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Abstract

In the future, the Internet of things will reduce the cell radius and increase the number of low-power nodes to support thousands of times of traffic growth under 5G. As a virtual multiple input multiple output technology, cooperative communication technology can solve these problems effectively. According to the evolution characteristics of cooperative communication networks, a multi-domain cooperative communication network evolution model with preferential attachment and random attachment is constructed in this article. And then, the network properties and robustness are analyzed using the mean-field method and different attacks. Aiming at the resource constraints and resource allocation problems of communication nodes, a relay selection strategy based on the combination of maximum degree and minimum clustering coefficient is proposed. The simulation results show that the relay node selection strategy based on the combination of maximum degree and minimum clustering coefficient has significant advantages in selection steps and selection time, which greatly enhanced the performance of relay selection in multi-domain cooperative communication networks. Through real-time monitoring and updating of the performance and security indicators of the multi-domain cooperative communication networks, it provides a strong guarantee for the node deployment and security management of the Internet of things cooperative communication system.

Keywords

Cooperative communication network degree distribution robustness relay selection limited resources

Introduction

The Internet of things (IoT) is a communication network between things established through the Internet, which can realize data transmission and management monitoring between things. In the 5G network environment with large capacity, low latency, and high speed, it can fully meet the data transmission requirements of the IoT, which will enable the application of the IoT to be comprehensively and deeply developed. The 5G network is to reduce the cell radius and increase the number of low-power nodes to support users’ demands for a 1000-fold increase in traffic. Among them, ultra-dense heterogeneous networks, self-organizing networks, and massive MIMO (multiple input multiple output) technologies have become several key technologies for improving data traffic in the future 5G network.¹ However, the deployment of ultra-dense networks will lead to denser interference distribution in the spatial domain, and cooperative communication technology can effectively overcome this interference with minimal changes to the original network architecture and provide a more uniform and continuous quality of service in the geographical area.^2,3 Cooperative communication technology is a virtual MIMO technology that shares single-antenna mobile devices with other mobile devices in the multi-user scenario, thereby achieving diversity gain of multi-antennas. It has the advantage of eliminating network coverage blind spots, expanding network coverage, saving construction costs, improving communication system performance, and rapidly deploying networks. It is widely used in cellular mobile communication networks,⁴ wireless local area networks,⁵ and wireless sensor networks.⁶ In the future 5G environment, the IoT will develop in the direction of broadband, diversification, intelligence, and integration. With the rapid development of various smart terminal devices, mobile data traffic will grow explosively. Due to the diversity and difference in performance and function of the increasingly large number of low-power nodes, the IoT integrated with cooperative communication technology forms a dynamic and complex multi-domain heterogeneous cooperative communication network.^7,8

The complex networks theory can conduct in-depth research on the network topology, functional properties, action laws, and predictive control of complex systems. It has penetrated into many disciplines and fields and set off an upsurge of research at home and abroad. The multi-domain heterogeneous cooperative communication network of the complex IoT is composed of a large number of nodes with functional diversity and performance differentiation, which is distributed in different geographical areas to form a network topology by self-organization and realize the advantages of cooperative communication diversity gain, so that the whole network presents a variety of complex network characteristics. There are many research results of complex network evolution models, such as Erdös and Rényi (ER) random graph model, Watts and Strogatz (WS) small-world model, Newman and Watts (NS) small-world model, and Barabasi and Albert (BA) scale-free model. The two key characteristics of BA scale-free model are growth⁹ and preferential attachment,¹⁰ which makes the degree distribution of the generated network show the characteristics of power-law distribution,¹¹ while the degree distribution of the network generated by the evolution of ER random graph model shows the exponential distribution. However, the degree distribution of most practical networks in nature falls between exponential and power-law distribution. According to the topological characteristics of the cooperative communication network in different scenarios, a variety of cooperative communication network evolution models are proposed.^12,13 However, the cooperative communication network models based on complex networks are all based on local homogeneous networks, and there are still some limitations when applied to actual multi-domain heterogeneous cooperative communication networks.

