2.1.1 Evaluation of Global Aircraft Combat Ring Node Contribution Based on Topics
TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) method[5] .It was first proposed by C. L. Hwang and K. Yoon in 1981. TOPSIS is a ranking method based on the closeness of limited evaluation objects to the ideal goal, and it is a relative evaluation of the advantages and disadvantages of the existing objects. As a sequence optimization technique of ideal target similarity, it is a very effective method in multi-objective decision analysis. Through the normalized data matrix, the optimal target and the worst target in multiple targets were found, and the distance between each evaluation target and the ideal solution and anti-ideal solution was calculated respectively to obtain the closeness degree of each target to the ideal solution. According to the closeness degree of the ideal solution, the targets were sorted, which was used as the basis for evaluating the quality of the targets. The value of closeness is between 0 and 1. The closer the value is to 1, the closer the corresponding evaluation target is to the optimal level, and vice versa.
In the evaluation of aircraft combat ring nodes, it is not difficult to know that there are the following types of nodes[6] :
1). Target Node (T Node): The target that needs to be attacked, destroyed, jammed, or intercepted during combat, which can be an aircraft node or an armed facility on the blue side.
2). Reconnaissance and early warning node (S node): equipment or facilities that collect various types of information on the battlefield, mainly referring to aircraft nodes or other entities that conduct reconnaissance, early warning, or monitor blue targets on the battlefield.
3). Command and control node (D node): the unit that makes decisions and commands combat equipment and personnel in the combat system, such as command posts at all levels, command and control systems, etc.
4). Impact node (type I node): an entity that can directly interfere with or damage the target node, mainly referring to the strike and jamming weapon system.
For different types of nodes, there are obvious differences in their measurement indicators. Taking the following quantitative table of capability indicators as an example, we can easily find that the evaluation of these nodes should first be on the same scale, abandoning the difference in the order of magnitude. Secondly, we should reasonably evaluate the importance of different indicators, so in this paper, we use the TOPSIS method to model this multi-index evaluation task. Some typical aircraft indicators are presented in the Table 1 below.
Table 1
Example table of multiple indicators of aircraft
Capability indicators | Value Range |
Coefficient of survivability | [0.75,0.82] |
Warning time/s | [0.4,0.9] |
Maneuvering speed/knots | [0,40] |
Stealth coefficient | [0.18,0.63] |
Initial distance/km | [500,2000] |
Relative interception capability | [0.5,0.9] |
Anti-optical coefficient | [0.44,0.74] |
Anti-radar coefficient | [0.2,0.71] |
Anti-infrared coefficient | [0.44,0.68] |
Response time/s | [5e-4,1e-3] |
Decision error | [0,1e-7] |
In a combat ring, there are usually m evaluation nodes D1, D2., Dm, and each target has n evaluation indexes X1, X2... Xn. Firstly, experts are invited to score the evaluation index, and then the scoring results are expressed in the form of a mathematical matrix to establish the following characteristic matrix: |
$$D=\left(\begin{array}{ccc}{x}_{11}& \cdots & {x}_{1n}\\ ⋮& \ddots & ⋮\\ {x}_{m1}& \cdots & {x}_{mn}\end{array}\right)$$
We further write:
$$D=\left[{\text{X}}_{1}\left({\text{x}}_{1}\right), \cdots , {\text{X}}_{\text{j}}\left({\text{x}}_{\text{i}}\right), \cdots , {\text{X}}_{\text{n}}\left({\text{x}}_{\text{m}}\right)\right]$$
The characteristic matrix is normalized to obtain a normalized vector, and a normalized matrix about the normalized vector is established:
$${r}_{ij}=\frac{{x}_{ij}}{\sqrt{{\sum }_{i=1}^{m}{x}_{ij}^{2}}}$$
$$i=\text{1,2}, \dots , m, j=\text{1,2}, \dots , n$$
By calculating the weight normalization value of the corresponding indicator \({l}_{j}\) ,establishing a weight normalization matrix with respect to the weight normalization values:
$${v}_{ij}={l}_{j}{r}_{ij},\text{i}=\text{1,2},\cdots ,\text{m},\text{j}=\text{1,2},\cdots ,\text{n}\text{. }$$
The ideal solution and the anti-ideal solution are determined according to the weight normalization values:
$${A}^{*}=\left(\underset{i}{\text{max}}{v}_{ij}∣j\in {J}_{1}\right),\left(\underset{i}{\text{min}}{v}_{ij}∣j\in {J}_{2}\right),\mid i=\text{1,2},\cdots ,m={v}_{1}^{*},{v}_{2}^{*},\cdots ,{v}_{j}^{*},\cdots ,{v}_{n}^{*}$$
$${A}^{-}=\left(\underset{i}{\text{min}}{v}_{ij}∣j\in {J}_{1}\right),\left(\underset{i}{\text{max}}{v}_{ij}∣j\in {J}_{2}\right),\mid i=\text{1,2},\cdots ,m={v}_{1}^{-},{v}_{2}^{-},\cdots ,{v}_{j}^{-},\cdots ,{v}_{n}^{-}$$
Calculate the distance scale; that is, calculate the distance from each target to the ideal solution and the anti-ideal solution. The distance scale can be calculated by the n-dimensional Euclidean distance. The target is at a distance \({S}^{*}\)from the ideal solution and a distance \({S}^{-}\)from the anti-ideal solution.
