Study on Cascading Failure and Elasticity of UAV Swarm Communication Network

Abstract

As an important new force in information warfare, the UAV swarm has attracted more and more attention from militarists and scholars of various countries. At present, large-scale and low-cost UAV swarms have been gradually applied to reconnaissance operations. The invulnerability and elasticity of its communication network are the key factors that determine whether the swarm can successfully complete the reconnaissance mission. The research object of this study is the UAV swarm communication network, which performs reconnaissance missions. The swarm network is extracted as a three-layer interdependent network model. We apply scale-free network theory and the new connection probability equation to establish three subnetworks. Then, a three-layer interdependent network is constructed by using the interlayer connection rules of the nodes with the largest degree. The classical M-L model is extended to an interdependent network. An interlayer load redistribution rule is proposed to simulate the cascading failure process of swarm networks. Then, we attack network nodes randomly and observe the invulnerability and elasticity of this network. The simulation results show that the performance of the interdependent network is better than that of the single-layer network due to the effect of the interdependent links. By analyzing the experimental results, some feasible suggestions are put forward to improve the invulnerability and elasticity of reconnaissance UAV swarm communication network.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

An unmanned aerial vehicle (UAV) swarm is an airmobile system composed of a certain number of single-function or multi-function UAVs [1]. After this concept was put forward, relevant theoretical research and verification tests emerged in an endless stream during the past decade. In 2012, the strategic capability Office of the U. S. Department of Defense proposed the “PARTRIDGE” project, which aimed to achieve the air coordination of multiple low-cost small UAVs through wireless Ad Hoc networks [2]. In 2015, the Defense Advanced Research Projects Agency of the U. S. (DAPAR) released the “GREMLINS” program, hoping to improve the UAV swarm capability through key technologies such as air recycling [3]. In 2017, the “OFFSET” project under the responsibility of DAPAR focused on urban operations on the basis of the accumulation of UAV swarm technology. Its reconnaissance capability and network robustness are the keys to the realization of combat tasks. In the same year, Yanjie Zhao’s team of China Electronics Technology Group completed the flight test and networking of 119 fixed-wing UAVs, and preliminarily conducted the ground reconnaissance mission test [4]. In 2020, Zhao’s team expanded the scale of UAVs, realized the networking flight and operational function verification of 200 UAVs.

On the basis of technology accumulation in the past decade, UAVs and UAV swarms have played an increasingly prominent role in recent local wars. On September 27th, 2020, a new round of armed conflict broke out between Azerbaijan and Armenia in the Naka region. Azerbaijan destroyed Armenia's air defense system through the comprehensive use of multiple UAVs such as AN-2 modified UAV and TB-2 Reconnaissance/Strike UAV [5]. This regional conflict has subverted people’s concept of local war, and it is also the first time that UAVs have appeared on the battlefield as the main combat equipment. While the first operational use of a UAV swarm took place in 2021, according to the Russian Satellite News Agency, in May 2021, during Israel's air strike against Hamas, the Israeli air force used an artificial intelligence-assisted UAV swarm to carry out reconnaissance and strike. These UAV swarms deployed near Gaza communicate with each other and share the situation through the communication network to accurately locate the target.

The UAV swarm depends deeply on its communication network when it performs aerial reconnaissance, target positioning, cooperative strike, and other combat tasks [6, 7]. The UAV swarm network is formed by connecting the UAV nodes through high-speed data link, which can realize the global situation sharing and data collaborative processing of UAVs in the network [8]. Communication network is the basic support of UAV swarm capability generation [9–11].

In order to reduce the dependence on ground facilities and improve the autonomous combat capability of UAV swarm, a self-organizing network is the preferred network mode for UAV swarm communication. In this networking mode, each UAV can be used as a network node of a mobile UAV swarm, and each node relies on the wireless communication device module to realize the independent management of wireless communication. Xiao et al. applied the complex network theory to design the UAV swarm self-organizing communication network and developed the network node control software, which provided a practical method for realizing the efficient autonomous communication of UAV swarm [12]. Wang and Gao analyzed the main characteristics and applicable environment of typical algorithms for autonomous management of UAV swarm communication based on summarizing the topology reconstruction of the UAV swarm communication network [13].