In this article, a multi-domain cooperative communication network evolution model with preferential attachment and random attachment is constructed. Using the rate equation and mean-field method, the topological properties of the network such as degree distribution are studied. Then, the robustness of the network model is analyzed. Finally, a relay node selection strategy combining maximum degree and minimum clustering coefficient is proposed. The main contributions of this article are as follows:

A multi-domain cooperative communication network evolution model with preferential attachment and random attachment is proposed, which can be properly applied to multi-domain heterogeneous cooperative communication networks in the actual IoT environment.

Five evolution steps of multi-domain cooperative communication network under different probability are given, and the corresponding rate equation of node degree is given.

The degree distribution of the network is analyzed by means of the mean-field method to calculate the rate equation and numerical simulation.

The robustness of a multi-domain cooperative communication network is studied, and the effects of three attack methods on network degree distribution, average degree, average path length, and average clustering coefficient are analyzed.

A relay selection strategy combining maximum degree and minimum clustering coefficient is proposed, and the optimal relay node selection scheme is obtained.

The rest of this article is organized as follows. The evolution model of the multi-domain cooperative communication network in detail and the degree rate equation corresponding to the five evolution steps are introduced in section “Related work.” The degree distribution of the network, which is consistent with the simulation results, is analyzed in section “Evolutionary modeling of multi-domain cooperative communication network.” Three attack strategies to analyze the robustness of the network are used in section “Analysis of network characteristics.” The superiority of the relay node selection strategy combining maximum degree and minimum clustering coefficient is proposed and verified in section “Robustness analysis.” Finally, it is concluded that the multi-domain cooperative communication network obtained in this article can effectively solve the problem of communication congestion. Then, the relay selection strategy combining maximum degree and minimum clustering coefficient has obvious advantages compared with the original strategy, and the best selection of relay nodes is obtained.

Related work

The multi-domain heterogeneous cooperative communication network is a dynamic behavior network, which needs to be analyzed and modeled from the dynamic behavior of the global cooperative communication network topology structure. Therefore, the multi-domain heterogeneous cooperative communication network evolution based on complex networks in the IoT environment model establishment and analysis still face great challenges. Based on the BA model, a multi-local world model was proposed¹⁴ to describe the topology of the Internet, showing its superiority compared to other existing models. A weighted group priority model is proposed,¹⁵ and various statistical properties such as the distribution of degree, intensity, and weight are analyzed and deduced.^16,17 A class of growing networks combining preferential attachment and random attachment is constructed,¹⁸ and the effects of random failure and intentional attack^19,20 on network performance are examined. To make the constructed network evolution model closer to the real network, researchers have made some deeper research on some topological properties in complex networks.^21,22 An information and communication network security management and control platform based on big data technology is built,²³ which can accurately collect sensor data and monitor network status in real time. A secure, efficient, and weighted access control scheme for cloud-assisted industrial IoT applications is proposed, which can better realize weighted access control.²⁴ The system composition of the new intelligent manufacturing system is proposed,²⁵ from industrial Internet system to artificial intelligence and then to the fifth-generation mobile communication; the system identified the application scenarios of information and communication technology–enabled manufacturing. The availability evaluation mechanism of the IoT system with malicious software propagation based on edge computing is constructed²⁶ to predict the probability of malicious software propagation and evaluate the steady-state availability of three typical IoT system topologies.

