Using the Gradient Method to Find the Optimal Resource Allocation Strategy

The two problems in our model are both a min-max formulation and both players are assumed that they could allocate continuous resources on each node in their own network and in the counterpart’s network in each round. Therefore, the gradient

Step2. To determine a positive or negative direction. If the problem needs to be solved is a maximization problem, the positive direction must be chosen.

On the other hand, when considering solving a minimization problem, the negative direction should be chosen.

Step3. The gradient method adopts a step-by-step method to find the optimization result. Therefore, the step size which is representative of the move size in each step must be determined.

Step4. To determine a dimension that wants to move. The derivative method would be used in the gradient method to find the most impact of all dimensions. And then, moving a step of the most impact of all dimensions and setting the new position as the next start point. And then, repeating step 4 until the stop criterion is satisfied

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method is extremely suitable to solve this kind of problem. However, the players’

respective two problems and the approaches to solve them are essentially the same except for considering the different networks of the two players. Therefore, in the following, we could consider both players’ objective functions simultaneously.

In our model, both players have a variety of defense and attack strategies which would influence the initial resources in each round, so all of these strategies must be taken into consideration when solving the problem by the gradient method. For both players’ objective functions, on one hand, since the inner problem is a maximization problem, the gradient ascent method should be adopted to solve; on the other hand, the outer problem is a minimization problem, and therefore the gradient descent method is suitable to solve. Nevertheless, there is still something needed to be discussed before using the gradient method to solve our problem:

 How many dimensions are there in this problem? Both players need to determine how to allocate defense and attack resources on each node in the network of their own and on the network of the counterpart, so the respective number of dimensions equals to the number of nodes of their own networks.

 What is the start point for both players? Both players are assumed that they

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would evenly allocate limited resources on each survival node, so the start point of player A and player B would be

1

survival node in their own networks (Where R1 and R2 are the total attack resources and r1 and r2 are defense resources in that round which respectively belong to player A and player B; N1 and N2 are the total number of the survival nodes of player A’s network under player B’s combined local view and of player B’s network under player A’s combined local view; n1 and n2

are the total number of the survival nodes of player A’s and player B’s network).

 How to calculate derivative of the Average DOD value? The derivative of the Average DOD value is difficult to be calculated, so the following method would be proposed：

 What is the stop criterion? If the impact degree of each dimension is the same meaning that it is stable, the gradient method therefore could stop calculating.

→ D means the Average DOD value

r

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In the following, the solution procedure of this problem would be introduced.

There are four steps in this approach and the detailed descriptions are as below: (In addition, the detailed process flow would be demonstrated inFigure 3-2.)

Step1. Initially, both players are assumed that they would uniformly allocate their limited defense and attack resources respectively on each survival node of their networks and on the survival nodes that in the combined local view of the counterpart’s network.

Step2. Player A and player B have limited attack resources in each round, so they would adopt gradient ascent method to maximize the damage degree of the counterpart’s network.

Step3. Besides, player A’s and player B’s defense resources are also limited in each round. Therefore, they would use the gradient descent method to minimize the damage degree of their networks.

Step4. Repeating step 2 and step 3 until the stop criterion is satisfied. As a result, the optimal resource allocation strategy for both players in each node could obtain. In addition, the Average DOD value would be used to evaluate the damage degree of the two networks.

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Figure 3-2: The Solution Procedure to Find the Optimal Resource Allocation on Each Node

Player A Player B

Allocate the defense budget on the nodes of player A’s network Repair the compromised nodes

Allocate the defense budget on the nodes of player B’s network Repair the compromised nodes

Reinforce defense budget to repaired nodes

Reinforce defense budget to repaired nodes

Allocate the attack budget on the nodes of player B’s network

Allocate the attack budget on the nodes of player A’s network

The exploration budget of updating information of unknown

vulnerabilities and private

information of player B’s network

The exploration budget of updating information of unknown

vulnerabilities and private

information of player A’s network

Using the gradient ascent method to find the maximization solution

Using the gradient descent method to find the

minimization solution

The optimal resource allocation solution for both networks, attack success probability of each node, and the Average DOD values would be separately obtained.

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