$${ S}^{*}=\sqrt{{\sum }_{j=1}^{n}{\left({V}_{ij}-{v}_{j}^{*}\right)}^{2}}$$
$${S}^{-}=\sqrt{{\sum }_{j=1}^{n}{\left({V}_{ij}-{v}_{j}^{-}\right)}^{2}}$$
$$\text{i}=\text{1,2},\cdots ,m$$
According to the closeness degree of the ideal solution, the higher the closeness degree of the sorting result is, the better the target is.
$${C}_{i}^{\text{*}}=\frac{{S}_{i}^{-}}{\left({S}_{i}^{\text{*}}+{S}_{i}^{-}\right)},i=\text{1,2},\cdots ,m$$
Defines the hierarchical weight of a node \({w}_{i}^{1}\) for
$${w}_{i}^{1}={{\alpha }}_{i}{C}_{i}^{\text{*}}$$
2.2 Aircraft node contribution evaluation based on the TOPSIS-PageRank multi-index fusion algorithm
Here we use PageRank[7] .The local weights of the nodes are evaluated, and then the TOPSIS hierarchical weights proposed above are used for weighted fusion. Finally, the comprehensive weight based on the TOPSIS-PageRank multi-index fusion is obtained[8] .
PageRank is a common algorithm to evaluate the relevance and contribution of aircraft nodes, which can be transferred to the research of node importance. In 1998, Sergey Brin and Lawrence Page, Ph.D. students at Stanford University[9] .In this paper, we proposed a new algorithm for network link analysis, which is based on the random surfer model. Specifically, if a command node follows the link for several steps of information transfer, then turns to a random starting point, and the aircraft node follows the link again for information transfer, then the value of an aircraft node is determined by the frequency with which the aircraft node is visited by the command node.
A simple description of the PageRank algorithm migrating to aircraft nodes is as follows: u is an aircraft node, F (u) is the set of aircraft nodes pointed to by node u, B (u) is the set of aircraft nodes pointed to u, N (u) = F (u) is the number of links pointed out by u, c is the normalization factor (generally 0.85).
The PR value of the importance of a node is formulated as follows, that is, the importance of a node is mainly determined by the importance that points to its node and the links that it points outward to:
$$PR\left(u\right)=c{\sum }_{v\in \left(B\left(u\right)\right)}^{n}\frac{PR\left(v\right)}{N\left(v\right)}$$
However, considering that there is no direct subordinate relationship between some nodes, that is, there are fewer external links, this does not mean that the combat node is not important, so in this combat system, it is necessary to introduce a new node weight evaluation method.