Due to the high speed and low latency of the 5G networks, it is proposed²⁷ to combine the 5G network and cognitive IoT to optimize available resources, providing technical help for the 5G network cognitive solutions on the IoT. To improve the transmission rate of mobile communication, a cooperative communication network model combining preferential attachment and random attachment was constructed,¹² and an in-depth study of its degree distribution and robustness was conducted. The cooperative communication fitness network model under the hybrid attachment mechanism is successively established,¹³ and four fitness distributions are considered: linear fitness distribution, unique fitness distribution, exponential fitness distribution, and Rayleigh fitness distribution. By controlling the proportion of random attachment and preferential attachment, the problem of communication congestion is alleviated to a certain extent. The security of the IoT is also a hot spot of concern. The demand for information storage is increasing in the era of big data. A complex network search strategy combining maximum and minimum clustering coefficients is proposed,²⁸ which excellently controls the occupation of network resources by some unnecessary information.²⁹ A game theory framework for Mobile two-dimensional MIMO communication networks is proposed³⁰ to realize optimal relay selection and cooperative control. A new cooperative Non-orthogonal multiple access (NOMA) transmission scheme is proposed³¹ for three-user downlink system, which is significantly superior to the prior art in terms of rate. A real-time scheme based on evolutionary fuzzy game is proposed,³² which is used to make cooperative decisions on data distribution strategies of neighboring nodes on the road. A defense scheme based on deep neural network Stackelberg game is proposed,³³ which is superior to other power allocation mechanisms. A cooperative defense mechanism based on evolutionary game is proposed³⁴ to enable physical sensor nodes to realize evolutionary stable strategy based on their own utility. In order to enhance the security of IoT communication, an elliptic curve cryptographic asymmetric information encryption algorithm is proposed³⁵ to protect the information data in the IoT. An evolutionary privacy protection learning strategy based on edge computing for IoT data sharing scheme is proposed³⁶ to effectively solve the privacy protection during the data sharing of IoT.

Compared with the above work, we construct a multi-domain cooperative communication network evolution model with preferential attachment and random attachment. These studies are aimed at the modeling of single-layer cooperative communication networks and cannot accurately describe the actual situation of multi-domain cooperative communication in the IoT environment. Based on this, this article establishes a temporal evolution model of multi-domain cooperative communication networks based on complex network theory and accurately describes the network structure and evolution mechanism of multi-domain heterogeneous cooperative communication networks. Next, we will compare our work with other works in Table 1 to further highlight our contribution.

Table 1.

Comparison between our work and others.

Paper	Scenario	Advances	Drawbacks
Liu et al.¹⁸	Scientific collaboration networks	Balances the power-law and exponential contributions to $p (k)$	The evolution is too simple and does not conform to reality
Du²³	Communication networks	Accurately collect sensor data and monitor network status in real time	Safety technology is not comprehensive enough
Shen et al.²⁶	IoT systems	Maximize the availability of IoT system under three typical IoT system architectures	Failure to allocate Industry Internet of things (IIOT) system resources reasonably during availability assessment
Wang et al.¹²	Cooperative communication networks	Effectively plan communication resources	Failure to completely solve the congestion problem of communication nodes
Wang et al.¹³	Cooperative communication fitness networks	Alleviate pressure of cooperative communication	Failed to achieve the best performance of Cooperative communication fitness networks(CCFN)
Feng²⁸	Complex networks	It is superior to other search strategies in time and steps	More complex network features such as weights are not considered
Liu et al.³⁰	Cooperative-relaying networks	Realize optimal relay selection and cooperative control	Evolutionary game framework to be improved
Fang et al.³¹	Communication systems	It is significantly superior to the prior art in speed	Practical application is too ideal
Liu and Zhang³⁵	IoT networks	Improve information storage speed in the IoT	Security and storage need to be further improved
Shen et al.³⁶	IoT networks	Effectively resist malicious intrusion and protect sensitive information from leakage	Poor consideration of data sender privacy protection
This article	IoT networks	The network structure and evolution mechanism of multi-domain heterogeneous cooperative communication network are accurately described A relay node selection strategy combining maximum degree and minimum clustering coefficient is proposed	Insufficient consideration of the diversity of nodes in the IoT

IOT: Internet of things.

Evolutionary modeling of multi-domain cooperative communication network

In the multi-domain cooperative communication network under the environment of the IoT, some nodes with better performance will oversaturate their communication capacity because they attract other nodes to communicate with them. Therefore, it is assumed that the nodes in the cooperative communication network will grow over time according to the attachment rules that are neither completely preferential nor completely random. Here, the probability expression of node attachment is as follows

Π (k_{i}) = \frac{(1 - p) k_{i} + p}{Σ_{j} [(1 - p) k_{j} + p]}

where $p$ is an adjustable parameter, and its value range is [0,1]. The proportion of random attachment and preferential attachment is controlled by adjusting the value of parameter $p$ . Here, $p$ is the communication probability that the new node is randomly attached to the node, and $1 - p$ is the communication probability that the new node is preferentially attached to the node. $k_{i}$ represents the degree of node $i$ .