UAV swarm warfare requires high reliability of data transmission. However, there are some problems in the self-organizing network applied to UAV swarm, such as wireless multihop routing, changeable network topology, limited bandwidth, and so on [14]. From the conclusion of theoretical reasoning and the results of many experiments in recent years, these problems may cause the failure of the UAV swarm network due to the damage to some UAV nodes. Liu and Zhang studied the connectivity of the UAV swarm Ad Hoc network, and pointed out that the communication connectivity between UAVs will greatly affect the ability of UAV swarm to perform tasks [15]. In addition, some nodes of UAV swarm may be attacked by the enemy in combat operations, causing cascading effects, affecting the generation of combat capability, and even leading to network collapse. Therefore, it has important theoretical significance and application value to establish the cascading failure model of the UAV swarm communication network and analyze its recovery ability. Cui et al. proposed a network elasticity analysis model based on node repair [16]. Referring to the classical “capacity & load model” invented by Motter and Lai (M-L model), this model constructs the main indicators such as recovery probability and average repair time. This new model can be applied to the evaluation of network elasticity. Wang et al. established a complex network model including a communication subnet, structure subnet, and task subnet. Later, he studied the invulnerability of UAV swarm networks with different configurations after being attacked from the perspective of network dynamics [17]. Khan et al. proposed an intelligent networking method. The UAV swarm communication network built according to this method can realize intelligent selection of connection mode. In this way, communication efficiency and network survivability are significantly improved compared with the traditional mode [18]. Based on the invulnerability theory and elasticity theory of complex networks, Gu et al. simulated the failure of some nodes due to the attack of a UAV swarm. This work explained the importance of the elasticity index to UAV swarm and proposed the elasticity optimization strategy for the network [19]. Sun et al. proposed a multidomain cooperative method for UAV swarm. Compared with the previous research results, this method can significantly improve the communication network elasticity of UAV swarm when they perform reconnaissance tasks, and make the disturbed UAV swarm performance quickly recover above the capability baseline [20].

Although the existing researches on the invulnerability and elasticity of UAV swarm communication networks are rich in content, there are still some theoretical and practical problems, which can be summarized as follows:

(a) Most researches on network invulnerability and elasticity focus on single-layer networks. However, when UAV swarms perform tasks, they often have regional division or network layering, and each layer is connected with others. This kind of network belongs to the category of interdependent network research. Single layer network can not accurately simulate the operation of a communication network when large-scale UAV swarm action. Especially, when the network is attacked, the invulnerability and elasticity of interdependent networks are greatly affected by their interdependent links [21].

(b) The research on invulnerability and elasticity of UAV swarm communication networks is often carried out separately. In order to avoid conceptual conflict, most scholars chose to study only one of the characteristics between network invulnerability and elasticity. For the recoverable UAV swarm communication network, invulnerability is the upstream performance of elasticity. In fact, these two properties should be studied at the same time, so as to comprehensively grasp the influencing factors of network elasticity [22].

(c) In most studies, the generation of UAV swarm communication networks refers to the traditional complex network model topology, rather than the connection mode in actual combat. Therefore, the simulation results have little guiding significance for combat application. Scholars are used to using classical regular networks, small world networks, random networks, and other models to carry out research. However, the UAV swarm network in combat often adopts the method of setting the connection probability to connect neighbor nodes considering the length of the links. This network topology is more conducive to the implementation of efficient communication. At the same time, some nodes have better invulnerability after being attacked [23].

This paper focuses on the UAV swarm reconnaissance mission and applies complex network theory and elasticity theory to study the invulnerability and elasticity of the UAV swarm communication network. We generate three UAV swarm communication subnets through scale-free network rules. Each subnet contains about 100 nodes. Then, according to the principle of maximum degree, three subnetworks are connected to form a three-layer interdependent network. The improved M-L model is applied to establish the cascading failure model of the interdependent network. Random attacks are carried out on network nodes. The effects of initial size, capacity coefficient, load coefficient, and other parameters on the network cascading invulnerability are observed. On this basis, the degree priority recovery rule is set to recover the network connection, then the relationship between network elasticity and various parameters is studied by observing the network performance.

The simulation results show that the interdependent network has better elasticity than the isolated network. For interdependent networks, increasing the number of interdependent nodes has a great impact on the network elasticity. At the same time, moderately increasing the network capacity coefficient and controlling the scale of UAVs can also improve the elasticity of UAV swarm communication network.

2. Construction of UAV Swarm Communication Network

2.1. Interdependent Network Model of Reconnaissance UAV Swarm

There is unidirectional coupling between the networks of each layer of the RTU network model. The main characteristics of the network are extracted, and the RTU network model is constructed as shown in the Figure 1.