We add an increment to prevent the node from being too low; that is, we add a damping coefficient called d (generally normalized to 0.85), which is like the paranoia in the neural network, to avoid sinking to 0 when calculating the weight, that is, to avoid some nodes not being considered.
$$PR\left(u\right)=\left(1-d\right)+d{\sum }_{v\in \left(B\left(u\right)\right)}^{n}\frac{PR\left(v\right)}{N\left(v\right)}$$
In the case of more data, we can improve it from the perspective of the damping coefficient. In the classical PageRank algorithm, the transition probability of the aircraft node is equally distributed to the out-link aircraft node, and the PR value of the new aircraft node is generally low due to fewer links. The classical PageRank algorithm calculates the PR value by linking without considering the practical significance of aircraft nodes, which has the problems of command weight drift and aircraft node weight equalization. Therefore, an authority degree p (vi) is introduced, which is determined by the ratio of the pointed links to the pointed links of the aircraft nodes:
$$\text{p}\left(\text{vi}\right)=\text{Q}\left(\frac{\text{ Linkein }}{\text{ Linkout }}\right)$$
After the introduction of the Ratio variable \(\text{p}\left(\text{vi}\right)\) :
$$PR\left(u\right)=\left(1-d\right)+d{\sum }_{v\in \left(B\left(u\right)\right)}^{n}\frac{\text{PR}\left(v\right)\cdot p\left(v\right)}{N\left(v\right)}$$
Finally, combined with the TOPSIS algorithm mentioned above, we integrate the weight meaning of the node itself with the node orientation relationship obtained by the PageRank we define the overall weight evaluation of the node as:
$${w}_{u}={k}_{1}\text{*}{w}_{u}^{1}+{k}_{2}\text{*}PR\left(u\right)$$
Where \({w}_{u}^{1}\) is the TOPSIS weight of the layer in which the u node is located, as represented above, and \({k}_{1}\) and \({k}_{2}\)represent normalization coefficients (initialization 0.15 and 0.85), respectively.
2. 3 Evaluation Method of Combat Capability Based on Information Entropy
The ability to complete combat tasks is often uncertain, and information entropy can well describe the uncertainty of information, so the operational network capability of the weapon equipment system can be measured by information entropy.[10]
The combat process can usually be decomposed into multiple nodes and edges, and each node and edge often have many factors that affect the combat capability. The smaller the uncertainty of these factors to meet the requirements of combat capability, the higher the combat capability. Conversely, the greater the uncertainty of meeting the capability requirements, the lower the combat capability. The uncertainty of various influencing factors in combat can be measured by the importance of its nodes. The greater the overall contribution of the nodes and the more distributed, it shows that the whole network has better invulnerability and better robustness in information transmission, and the more it can meet the needs of combat capability, the less uncertainty it brings to combat. Transforming the entropy function to obtain the weighted self-information quantity of the node, namely:
$${\text{I}}_{\text{u}}=\frac{{\text{w}}_{\text{u}}}{\text{N}}$$
Among \({\text{W}}_{\text{u} }\)Represents the comprehensive contribution of nodes evaluated in Section 2.2, and N is the number of nodes in the entire battle graph. We evaluate the amount of information of nodes in a way like K-shell, but we need to evaluate the information entropy. Further, we integrate the amount of information of other nodes about a node to obtain the information entropy function of a node:
$${\text{e}}_{\text{u}}=-{\sum }_{\text{j}\in {\Gamma }\left(\text{u}\right)}{\text{I}}_{\text{u}}\cdot \text{ln}{\text{I}}_{\text{u}}$$
Where J (\(\text{j}\in {\Gamma }\left(\text{u}\right)\)) is the set of neighbors of node Vu. The information entropy of a node considers the propagation effect of its neighbors, and the greater the information entropy of a node, the greater its influence.
Then we define the operational capability of the operational graph u as Ku, which can be measured by the amount of self-information, and we define the operational capability as the sum of the information entropy of nodes on all operational rings in the operational graph:
$${\text{K}}_{\text{u}}={\sum }_{\text{i}=1}^{\text{M}}{\sum }_{\text{j}\in {\sigma }\left(\text{i}\right)} {\text{e}}_{\text{j}}$$
Among \({\sigma }(i)\)Is the set of nodes contained in the combat ring I, and M is the number of combat rings owned by the combat graph.
Then the combat capability F of the joint combat system for the multi-target combat mission is
$$\text{F}={\sum }_{\text{i}=1}^{\text{M}}{\text{q}}_{\text{i}}{\text{K}}_{\text{i}}$$
In the formula, qi represents the weight value of each target, and the target weight is determined mainly based on the following aspects:
(1) The urgency of the military task (the threat level of the target to our side).
(2) The importance of the target in the enemy equipment system (the key node in the enemy system).