Research on an intelligent game simulation system based on road network

Site location of security resource emergency facilities can be described as below. In a given regional traffic network M(V, E), V (v₁,v₂,…,v_n) is a set of vertices representing a collection of important facilities or sites in the region. It is a collection of potential attack targets and an alternative collection of selected points for security resource facilities. When E(e₁,e₂,…,e_n) is the edge set, it represents the set of sections in the regional traffic network. We set d(i, j) as the shortest path distance between the vertex v_i and the top point v_j, and set w_i as the security resource required after each vertex v_i which suffers an attack. It is directly proportional to the importance of the vertex v_i and also represents the weight of the vertex v_i. There are two types of participants in attack and defence now. Attackers are divided into three types based on the degree of rationality, attacking respectively the vertices in the vertex set V with different attack strategies. We assume that the attack must succeed. In order to reduce the loss of the attack, the defenders need to select several vertices in the vertex set V to build the security resource placement points for the security resource rescue for the attack points so as to improve the emergency efficiency after the attack. Because the attack strategies of different types of attackers vary, we need to consider the classification and construct the corresponding site selection model in turn. In this paper, we first define the specific types of attackers. In the current research studies on security resource allocation, Hao et al. [15] and Xiang Yin and Wei Hang [12] have roughly divided the attackers into three categories: (1) completely rational attacker, (2) irrational attacker and (3) incompletely rational attacker. We will discuss this content in Section 3. the higher the degree of rationality of the attacker, the higher the information level and computing power, with the goal of maximising the attack revenue. Among the above three types of attackers, completely rational and irrational attack decision and responses are partial to two extremes, which are rare in reality. The incompletely rational attacker has the characteristics of both completely rational and irrational attackers, which is more in line with the real scene. Therefore, in this paper, we assume that the attacker is incompletely rational.

The hypothesis on security resources path planning can be described as follows. On the basis of security resource emergency facilities site problem, we assume that the attack point v_i is attack and needs certain security emergency resources rescue. Meanwhile, as there are multiple security resources emergency facilities to meet the conditions, we need to select a point for the vertex v_i to have emergency rescue and plan the path. Among them, if the road situation is known, there is an effective connection between any two nodes, and each emergency facility is subject to the deployment of the system.

As it can be seen from the above description of the location of security resource emergency facilities, the defender's decision goal is to minimise the loss, while the attacker's is to maximise the perceived defender loss. The ultimate goal of path planning is to solve a point to rescue the vertex v_i from multiple security resource emergency facility points and plan the path.

Berman [8, 9] and Han et al. [10] discussed the facility site location model, where security information is that the attacker is completely rational, and abstracted the site selection problem into a Stackelberg game model. The defenders first set up k facility points and form the strategy space s = {s₁,s₂,…,s_k}. After observing the pioneer's strategy, the attackers make mixed strategies, attacking the vertex in the vertex set and forming the strategy space set π = {π₁,π₂,…,π_k}. π_k represents the probability of attacking point k, which considers only a single attack, meaning that the sum of all attack probabilities is equal to 1.

In the relevant literature of Berman [8, 9] and Han et al. [10], the sum of defender disutility is defined as the sum of attack loss and facility construction costs, and attacker utility is defined as attack loss. Xiang Yin and Wei Hang [12] improved the utility function and obtained the defensive negative utility function in the general attack case. Based on the above research, considering the loss caused by the event diffusion of a road network in a public safety area, the paper sets the improved defensive negative utility function of τ as follows: (1) UGR(s,π)=∑i=1nπi[α⋅d(s,i)+βlnwi−γ]wi+cm+τ {U_{GR}}\left( {s,\pi } \right) = \sum\limits_{i = 1}^n {\pi _i}\left[ {\alpha \cdot d\left( {s,i} \right) + \beta \ln {w_i} - \gamma } \right]{w_i} + {cm} + \tau

U_GR represents the defender's disutility function and π_i [α ·d (s,i) + β ln w_i − γ]w_i the loss of the individual vertices after suffering an attack. When i is not attacked, that is, when the attack loss at the π_i = 0, α · d (s,i) · w_i represents the loss arising during transport and α represents the loss caused by the time delay of the unit of materials during the transportation of rescue materials while c represents the cost of setting each vertex security resource point, m indicates the number of emergency facilities, τ represents the sum of losses resulting from the diffusion of individual vertex events, β is the loss caused by the time delay of unit material unit and γ is the pioneer's ability to manage unit materials.