Carrying out battlefield reconnaissance missions is an important application form of UAV swarm in combat. UAV swarm also has significant advantages in reconnaissance operations because of its flexibility, good concealment, and low cost. According to the characteristics of UAV swarm to a ground reconnaissance mission, a UAV swarm is dispatched to target groups composed of three reconnaissance areas on average. The UAV swarm can be divided into three subgroups. In order to ensure the air coordination and intelligence sharing of the UAV swarm, the communication within each subgroup needs to be kept smooth, that is, the communication network is stably connected. At the same time, in order to ensure that the UAVs in the whole swarm share the battlefield situation, the three subgroups should also maintain smooth communication. Thus, connections between three layers of networks should be established. As shown in Figure 1, we build the model and divide the UAV swarm into three subgroups, namely UAVS1, UAVS2, and UAVS3, to conduct reconnaissance on three areas Ω 1, Ω 2, and Ω 3 respectively.

[figure(s) omitted; refer to PDF]

We build a three-layer complex network mathematical model to represent the complex node motion behavior and connection topology of the UAV swarm communication network effectively. Because of the connection between subnets, the three-layer network is an interdependent network [24–26]. Each UAV is regarded as a node of this interdependent network, and its connection relationship is regarded as the link of the network. Assuming that a total of $N$ UAVs are divided into $m$ groups and perform reconnaissance missions in $M$ areas at the same time, where $m=M$, the reconnaissance UAV swarm interdependent network model can be established.$$\begin{array}{}\text{(1)}& G=G^1\cup G^2\cup \dots \cup G^m,\end{array}$$where $G$ is the reconnaissance UAV swarm communication interdependent network composed of $N$ UAVs, $G^1,G^2,\cdot \cdot \cdot ,G^m$ are the 1st, 2nd, . . . , and $m$th reconnaissance UAV swarm communication subnetworks, respectively [27–29].

The network model of the $l$ st reconnaissance UAV swarm communication network is:$$\begin{array}{}\text{(2)}& G^l=V^l,E^l,l=\mathrm{1,2},\cdot \cdot \cdot ,m,\end{array}$$where $V^l$ is the UAV node set in the $l$ st subnetwork, and $E^l$ is the set of connection relations between UAV nodes in the $l$ st subnetwork.

In the reconnaissance UAV swarm communication network model, the sum of nodes is $N$, and the nodes of each subnetwork do not intersect, as shown in equations (3) and (4).$$\begin{array}{}\text{(3)}& {\displaystyle \sum _{l=1}^m{V}^l}=N,\text{(4)}& V^1\cap V^2\cap \dots \cap V^m=\varnothing .\end{array}$$

Furthermore, the interlayer coupling relationship between the three subnetworks is established, as shown in Figure 2. In order to increase the communication efficiency of interlayer connections as much as possible, the nodes with a large degree of subnetworks are selected to establish interlayer connections. In Figure 2, TE ${\text{u}}_{\text{i}}\text{i}=\mathrm{1,2,3}\dots \text{n}$ are the interlayer dependence links between networks of UAVS1 and UAVS2, as well as $w_{j}j=\mathrm{1,2,3}\dots m$ are the interlayer dependence links between networks of UAVS2 and UAVS3. Values of $u_i$ and $w_j$ indicates the dependence of the links between two layers. Where $u_i\in \mathrm{0,1}$ and $w_j\in \mathrm{0,1}$. When $u_i=0\text{\hspace{0.17em}or\hspace{0.17em}}{w}_j=0$, there is no interlayer dependence between the two nodes linked by $u_i$ or $w_j$, then the links can be deleted. When $u_i=1\text{\hspace{0.17em}or\hspace{0.17em}}{w}_j=1$, the dependence between the two layers is connected by the link of $u_i$ and $w_j$ is the strongest, and the failure of a certain node can cause the complete failure of the connecting nodes.

[figure(s) omitted; refer to PDF]

2.2. Generation Rules of UAV Swarm Interdependent Network

To build the model of a real network, it is necessary to consider the main characteristics of the network and the interaction between nodes. In the early research of complex networks, people used to build regular network models, such as ring networks, star networks, and so on. However, because the physical network is often more complex than the regular network, this modeling method is gradually replaced by other network models. At present, Erd s-Rény (ER) random network, small-world network, and scale-free network are several commonly used network models [30–32].

When modeling ER random network, it is no longer simple connection rule like regular network, but a certain connection probability is given to describe the relationship between nodes. The connection probability of ER random network follows the following rules:$$\begin{array}{}\text{(5)}& P=\frac{2n}{NN-1},\end{array}$$where $N$ represents the total number of nodes, and $n$ represents the total number of links. $NN-1/2$ is the maximum number of possible connected links, where $n<NN-1/2$.