Initial state

Suppose that there are independent single-domain cooperative communication networks, there are $m_{0}$ nodes and $e_{0}$ links in each communication network. Complete the following five steps in each unit time $t$ .

Step 1

Select a cooperative communication network $W_{y}$ , randomly. And then, add a new equipment node with probability $p_{1}$ and connect to the $m_{1}$ existing nodes in the same network with the probability $Π$ . The rate equation for $k_{i}$ can be obtained as shown in equation (1)

\frac{\partial k_{i}}{\partial t} = \frac{m_{1} p_{1}}{n} Π

(1)

Step 2

Select a cooperative communication network $W_{y}$ , randomly. And then, add a new link into it with probability $p_{2}$ . One end of the added link randomly selects a node in $W_{y}$ , and another end of the added link is linked with the other node in $W_{y}$ with probability $Π$ . The rate equation for $k_{i}$ can be obtained as shown in equation (2)

\frac{\partial k_{i}}{\partial t} = \frac{p_{2}}{n} (\frac{1}{N_{y} (t)} + (1 - \frac{1}{N_{y} (t)}) Π)

(2)

Step 3

Select a cooperative communication network $W_{y}$ , randomly. And then, delete a weak node with degree $m_{2}$ with probability $p_{3}$ . Meanwhile, the links of the deleted node are also deleted from $m_{2}$ nodes with the probability $Π^{'}$ . The rate equation for $k_{i}$ can be obtained as shown in equation (3)

\frac{\partial k_{i}}{\partial t} = - \frac{m_{2} p_{3}}{n} Π^{'}

(3)

where $Π^{'} = \frac{1}{N_{y} (t) - 1} (1 - Π)$ .

Step 4

Select a cooperative communication network $W_{y}$ randomly. And then, delete a link with probability $p_{4}$ . One end of the deleted link randomly selects a node in $W_{y}$ , and another end of the deleted link is deleted with the other node with probability $Π$ . The rate equation for $k_{i}$ can be obtained as shown in equation (4)

\frac{\partial k_{i}}{\partial t} = - \frac{p_{4}}{n} (\frac{1}{N_{y} (t)} + (1 - \frac{1}{N_{y} (t)}) Π^{'})

(4)

Step 5

Randomly select two cooperative communication networks and add a long link with probability $p_{5}$ . One end of the added link randomly selects a node in one network to link, and the other end of the added link connects with a node in another network with probability $Π$ . The rate equation for $k_{i}$ can be obtained as shown in equation (5)

\frac{\partial k_{i}}{\partial t} = \frac{p_{5}}{n} \frac{1}{N_{y} (t)} + \frac{p_{5}}{n - 1} Π

(5)

The parameters satisfy $0 \leq p_{1}, p_{2}, p_{3}, p_{4}, p_{5} \leq 1$ , and $p_{1} + p_{2} + p_{3} + p_{4} + p_{5} = 1$ . Repeat the above steps until the target network scale is reached.

Analysis of network characteristics

Degree distribution calculation

First, the degree distribution $p (k)$ characteristics of the multi-domain cooperative communication networks are analyzed.

The number of nodes in a cooperative communication network $W_{y}$ can be obtained first. In unit time $t$ , the average value is shown as

N_{y} (t) = \frac{n m_{0} + p_{1} t - p_{3} t}{n} = m_{0} + p_{1} t - p_{3} t

(6)

where $y \in [1, n]$ , let $t \to \infty$ and $n$ is countable.

In unit time $t$ , the average value of the total degree of nodes in a cooperative communication network $W_{y}$ is shown as

\begin{matrix} \sum_{j \in y} [(1 - p) k_{j} + p] \\ = \frac{2 n e_{0} + 2 p_{1} m_{1} t + 2 p_{2} t - 2 p_{3} m_{2} t - 2 p_{4} t + 2 p_{5} t}{n} \\ = 2 p_{1} m_{1} t + 2 p_{2} t - 2 p_{3} m_{2} t - 2 p_{4} t + 2 p_{5} t \end{matrix}

(7)

Here, $t \to \infty$ , and the constant $2 n e_{0}$ is ignored. Let $c = 2 (p_{1} m_{1} + p_{2} - p_{3} m_{2} - p_{4} + p_{5})$ .