The utility formula of the attacker Eq. (2) lacks the item cm when compared with Eq. (1). As shown by Eq. (2), the utility obtained by the attacker is also directly affected by the spread of the event and the defender's strategy s as well as the attacker's attack strategy. (2) UTR(s,π)=∑i=1nπi[α⋅d(s,i)+βlnwi−γ]wi+τ {U_{TR}}\left( {s,\pi } \right) = \sum\limits_{i = 1}^n {\pi _i}\left[ {\alpha \cdot d\left( {s,i} \right) + \beta \ln {w_i} - \gamma } \right]{w_i} + \tau

Therefore, in the face of completely rational attackers, the integrated model formula of Eqs (1) and (2), Eq. (3) is written as follows: minsUGR(s,π*)=∑i=1nπi*[α⋅d(s,i)+βlnwi−γ]wi+cm+τ \mathop {\min }\limits_s {U_{GR}}\left( {s,{\pi ^*}} \right) = \sum\limits_{i = 1}^n \pi _i^*\left[ {\alpha \cdot d\left( {s,i} \right) + \beta \ln {w_i} - \gamma } \right]{w_i} + cm + \tau (3) s.t.{∑i=1nπi*=1π*(s)∈argmaxiUGR(s,π)=∑i=1nπi[α⋅d(s,i)+βlnwi−γ]wi+τπi≥0 s.t.\left\{ {\matrix{ {\sum\nolimits_{i = 1}^n {\pi _i^* = 1} } \hfill \cr {{\pi ^*}(s) \in argma{x_i}{U_{GR}}\left( {s,\pi } \right) = \sum\nolimits_{i = 1}^n {{\pi _i}\left[ {\alpha \cdot d\left( {s,i} \right) + \beta \ln {w_i} - \gamma } \right]{w_i} + \tau } } \hfill \cr {{\pi _i} \ge 0} \hfill \cr } } \right.

In Eq. (3), there are three constraints. First, there is only one attack. Second, each vertex has an attack probability ≥ 0. And third, the attacker responds optimally to the defender's location strategy to maximise its utility function.

In allusion to the completely rational situation, the game problem becomes a single constraint linear planning problem. Assuming that k(i) is the node nearest to the attack point v_i in the site selection strategy s, then the optimal attack solution of the perfectly rational attacker and the unique equilibrium solution s^*, π_z^*(s^*) is obtained: (4) z*(s)=argmaxi∑i=1nπi*[α⋅d(s,i)+βlnwi−γ]wi {z^*}\left( s \right) = \mathop {\arg \max }\limits_i \sum\limits_{i = 1}^n \pi _i^*\left[ {\alpha \cdot d\left( {s,i} \right) + \beta \ln {w_i} - \gamma } \right]{w_i}

And the optimal solution of the defence's location strategy is written as follows: (5) s*=argmins[α⋅d(s,z*(s))+βlnwz*(s)−γ]wz*(s)+cm+τ {s^*} = \mathop {arg \min }\limits_s \left[ {\alpha \cdot d\left( {s,{z^*}\left( s \right)} \right) + \beta \ln {w_{{z^*}\left( s \right)}} - \gamma } \right]{w_{{z^*}\left( s \right)}} + cm + \tau

Since this situation will hardly happen in reality, there is no literature focussing specifically on irrational attack game models. By the definition of irrationality through game theory, Xiang Yin and Wei Hang [12] described the location selection problem when the attacker is irrational as follows: in a regional traffic extension network M(V, E), V (v₁,v₂,…,v_n) is the vertex set and E(e₁,e₂,…,e_n) is the edge set; there exists only a main body, defenders, who have known that attackers will attack each vertex with a fixed probability σ = (σ₁,σ₂,…,σ_n), where σi=wi/∑i=1nwi {\sigma _i} = {w_i}/\sum\nolimits_{i = 1}^n {w_i} , and the attack probability of each vertex is directly proportional to the vertex weight. Consequently, in this game case, the defence side only needs to reasonably select m vertices to build safety resource emergency facilities to reduce the loss, and the attackers will not react according to the defence's strategy. It can be concluded that the game model in irrational cases is as in Eq. (6). The strategy has two constraints. On the one hand, the total attack probability of all vertices is 1, and on the other hand, the attack probability of each vertex is ≥ 0. mins UGNR(s,σ)=∑i=1nσi[α⋅d(s,i)+βlnwi−γ]wi+cm+τ \mathop {\min }\limits_s \;{U_{GNR}}\left( {s,\sigma } \right) = \sum\limits_{i = 1}^n {\sigma _i}\left[ {\alpha \cdot d\left( {s,i} \right) + \beta \ln {w_i} - \gamma } \right]{w_i} + cm + \tau (6) s.t.{σi=wi/∑i=1nwiσi≥0 s.t.\left\{ {\matrix{ {{\sigma _i} = {w_i}/\sum\nolimits_{i = 1}^n {{w_i}} } \hfill \cr {{\sigma _i} \ge 0} \hfill \cr } } \right.