The small world network is derived from a famous paper published by Watts and Strogatz in Nature. This paper has epoch-making significance for the research and development of complex network science. This network model filled the research gap between “regular network” and “random network.” The small world network describes the network form of a certain regular network after adding random links, but it has not yet reached the random state. This kind of network is more accurate than a regular network and random network in describing various network relationships.

As the research goes deeper, people found that using small-world network model to study interpersonal networks and public opinion communication network has unique advantages. However, the “small world” characteristics of many real networks are not significant. In their research, Barabasi and Albert found that the degree of most networks, in reality, obeys power-law distribution, and they called this characteristic “scale-free.” Based on this discovery, they proposed a new network model, the scale-free network model. Scholars gradually discovered that the degree distribution of the Ad Hoc network that realizes the swarm communication will show significant scale-free. Therefore, using a scale-free network model to simulate the UAV swarm communication network is more in line with the actual situation [33].

The core of scale-free network modeling is to define the network connection rules. Tran et al. proposed a method to build a scale-free network using a preference connection algorithm [34]. At the same time, considering the cost of network connection and reconnection, the different network connection modes are compared. Assuming that the number of initially fully connected nodes is $m_0$, a new node $j$ creates $m$ connections with existing nodes in the network $m\le m_0$, then the connection probability between node $j$ and existing node $i$ is defined as:$$\begin{array}{}\text{(6)}& P_{j⟶i}=\frac{k_i}{{\displaystyle {\sum }_{i=1}^{N_t}{k}_i}},\end{array}$$where $N_t$ represents the number of existing nodes in the network at time $t$, and $k_i$ represents the number of existing connections of node $i$, that is, the degree of node. In the recovery stage, the probability of node $j$ losing connection due to the disappearance of other nodes to connect with other nodes is also shown in equation (6). In the attack phase, the scale-free network has high survivability for randomly removed attack modes. However, scale-free networks are vulnerable to targeted attacks. Based on the above research, Barthelemy introduced the distance factor into the connection probability model and proposed different adjustment variables and distance functions [35].$$\begin{array}{}\text{(7)}& P_{j⟶i}=\theta k_{i}F{d}_{j⟶i},\end{array}$$where $\theta$ represents the adjustment factor, and $k_i$ represents the number of existing connections of node $i$. $F{d}_{j⟶i}$ is the distance function, representing the influence of physical distance on the connection status and information transmission between UAVs in the swarm, where $d_{j⟶i}$ is the distance between node $j$ and node $i$.

Considering the self-organization and self-adaptive characteristics of UAV swarm, Bai proposed a new network model [36]. He pointed out that if there are isolated nodes or branches in UAV swarm, the overall performance of the swarm will be reduced. The connection probability equation for this model is as follows [37–39].$$\begin{array}{c}\text{(8)}& P_{j⟶i}=\alpha k_i+\epsilon F{d}_{j⟶i}=\begin{array}{c}\frac{k_i+\epsilon F{d}_{j⟶i}}{{\displaystyle {\sum }_{i=1}^{N_t}F{d}_{j⟶i}{k}_i+\epsilon }},{\displaystyle \sum _{i=1}^{N_t}F{d}_{j⟶i}>0}\\ 0,\text{otherwise}\end{array},\end{array}$$where $N_t$ represents the number of existing nodes in the network, $\epsilon$ represents the adjustment factor of node degree value, and $k_i$ represents the number of existing connections of node $i$, $i$, that is, the node degree value [40].

The function $F{d}_{j⟶i}$ is as follows:$$\begin{array}{}\text{(9)}& F{d}_{j⟶i}=\begin{array}{c}1,d_{j⟶i}<\eta r_c\\ \frac{r_c-d_{j⟶i}}{1-\eta r_c},\eta r_c<d_{j⟶i}<r_c,\\ 0,d_{j⟶i}>r_c\end{array}\end{array}$$where $r_c$ represents the communication range, $d_{j⟶i}$ represents the distance between nodes $j$ and $i$, and $\eta$ represents the distance influence factor. It should be noted that when ${\displaystyle {\sum }_{i=1}^{N_t}F{d}_{j⟶i}}=0$, the probability of node $j$ connecting with any existing node in the network is 0. That is, there are no nodes connected to node $j$ within the maximum communication range [41–44].