By adding the degree rate equations of the above five steps (1)–(5), the following equation is obtained as

\begin{matrix} \frac{\partial k_{i}}{\partial t} = \frac{m_{1} p_{1}}{cn} \frac{(1 - p) k_{i} + p}{t} + \frac{p_{2}}{cn} \frac{(1 - p) k_{i} + p}{t} \\ - \frac{p_{2}}{cn} \frac{(1 - p) k_{i} + p}{t (m_{0} + p_{1} t - p_{3} t)} + \frac{p_{2}}{n (m_{0} + p_{1} t - p_{3} t)} \\ + \frac{m_{2} p_{3}}{cn} \frac{(1 - p) k_{i} + p}{t (m_{0} + p_{1} t - p_{3} t - 1)} \\ - \frac{m_{2} p_{3}}{n (m_{0} + p_{1} t - p_{3} t - 1)} \\ + \frac{p_{4}}{cn} \frac{(1 - p) k_{i} + p}{t (m_{0} + p_{1} t - p_{3} t - 1)} \\ - \frac{p_{4}}{cn} \frac{(1 - p) k_{i} + p}{t (m_{0} + p_{1} t - p_{3} t) (m_{0} + p_{1} t - p_{3} t - 1)} \\ - \frac{p_{4}}{n (m_{0} + p_{1} t - p_{3} t)} - \frac{p_{4}}{n (m_{0} + p_{1} t - p_{3} t - 1)} \\ + \frac{p_{4}}{n (m_{0} + p_{1} t - p_{3} t) (m_{0} + p_{1} t - p_{3} t - 1)} \\ + \frac{p_{5}}{c (n - 1)} \frac{(1 - p) k_{i} + p}{t} \\ + \frac{p_{5}}{n (m_{0} + p_{1} t - p_{3} t)} \end{matrix}

(8)

Let

\begin{matrix} c_{1} = \frac{m_{1} p_{1} + p_{2}}{cn} + \frac{p_{5}}{c (n - 1)} \\ c_{2} = \frac{p_{2} - m_{2} p_{3} - 2 p_{4} + p_{5}}{p_{1} - p_{3}} \end{matrix}

From the above two assignment equations, when $t \to \infty$ , we obtain

\frac{\partial k_{i}}{\partial t} = c_{1} \frac{(1 - p) k_{i} + p}{t} + c_{2} \frac{1}{t}

(9)

The analysis shows that $c > 0$ and then $c_{1} > 0$ .

When $c_{2} = 0$ , equation (9) becomes

\frac{\partial k_{i}}{\partial t} = c_{1} \frac{(1 - p) k_{i} + p}{t}

(10)

When $p = 0$ , equation (10) becomes

\frac{\partial k_{i}}{\partial t} = c_{1} \frac{k_{i}}{t}

(11)

According to the above evolution steps, the generated multi-domain communication network has an obvious scale-free property.

When $p = 1$ , equation (10) becomes

\frac{\partial k_{i}}{\partial t} = c_{1} \frac{1}{t}

(12)

Obviously, the network generated in this way is still a completely random cooperative communication network.

When $c_{1}, c_{2} \neq 0$ , by analyzing equation (9), we can get

k_{i} = c t^{(1 - p) c_{1}} - \frac{p c_{1} + c_{2}}{(1 - p) c_{1}}

(13)

Since $c_{1}, c_{2} \neq 0$ , the generated cooperative communication network is not completely random or scale-free but evolves between them. Here, we substitute the initial condition $k_{i} (t_{i}) = m_{1}$ into the above formula, and the following equation is obtained

\begin{matrix} k_{i} (t) = (m_{1} + \frac{p c_{1} + c_{2}}{(1 - p) c_{1}}) {(\frac{t}{t_{i}})}^{(1 - p) c_{1}} \\ - \frac{p c_{1} + c_{2}}{(1 - p) c_{1}} \end{matrix}

(14)