The site selection model in the irrational attack mode Eq. (6) is equivalent to the p-median model, where the attacker is completely irrational, and the attack probability is also given externally. Thus, the defensive location problem can be transformed into a 0–1 planning problem, namely a vertex placed facility point j or no facility point, j rescue attack point i or no rescue. Let the d_ij be the shortest distance between i and j. The most optimal solution obtained is s^**: (7) s**=argmins∑i=1nσi[α⋅d(s,i)+βlnwi−γ]wi+cm+τ {s^{**}} = \mathop {arg \min }\limits_s \sum\limits_{i = 1}^n {\sigma _i}\left[ {\alpha \cdot d\left( {s,i} \right) + \beta \ln {w_i} - \gamma } \right]{w_i} + cm + \tau

Many real attackers are not completely rational; therefore, Hao et al. [15] and Xiang Yin [7,14] all considered the allocation model of incomplete rational attackers in the process of studying the security resource allocation problem. This paper learns from the introduction of a rational degree factor of modelling literature in the analysis of incomplete rational game situation from the documents of Xiang Yin and Wei Hang [12].

Let p ∈ [0,1], the obtained corresponding model is Eq. (8). When p = 1, the attacker is completely rational. Simplifying Eq. (8) to be equivalent to Eq. (3), σ_i's constraint on Eq. (8) won’t be effective or affect the decision result. When p = 0, that is, when the attacker is irrational, formula simplifying Eq. (8) to be equivalent to Eq. (6), π_i's constraint on Eq. (8) will neither work nor affect the decision result. mins UGLR(s,π*,σ)=p⋅∑i=1nπi*[α⋅d(s,i)+βlnwi−γ]wi+(1−p)∑i=1nσi[α⋅d(s,i)+βlnwi−γ]wi+cm+τ \mathop {\min }\limits_s \;{U_{GLR}}\left( {s,{\pi ^*},\sigma } \right) = p \cdot \sum\limits_{i = 1}^n \pi _i^*\left[ {\alpha \cdot d\left( {s,i} \right) + \beta \ln {w_i} - \gamma } \right]{w_i} + (1 - p)\sum\limits_{i = 1}^n {\sigma _i}\left[ {\alpha \cdot d\left( {s,i} \right) + \beta \ln {w_i} - \gamma } \right]{w_i} + cm + \tau (8) s.t.{∑i=1nπi*=1π*(s)∈argmaxiUGR(s,π)=∑i=1nπi[α⋅d(s,i)+βlnwi−γ]wi+τπi≥0,σi≥0σi=wi/∑i=1nwi s.t.\left\{ {\matrix{ {\sum\nolimits_{i = 1}^n {\pi _i^* = 1} } \hfill \cr {{\pi ^*}(s) \in \arg {{\max }_i}{U_{GR}}\left( {s,\pi } \right) = \sum\nolimits_{i = 1}^n {{\pi _i}\left[ {\alpha \cdot d\left( {s,i} \right) + \beta \ln {w_i} - \gamma } \right]{w_i} + \tau } } \hfill \cr {{\pi _i} \ge 0,{\sigma _i} \ge 0} \hfill \cr {{\sigma _i} = {w_i}/\sum\nolimits_{i = 1}^n {{w_i}} } \hfill \cr } } \right.

In the above two parts, the irrational attacker's attack probability is determined by exogenous factors, while the completely rational attacker's is completely determined by endogenous factors. Bringing the above reaction function into the Eq. (8), we can then work out the optimum solution of the only equilibrium solution s′,π_z^*(s,) of the defender location selection strategy in the incompletely rational case: (9) s'=argmins p⋅[α⋅d(s,z*(s))+βlnwz*(s)−γ]wz*(s) +(1−p)⋅∑i=1nσi[α⋅d(s,i)+βlnwi−γ]wi+cm+τ \matrix{ {{s^'} = \mathop {arg \min }\limits_s p \cdot \left[ {\alpha \cdot d\left( {s,{z^*}\left( s \right)} \right) + \beta \ln {w_{{z^*}\left( s \right)}} - \gamma } \right]{w_{{z^*}\left( s \right)}}} \hfill \cr {\;\;\; + \left( {1 - p} \right) \cdot \sum\limits_{i = 1}^n {\sigma _i}\left[ {\alpha \cdot d\left( {s,i} \right) + \beta \ln {w_i} - \gamma } \right]{w_i} + cm + \tau } \hfill \cr }