Based on the network generation rules of equations (8) and (9), the simulation software is used to generate three complex UAV swarm communication networks with different parameters. The network generation parameters are shown in Table 1. In order to ensure that the experimental data conform to the actual scale of future UAV swarm operations, 100 UAVs are set in each swarm to form three UAV swarm networks to scout the target area, as shown in Figures 3–5.

Table 1

Parameters of UAV communication network.

Parameter	Meaning	Value
N	Swarm size	100
m	Number of connections with original nodes in the network	2
m0	Initial full connections	5
D	UAV mission area	100 × 100
ε	Adjustment factor of node degree value	1
rc	Communication distance	20
η	Distance influence factor	0.25

[figure(s) omitted; refer to PDF]

The interlayer communication among the three networks depends on the five nodes with high node degree in each subnetwork. The first and third networks are respectively connected with the second, while the first and third networks rely on the second network for information exchange. Therefore, the three UAV networks are connected into a three-layer interdependent network, as shown in Figure 6.

[figure(s) omitted; refer to PDF]

3. Cascading Failure Model of UAV Swarm Interdependent Network

When the UAV swarm communication network is attacked or some UAV nodes fail because of their own breakdowns, the faults will spread through the network due to the connection relationship. In order to accurately describe the propagation of faults, the network cascading failure model is selected. Based on the classical model, it can accurately describe the failure propagation process in three-layer swarm communication interdependent network.

3.1. Cascading Failure Model

The M-L model proposed by Motter and Lai is a classical model for cascading failure of complex networks [45]. According to the model establishment rules, set the capacity of node $i$ to $C_i$, the initial load to $L_i$. And $\lambda$ is the capacity coefficient. According to the M-L model, $C_i$ and $L_i$ have the following relationship:$$\begin{array}{}\text{(10)}& C_i=1+\lambda L_i,i\in \mathrm{1,2},\dots ,N,\end{array}$$where $L_i$ is determined by the pitch number.$$\begin{array}{}\text{(11)}& Q_i=1+\mu C_i,\end{array}$$where $Q_i$ is the load of the node and $\mu$ is the load factor.

When a node fails, the load of the failed node is distributed to the node directly connected to the failed node. The load obtained by the node directly connected to the failed node after one load distribution is as follows:$$\begin{array}{}\text{(12)}& {\overline{M}}_R=\frac{R{M}_R}{n_R},\end{array}$$where $R$ is the adjustment factor of load distribution of failed nodes, $n_R$ is the number of nodes connected to the failed node, and $M_R$ is the load of the failed node at the moment of load distribution [46].

Based on the M-L model of single-layer network, the cascading failure model of three-layer interdependent network is considered. If node $j$ in network B is connected to some nodes in network A to form interdependent links, the following principles should be followed when cascading failure propagates between two layers of subnetworks:

(a) When node $i$ connected with node $j$ in network A fails, the load of node $i$ is only transferred in network A, not to node $j$.

(b) When all the interdependent nodes of node $j$ in network A fail, node $j$ fails immediately, and its load is transferred to node $m$ in network B, which is the interdependent node of node $j$, according to the following equation:$$\begin{array}{}\text{(13)}& {\eta }_m=\frac{C_m^B}{{\displaystyle {\sum }_{n\in {\Gamma }_L}{C}_n^B}}.\end{array}$$

(c) When the load of node $m$ exceeds the capacity ($L_m+{\eta }_j{L}_j>C_m$), node $m$ fails, and cascading failure propagates in network B.

In the swarm communication interdependent network studied in this paper, the failure propagation from network B to network C is the same as that from network A to network B. And the failure can spread two-way between the three layers of networks. In particular, when node $m$ in network B is an interdependent node between networks A and B as well as networks B and C. Meanwhile, node $t$ in network C has only one interdependent node $m$ in network B, so the failure of node $m$ and node $t$ occurs at the same time. The cascading failure caused by the failure of node $m$ and node $t$ in networks B and C also begins to spread at the same time.

3.2. Cascading Failure Experiment

Attacks on complex networks can be divided into random attacks and targeted attacks. Random attack refers to the equal probability attack on each node in the network, which does not distinguish the importance of nodes. Targeted attacks are attacks on nodes according to certain rules, such as attacking nodes from high to low degrees. Considering that the reconnaissance UAV swarm communication network is not easy to be identified by the enemy. The enemy attacks the swarm at random mostly. Therefore, in this experiment, random attack is used to attack the UAV swarm communication network. First, when the three-layer interdependent network is not connected, the network is randomly attacked by setting a single parameter change. Observe the performance change trend of the UAV swarm communication network after three isolated networks are attacked, and analyze its invulnerability. Then, the interdependent links are introduced to construct the interdependent network, and the whole swarm network is attacked randomly. Observe the performance change trend of the network and conduct a comparative analysis. The initial experimental parameter settings are shown in Table 2.

