This section focuses on evaluating the effectiveness of NIGA within the context of attacker–defender interactions. Initially, NIGA’s efficacy was tested compared with other related algorithms. Subsequently, to prove NIGA’s optimization ability, its performance was tested across a variety of network sizes. Moving forward, NIGA was utilized to determine the Nash equilibrium at varying initial strategy ratios, denoted by r. To conclude the analysis, the algorithm was used to address the Nash equilibrium problem with different attack () and defense resources (). The experiments were based on power-law networks and were characterized by a heavy-tailed degree distribution. The probability that a node had k connections followed a power-law distribution, typically represented as . This property implies that a few nodes, known as “hubs”, possess a disproportionately large number of connections compared with the majority of nodes, which have only a few links. The structure of power-law networks has significant implications for various phenomena observed in real-world networks, such as robustness to random failures and the potential for the rapid spread of information or diseases.
Our analysis was conducted on a system equipped with a 12th Gen Intel Core i7-12700H processor, 32.0 GB of RAM, and a 64-bit operating system running on an x64-based processor. The data originated from randomly generated scale-free networks.
5.1. Performance Test of NIGA
In this part, we report on two series of performance evaluation experiments. The first set was based on different initial strategies, with 30 runs recorded for each algorithm’s results. The second set was based on varying network sizes, specifically networks with 100, 200, and 300 nodes. We examined the effectiveness of each algorithm under these conditions. For NIGA, performance was gauged according to the attainment of the lowest possible value, signifying the smallest possible loss for the defender. Optimally, an algorithm’s performance is indicated by the smallest loss it can identify.
For analysis, we conduct an analysis that encompasses the worst case, average case, and best case scenarios.
We have defined the worst case scenario as the situation where the attacker is most successful in causing damage to the network, resulting in the maximum possible loss for the defender. This scenario tests the resilience and robustness of the model under the most adverse conditions.
The average case represents a typical interaction between the attacker and defender, where the outcomes are neither the best nor the worst but fall within a range of expected results. This scenario provides insight into the model’s performance under normal operating conditions.
The best case scenario is characterized by the optimal performance of the model, where the defender’s strategies are most effective in minimizing the impact of the attacker’s actions, resulting in the lowest possible loss.
The performance of NIGA was compared with that of other traditional algorithms within the attacker–defender game model, including random strategy (RS), degree strategy (DS), betweenness strategy (BS), closeness strategy (CS), and eigenvalue strategy (ES). In RS, the defender randomly updates strategies at each iteration, and in DS, they update strategies based on the degree of nodes, favoring strategies with the highest degree. In BS, the defender favors strategies with the highest betweenness centrality of nodes; in CS, those with the highest closeness centrality of nodes; and in ES, those with the highest eigenvalue of nodes.
Firstly, the performance of NIGA was assessed relative to the conventional methods of random strategy (RS), degree strategy (DS), betweenness strategy (BS), closeness strategy (CS), and eigenvalue strategy (ES). This comparison was based on their respective optimization capabilities when utilizing various initial strategies. The experimental data can be found in
Table 3, revealing that NIGA was capable of identifying superior solution paths in comparison with the other algorithms across a series of 30 trials with randomized initial strategies. To further elucidate the comparative optimization effectiveness among RS, DS, BS, CS, ES, and NIGA,
Figure 2 presents nine diagrams that depict their optimization processes, highlighting their notable performance aspects.
Table 3 offers a detailed performance comparison among several strategies within an attacker–defender game model. The NIGA stands out as it consistently demonstrates superior performance across all trials, indicating its potential as a robust method for optimizing defense strategies against potential attacks. The uniformity in the performance of strategies such as the RS, DS, BS, CS, and ES suggests that these approaches might be similarly influenced by certain network metrics.
The table also reveals that strategies based on network centrality metrics, such as DS, BS, and CS, do not invariably outperform the RS. This observation might suggest that these metrics alone might not sufficiently capture the intricacies of the interactions within the attacker–defender game. The consistent lower values recorded for NIGA imply that it is more adept at minimizing the defender’s loss.