In existing research studies, the shortest path refers to the shortest path from the starting point and the end point and the smallest sum of the path weights. However, after the occurrence of the emergency, besides the shortest path to the demand point, the effective time of delivery materials and other restrictive condition are added. Comprehensively analysing, the so-called ‘optimal path of security resource scheduling path’ refers to the shortest one, ensuring the supplies to be delivered in order to reduce the more unnecessary losses incurred during the transportation process and arrived within the scope of time requirements when a security event occurs at a certain point in the road network of the park. Since there are more than one site in the campus for storing materials, it involves multiple material point, that is, the ‘optimal path’ of the campus security material scheduling. Therefore, it is necessary to propose the u-Dijkstra algorithm as shown in the following Eqs (10)–(12) on the basis of the existing improved Dijkstra algorithm proposed by Zhang Yajing et al. [16]. (10) d1=0 {d_1} = 0 (11) dj=mini≠j,(vi,vj)∈E{dj+lijwij}=mini≠j,(vi,vj)∈E{dj+ntijkijwij}(j=2,3,…,n) {d_j} = \mathop {\min }\limits_{i \ne j,({v_i},{v_j}) \in E} \left\{ {{d_j} + {l_{ij}}{w_{ij}}} \right\} = \mathop {\min }\limits_{i \ne j,\left( {{v_i},{v_j}} \right) \in E} \left\{ {{d_j} + {{n{t_{ij}}} \over {{k_{ij}}}}{w_{ij}}} \right\}(j = 2,3, \ldots ,n) (12) dj<dmax {d_j} < {d_{max }} where v_i is the node in the map network, E is the edge set of v_i to v_j, l_ijw_ij is the effective path length, l_ij is the equivalent weight factor of the roadway from node v_i to v_j laneway, d_j is the optimal path, d_max is the longest path, t_ij is the impact factor of length, k_ij is the traffic efficiency and n is the number of trips required for transporting materials. Among the many storage points on the map, one is selected to mobilise supplies for the attack point.

In this paper, Xinjiang Normal University is treated as an example. The campus map is shown in Figure 1a. Campus network number map and important campus places are abstract as random numbers, and roads between important sites are edges. The park network map is shown in Figure 1b. The map is composed of 44 nodes and 64 edges.

Fig. 1

In this paper, considering the real data and importance data on the campus security, data demensitisation processing, select the vertex 1–22 importance value as shown in Table 1, the actual test distance between vertices, exhaustive calculation, and then according to a certain proportion, get the shortest path distance of each vertex to other vertices, as shown in Figure 2 below:

Fig. 2

First, in order to intuitively observe the impact of the number of parameter security resource facility points m on the loss of defenders, this paper uses mathematical modelling software MATLAB to experiment. The results are shown in Figure 3, and the number of security facility points increases after the end of the second genetic algorithm.

Fig. 3

When m=3, α = 2, β = 0.5 and γ = 0.2, the obtained p-value is 0, 0.3, 0.6, and 0.9, respectively. In Figure 4, as the degree of rationality of the attacking party increases, the loss of the attacker also increases significantly.

Fig. 4

The results shown in Figure 5 show that in the fixed number of fixed emergency facilities, α = 2, β = 0.5, γ = 0.2, as the degree of rationality of the attacker decreases, the completely rational attacker causes the most loss to the defence, while in Figure 3, the irrational attacker causes the least loss to the defender. This is because irrational attackers always fail to adopt optimal solutions after observing defensive actions and maximise their utility. A perfectly rational attacker is quite the opposite. Therefore, the attack consequences of a completely rational attacker are the worst case, while the attack consequences of an irrational attacker are the best case. This point can be used in daily management: (1) In the case of funds allowed, increasing the number of security resource facilities points can effectively reduce the loss. (2) The judgement of the attacker can be affected by releasing interference information so as to reduce the rationality of the attacker. (3) Due to the relevant value of public opinion management introduced in the loss item, the resulting loss will also increase; so, the hidden loss can be reduced by controlling the spread influence of preventive events.

Fig. 5

Secondly, in Figure 5, when the degree of rationality is fixed, the defensive loss gradually decreases as the number of facilities increases, but obviously, when the number of facilities increases to a certain extent, increasing the number of facilities has little impact on the defensive loss. And by changing the rationality of the attacker, the relationship between the number of facilities and the defensive utility remains.

In this paper, the simulation is conducted at p=0.5, p = 0.5, m=3 and m=5. According to the results of the site selection strategy, the blue route planning is shown in Figures 6 and 7. After calculating the verification of the weights of the path and the shortest path in the table, the results prove that the planning path is accurate.

Fig. 6

Fig. 7