Table 2

Initial parameters of cascading failure propagation experiment.

Parameter	Meaning	Value
N	Swarm size	100
Nh	Number of damage UAVs in a single attack	20
m	Number of connections with original nodes in the network	2
m0	Initial full connections	5
D	UAV mission area	100 × 100
ε	Adjustment factor of node degree value	1
rc	Communication distance	20
η	Distance influence factor	0.25
λ	Capacity factor	0.2
μ	Load factor	0.2

3.3. Results and Analysis of Cascading Failure Experiment

The three-layer interdependent network model established in this paper is attacked randomly to observe the network invulnerability under different parameters. Considering that the communication performance is the most important capability of swarm communication network, this paper chooses network communication performance as the measure of network invulnerability.

Node state $s_{i}t$ indicates the performance of node $i$ at time $t$. Node overload will cause the state of the node to drop, that is, $s_{i}t$ becomes smaller. When a node is overloaded, the working efficiency of the node will be reduced, resulting in changes in the node state. Set the status of node $i$ at different times as $s_{i}t$:$$\begin{array}{}\text{(14)}& s_{i}t=\begin{array}{c}1,M_i<C_i,\\ \frac{C_i}{M_i},C_i<M_i<Q_i,\\ 0,Q_i<M_i.\end{array}\end{array}$$

The network communication performance is the sum of the performance of each node, expressed in $St$ [47–49]:$$\begin{array}{}\text{(15)}& St={\displaystyle \sum _{i\in N}{s}_{i}t}.\end{array}$$

The experimental results are shown in the following figures. It can be seen from (13) and (14) that $St$ is obtained by superimposing the normalized magnitude $s_{i}t$. Therefore, when the vertical axis in the following is Network Communication Performance, it has no unit.

Random attacks are carried out on the three-layer UAV swarm network. The average communication performance changes of the three subnetworks in the interdependent network are shown in Figures 7–9.

[figure(s) omitted; refer to PDF]

Figure 7 shows the impact of the number of attacked nodes on the network communication performance. It can be seen from the figure, after being attacked, cascading failure occurs in the three-layer interdependent network. As the number of damaged nodes increases after being attacked, the time of cascading failure process increases, and the network communication performance decreases gradually. The more the damage, the worse the network performance is after random attack.

Figures 8 and 9 show the trend of network communication performance as parameter λ and parameter μ change, respectively. It can be observed that during the initial stage of cascading failure, there is no obvious rule between capacity factor and load on the rate of network performance degradation. However, after the cascading failure, the larger the capacity factor and load value factor, the better the network communication performance of the UAV swarm network will be the more destructive they will be.

4. Elasticity of Swarm Communication Network

4.1. Establishment of Elasticity Index

Elasticity generally refers to the performance of an individual or system to restore its original ability and state after being disturbed. For the reconnaissance UAV swarm communication network, after the network is disturbed, it will experience three stages: performance degradation, performance recovery, and performance stability. The specific performance changes are shown in Figure 10 below [50].

[figure(s) omitted; refer to PDF]

At time $t_0$, when the reconnaissance UAV swarm communication network operates normally, it can be considered that the network performance is 100%. At time $t_1$, the swarm communication network is disturbed by random attack, and the network performance is degraded. At time $t_{\min }$, the network performance reaches the lowest value $S_{\min }$. By implementing the recovery strategy, the performance gradually rises, and recovers to the stable level $S_{final}$ at time $t_2$ until the end of the observation period at time $t_{end}$.

The quantitative measurement of network elasticity mainly includes the definition of elasticity index based on network topology parameters, the definition of elasticity index based on cumulative performance changes, and the elasticity index considering invulnerability and rapidity. Defining the elasticity index with the network topology parameters needs to calculate the maximum connectivity subgraph size after the network is attacked or disturbed. This method is more intuitive and can reflect the change trend of relevant parameters of the network after being disturbed. However, this index is only applicable to the rough analysis of the performance of simple network models. Therefore, this index can not accurately analyze the elasticity of large-scale UAV swarm network with complex structures [50, 51].