Figure 2 demonstrates that NIGA can find smaller values than RS, DS, BS, CS, and ES. Moreover, it converges more rapidly. In other words, the proposed algorithm possesses superior optimization capability compared with RS, DS, BS, CS, and ES.
Secondly, NIGA was compared with RS, DS, BS, CS, and ES in terms of optimization capability in different networks. The results of the experiment, shown in
Table 4, show that compared with the other traditional algorithms, NIGA could achieve better results on the networks with 30, 100, 200, and 300 nodes. In the 50-node network, NIGA, RS, DS, and BS had the same optimization effect.
Table 4 provides an insightful comparison of the performance of different strategies across varying network sizes. The results are particularly revealing in terms of how each strategy scales with the increase in the number of nodes in the network. Initially, it is noticeable that NIGA consistently performs well across all node sizes, indicating its robustness and effectiveness in different network conditions. The scores for NIGA are the lowest among the strategies compared, which in this context suggests a more optimal performance as the objective is to minimize the defender’s loss.
As the network size increases from 30 to 300 nodes, the performance of most strategies, except for NIGA, tends to converge. This could imply that as networks grow larger, the strategies based on network centrality metrics such as DS, BS, and CS, which initially show variability, start to perform similarly. This convergence might be due to the increased complexity and interconnectivity in larger networks, which could diminish the impact of individual node characteristics that these strategies focus on.
In order to better show the optimization performance of RS, DS, BS, CS, ES, and NIGA, three optimization process diagrams showing noticeable effects are shown in
Figure 3.
Figure 3 illustrates that across various network sizes, NIGA was capable of identifying lower values than those obtained with random strategy (RS), degree strategy (DS), betweenness strategy (BS), closeness strategy (CS), and eigenvalue strategy (ES). This suggests that NIGA generally outperforms these strategies in terms of optimization efficiency.
5.2. Comparison with Smart Methods
While the comparison in
Section 5.1 demonstrates NIGA’s superior performance against traditional algorithms such as RS, DS, BS, CS, and ES, it is also essential to evaluate its effectiveness in comparison with more advanced, state-of-the-art methods. These advanced methods often incorporate machine learning, artificial intelligence, or other sophisticated computational techniques to address the complexities of attacker–defender games in large-scale networks.
One such state-of-the-art method is the deep reinforcement learning approach (DRL), which has been applied to security games to learn optimal defense strategies through interaction with the environment. DRL methods have shown promise in dynamic and uncertain conditions but can be computationally intensive and require extensive training periods.
Another advanced method is the multi-agent reinforcement learning (MARL) framework, which models the interactions between multiple defenders and attackers. MARL provides a more realistic representation of real-world scenarios but faces challenges in terms of convergence and the computational complexity of handling multiple agents.
A third approach includes the use of evolutionary algorithms (EA), which simulate the process of natural selection to evolve optimal strategies. EAs are known for their robustness in searching large and complex solution spaces but may suffer from slow convergence and the need for fine-tuning of various parameters.
In our comparison, we found that while these advanced methods offer unique advantages, NIGA demonstrates a better balance between computational efficiency and solution quality, especially for large-scale networks. NIGA’s node-incremental approach allows for a more manageable exploration of the strategy space, reducing the computational burden compared to methods that require simultaneous consideration of all possible strategies.
5.3. Overview of Simulation Setting and Criteria for Performance Assessment
The attacker has the capability to target infrastructures through various means, including the use of explosives, the employment of chemical and biological weapons, and the initiation of cyber attacks. Conversely, the defender can safeguard infrastructures by bolstering physical security measures and taking other protective actions. Furthermore, the model presented in this study is also applicable for the analysis of individual attack instances.
The simulation experiment diagrams of this study are depicted in
Figure 4.
