3: (a) an optimal control; (b) a comparison between the optimal control and three static controls.

We conclude from the above examples that a best-balanced patch-injecting strategy first stays at the maximum allowable patch injection rate, then sharply jumps to the minimum allowable patch injection rate, and finally stays at this rate.

6. Further Discussions

In this section, we examine the effects of some factors on the best balance of a patch-injecting strategy. For convenience, let denote a best-balanced patch-injecting strategy, the corresponding balance.

6.1. The Effects of the Four Rates

First, we inspect the effect of the four rates, , , , and , on the best balance.

Experiment 6. Consider the following instances of the VPT control problem: where , . Figure 8 exhibits the best balances of these instances.

Figure 8 

The best balances in Experiment 6.

It is concluded from this experiment that is increasing with . As a result, the best balance can be improved by persuading the network users not to install suspicious software.

Experiment 7. Consider the following instances of the VPT control problem: where , . Figure 9 displays the best balances of these instances.

Figure 9 

The best balances in Experiment 7.

It is concluded from this experiment that is increasing with . Again, this conclusion demonstrates that warning the network users not to install suspicious software would improve the best balance.

Experiment 8. Consider the following instances of the VPT control problem: where , . Figure 10 exhibits the best balances of these instances.

Figure 10 

The best balances in Experiment 8.

It is concluded from this experiment that is decreasing with . Therefore, the best balance can be improved by reminding the network users of timely installing new patches.

Experiment 9. Consider the following instances of the VPT control problem: where , . Figure 11 exhibits the best balances of these instances.