Elasticity index defined by cumulative performance change interprets the elasticity index as the overall performance change of the network during the observation period, which is expressed in the form of integral as follows:$$\begin{array}{}\text{(16)}& R_1=\frac{{\displaystyle {\int }_{t_0}^{t_{end}}Stdt}}{t_{end}-t_0},\end{array}$$where $St$ can be obtained from discrete time series during simulation. So (15) can be expressed as follows:$$\begin{array}{}\text{(17)}& R_2=\frac{{\displaystyle \sum _{t_0}^{t_{end}}St}}{t_{end}-t_0}.\end{array}$$

The dynamic performance of the network can reflect the dynamic change of the performance through the elastic index. However, the simple index value can not accurately reflect the invulnerability and rapidity, which needs to be investigated in combination with the performance change curve [52].

The elasticity index considering invulnerability and rapidity is evolved from the definition of elasticity index based on the cumulative change of performance. Based on the performance change curve, the index comprehensively considers the overall performance change $\alpha$, invulnerability $\beta$, rapidity $\tau$, and recovery $\gamma$ of the swarm communication network. Where,$$\begin{array}{}\text{(18)}& \alpha =\frac{{\displaystyle {\int }_{t_0}^{t_{end}}Stdt}}{S{t}_0{t}_{end}-t_0},\text{(19)}& \beta =\frac{S_{\min }}{S_0},\text{(20)}& \tau =\frac{t_2-t_1}{t_{end}-t_0},\text{(21)}& \gamma =\frac{S_{\text{final}}}{S_0}.\end{array}$$

For elastic analysis, the perturbation and recovery phases are considered. The $\alpha$ is divided into performance change ${\alpha }_1$ under perturbation and performance change ${\alpha }_2$ under recovery. The $\tau$ is expressed as the absorption rapidity parameter ${\tau }_1$ and recovery rapidity parameter ${\tau }_2$. Then the elasticity index is defined as follows:$$\begin{array}{}\text{(22)}& R_3={\alpha }_1{\tau }_1\beta +{\alpha }_2{\tau }_2\gamma .\end{array}$$

Tran et al. considered the volatility of the network, proposed the volatility factor $\theta$, and expressed the elasticity index as follows:$$\begin{array}{}\text{(23)}& R_4=\begin{array}{c}\alpha \gamma \beta +\theta +1-{\tau }^{\gamma -\beta },\gamma \ge \beta \\ \alpha \gamma \beta +\theta \text{,otherwise}.\end{array}\end{array}$$

For the case of multiple disturbances, the total elastic value can be obtained by weighting the system elasticity value at each disturbance [53]. This kind of index can completely and accurately reflect the definition of UAV swarm elasticity. Therefore, this index is selected to test the elasticity of UAV swarm communication network in this paper.

4.2. Elasticity Experiment of UAV Swarm Communication Network

First, a single network without interlayer connection relationship is randomly attacked according to the parameters in Table 2. Then, a three-layer interdependent network with interlayer connection based on the network generation rules in this paper (the network shown in Figure 6) is randomly attacked. The network elasticity under different structures and different parameters is compared and analyzed.

Observe the network performance degradation and recovery process after an attack when three subnetworks are not connected to interdependent networks. The test results are shown in Figures 11–13:

[figure(s) omitted; refer to PDF]

The initial number of UAVs in swarm communication network in Figure 11 is 100. When 20 UAVs are attacked randomly, the communication performance of the UAV network is degraded to 53.0% of the initial state. The number of UAVs damaged due to cascading failure was 27, accounting for 57.5% of the total damage. Conduct network recovery after the attack process. When 10 UAVs are restored, the network communication performance reaches 76.0% of its initial state, and the network elasticity R = 0.9947.

The initial number of UAVs in swarm communication network in Figure 12 was 100. Some nodes were not connected when the network was generated, and 94 UAVs were successfully connected. When 20 UAVs were attacked randomly, the UAV network communication performance was degraded to 54.0% of the initial state. The number of UAVs damaged due to cascading failure is 20, accounting for 50.0% of the total damage. Conduct network recovery after the attack process. After 15 UAVs are recovered, the UAV network communication performance reaches 76.0% of the initial state, and the network elasticity R = 1.0933.

The initial number of UAVs in swarm communication network in Figure 13 is 100. When 20 UAVs are attacked randomly, the communication performance of the UAV network is degraded to 64.0% of the initial state. The number of UAVs damaged due to cascading failure was 16, accounting for 44.4% of the total damage. Conduct network recovery after the attack process. After 15 UAVs are recovered, the UAV network communication performance reaches 81.0% of the initial state, and the network elasticity R = 1.1184.