Figure 4a represents a power-law network with 30 nodes, while
Figure 4b illustrates one with 100 nodes. To provide a comprehensive demonstration, simulation experiments were conducted based on these two networks. In
Figure 4, the nodes and edges represent the infrastructures and their interconnections within the infrastructure network, respectively. All nodes depicted in
Figure 4 are susceptible to attack and can also be defended.
The simulation experiment (
Figure 4) utilizes networks of typical sizes. However, the range of strategies for attack and defense is vast. We assume that the cost to attack or defend a single node is one unit.
Table 5 shows the number of strategies available given different resource allocations. In a network of 30 nodes, there are 5.8 million possible strategies. This number grows significantly for a 100-node network, reaching 18.609 billion strategies.
Next, the influence of different scales of the initial strategies in the attacker–defender game model and the influence of the attack-defense resources on the Nash equilibrium were analyzed.
5.4. The Influence of the Scale of the Initial Strategies on the Nash Equilibrium
In this section, with the attacker’s and defender’s resources fixed, we report the analysis of the impact of the initial strategy scales on the experimental outcomes. In the 30-node network, due to the relatively small total number of strategies, the proportions of the initial strategy count to the total strategy count were set to 0.05, 0.10, and 0.15. In the 100-node network, given the large total number of strategies, the proportions were set to 0.01, 0.02, and 0.03.
Table 6 and
Table 7 present the results of the two sets of experiments. To provide a more intuitive display of the experimental results, we created the graphs in
Figure 5.
In
Table 6, for
, the results vary from 1.8182 to 1.8333 to 1.8519 as
r increases from 0.05 to 0.10 to 0.15, respectively. As
r increases from 0.05 to 0.15, the number of initial strategies increases, and the optimization result changes. The convergence iterations decrease from six to three, which might suggest that as more strategies are available at higher ratios, the optimization algorithm can find a solution more quickly or it might reach a point of diminishing returns where additional strategies do not significantly improve the result.
In
Table 7, for
, the optimization results are 1.9130 for
, 1.9444 for
, and 1.9545 for
. However, the convergence iterations do not show a clear trend, varying between four and nine. This could mean that the optimization landscape is more complex with more initial strategies, or it could be that the algorithm’s performance is affected by other factors.
5.5. The Influence of Attack-Defense Resources on the Nash Equilibrium
In this subsection, we present further experiments to analyze the defender’s loss where the scale of the initial strategies was fixed. The attack–defense resources were set to two, four, six, and eight. The defender’s losses for different attack–defense resource allocations are reported in
Table 8 and
Table 9. The results in
Table 8 are based on a 30-node network, while those in
Table 9 are based on a 100-node network.
Figure 6 provides a more intuitive representation of the changes in the defender’s profit under varying attack–defense resource conditions across different network sizes.
Table 8 illustrates the defender’s loss for a smaller network with 30 nodes. The data reveal a significant reduction in the defender’s loss as the defense resources increase, regardless of the attacker’s resources. This trend underscores the importance of resource allocation in defense strategies. Notably, when the defender has more resources, the loss decreases, demonstrating the direct relationship between resource availability and the effectiveness of defense. Conversely, as the attacker’s resources increase, the defender’s loss escalates.
Table 9, which pertains to a larger network of 100 nodes, presents a similar pattern. However, the magnitude of the defender’s loss is generally higher in this case, reflecting the increased complexity and potential attack vectors in larger networks. The results here also indicate that as defense resources increase, the defender’s loss decreases, but the rate of decrease is less pronounced than in the smaller network. This could be attributed to the larger network’s greater capacity for potential attack combinations, which might dilute the impact of increased defense resources.
A comparative analysis of
Table 8 and
Table 9 shows that the defender’s loss is generally lower in the smaller network (N = 30) across all resource conditions. This suggests that smaller, more manageable networks might be easier to defend effectively. However, as network size increases, the defender faces a more challenging task, necessitating more sophisticated defense strategies and potentially more resources.
Figure 6 show that when the attack resources remain unchanged, the more numerous the defense resources, the smaller the loss of the defender, and when the defense resources remain unchanged, the more numerous the attack resources, the greater the loss of the defender.