Add interdependent links to three isolated networks to form a three-layer interdependent network. The elasticity of the interdependent network is observed after being attacked through experiments.

We select five nodes with the largest degree in each of the three subnetworks UAVS1, UAVS2, and UAVS3, and establish interlayer links between these nodes. In this way, the UAV swarm communication network is established. The initial number of UAVs in swarm communication network in Figure 14 is 300. Except for some isolated nodes, a total of 294 have been connected. When 60 UAVs are attacked randomly, the communication performance of UAV network is degraded to 56.1% of the initial state, which is close to that of a single layer. The number of UAVs damaged due to cascading failure is 69, accounting for 53.5% of the total damage, which is similar to that of single layer. After the restoration of 40 UAVs, the communication performance of UAVs reached 86.1% of the initial state, which was significantly higher than that of single-layer network. It shows that the establishment of three-layer interdependent network is helpful to improve the performance of swarm communication network. The elasticity of the three-layer network is R = 1.1151, which is higher than the average value of the three single-layer elasticity, indicating that the three-layer interdependent network helps to improve the network elasticity.

[figure(s) omitted; refer to PDF]

We further studied the elastic changes of UAV swarm communication interdependent networks under different network sizes and parameters. The test results are shown in Figures 15–18.

[figure(s) omitted; refer to PDF]

It can be clearly seen from the above figure that the elasticity of the UAV swarm is declining with the increase in network scale. When the number of UAVs in the swarm is 50, the network elasticity is 1.483, while when the number of UAVs increases to 250, the network elasticity is only 0.863. This is because, with the increase of the network scale, the nodes are easier to be attacked when the swarm is attacked. After the number of attacked nodes increases, it takes more time to recover nodes, so the elasticity becomes lower and lower.

It can be seen from the above figure that with the increase of capacity coefficient, the elasticity of UAV swarm network gradually increases, but the rising rate gradually decreases. When the capacity coefficient increases to 0.9, the elasticity no longer increases and decreases slowly. This indicates that when the node capacity is too large, the elasticity will appear extreme value instead of increasing all the time.

Figure 17 shows the variation of network elasticity with the number of damaged nodes. As the number of destroyed nodes increases, the swarm elasticity decreases gradually. The increase in the number of damage directly increases the impact on the network, making it more difficult to recover, so the elasticity is lower.

Furthermore, according to the characteristics of interdependent networks, the effect of interdependent nodes’ number on the network elasticity is observed. As the number of interdependent nodes increases, the network is more closely connected, and the network elasticity increases quasi-linear. Thus, increasing the number of network interdependent links can promote the network to complete reconnection more quickly after being destroyed.

5. Summary and Outlook

In this paper, the performance of swarm communication is studied based on the background of the UAV swarm reconnaissance mission. A three-layer interdependent network model is constructed by setting the scale-free network generation rules. The M-L model is improved and applied to describe the cascading failure process in the interdependent network. Through simulation experiments, it is found that when random attacks are carried out on three-layer interdependent network, with the increase of capacity coefficient and load value coefficient, the communication performance retention of the UAV swarm network gradually increases, and the invulnerability is improved. The influence of different network generation parameters and rules on the network elasticity is explored. We conclude that the interdependent network has higher elasticity than the three single-layer networks. And the more interdependent nodes the network has, the higher the swarm network elasticity will be.

The integrity of the communication network is an important guarantee for UAV swarm to carry out operational tasks. When the UAV swarm is attacked, the resilience of the communication network, that is, network elasticity, is particularly critical.

It can be seen from the research conclusion of this paper, in order to ensure the maximum recovery of communication ability after the reconnaissance UAV swarm is attacked, the swarm network should be dissociated into subnetworks according to the reconnaissance areas. Then the interdependent network should be constructed by establishing the links between layers. Furthermore, the interdependent network elasticity can be enhanced by controlling the swarm size, appropriately increasing the capacity coefficient, and increasing the number of interdependent links, so as to obtain better communication reliability and improve the reconnaissance operation capability of the swarm.

Authors’ Contributions

The authors confirm their contribution to the paper as follows. Boyu Feng and Zhihao Zhang conceptualized and designed the study; Zhihao Zhang collected the data; Lin Zhou and Boyu Feng analysed and interpretated the results; Lin Zhou prepared the original draft. All authors reviewed the results and approved the final version of the manuscript.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants 05171101 and 71901216.

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