The average field method is used to calculate the degree distribution of the generated network

\begin{matrix} P (k_{i} (t) < k) \\ = P [(m_{1} + \frac{p c_{1} + c_{2}}{(1 - p) c_{1}}) {(\frac{t}{t_{i}})}^{(1 - p) c_{1}} \\ - \frac{p c_{1} + c_{2}}{(1 - p) c_{1}} < k] \\ = P [t_{i} > t {(k + \frac{p c_{1} + c_{2}}{(1 - p) c_{1}})}^{- \frac{1}{(1 - p) c_{1}}} \\ \times {(m_{1} + \frac{p c_{1} + c_{2}}{(1 - p) c_{1}})}^{\frac{1}{(1 - p) c_{1}}}] \end{matrix}

\begin{matrix} = 1 - P [t_{i} \leq t {(k + \frac{p c_{1} + c_{2}}{(1 - p) c_{1}})}^{- \frac{1}{(1 - p) c_{1}}} \\ \times {(m_{1} + \frac{p c_{1} + c_{2}}{(1 - p) c_{1}})}^{\frac{1}{(1 - p) c_{1}}}] \\ = 1 - {(k + \frac{p c_{1} + c_{2}}{(1 - p) c_{1}})}^{\frac{1}{(1 - p) c_{1}}} \\ \times {(m_{1} + \frac{p c_{1} + c_{2}}{(1 - p) c_{1}})}^{\frac{1}{(1 - p) c_{1}}} \end{matrix}

(15)

\begin{matrix} P (k) = \frac{\partial P (k_{i} (t) < k)}{\partial k} \\ = \frac{1}{(1 - p) c_{1}} {(m_{1} + \frac{p c_{1} + c_{2}}{(1 - p) c_{1}})}^{\frac{1}{(1 - p) c_{1}}} \\ \times {(k + \frac{p c_{1} + c_{2}}{(1 - p) c_{1}})}^{- \frac{1}{(1 - p) c_{1}} - 1} \end{matrix}

(16)

Let the exponent $μ = 1 + \frac{1}{(1 - p) c_{1}}$ and assume $p = 0$ , then the degree distribution of the cooperative communication network satisfies the power-law distribution $p (k) = k^{- γ}$ , where $γ = μ = 1 + \frac{1}{c_{1}}$ , the generated network has scale-free characteristics.

Assume $p \to 1$ , then $μ \to \infty$ , the degree distribution of the network follows the exponential distribution, and the generated network has the characteristics of random network.

To sum up, assuming that $0 < p < 1$ , the multi-cooperative communication network both has scale-free characteristic and random characteristic. By adjusting the value of $p$ , the link of nodes is controlled, and some nodes in the communication network will not reach saturation communication, thus avoiding traffic congestion of communication nodes.

Numerical simulation

Next, the network characteristics are analyzed through numerical simulation.

The topology diagram of the multi-domain cooperative communication network after evolution is shown in Figure 1. Set parameter $p_{1} = 0.2$ , $p_{2} = 0.25$ , $p_{3} = 0.3$ , $p_{4} = 0.05$ , $p_{5} = 0.2$ , $p = 0.5$ , $m_{0} = 5$ , $m_{1} = 3$ , $m_{2} = 3$ , $e_{0} = 8$ , $n = 10$ , $N = 200$ . The size of the mark represents the degree value of the node. The larger the mark, the greater the degree value, which means the stronger the communication ability of this node. It can be seen that some relay nodes attach multi-communication networks to form a huge cooperative communication network. Obtain spatial diversity gain with the advantages of cooperation, improve the transmission performance of the network system, and make effective use of the whole network resources.

Figure 1.

The evolution diagram of multi-domain cooperative communication network with network scale $N = 200$ .

When the network scale $N = 200$ , the relationship diagram of the degree value corresponding to each node after network evolution is shown in Figure 2. Set other parameters $p = 0.5$ , $p_{1} = 0.2$ , $p_{2} = 0.25$ , $p_{3} = 0.3$ , $p_{4} = 0.05$ , $p_{5} = 0.2$ , $m_{0} = 5$ , $m_{1} = 3$ , $m_{2} = 3$ , $e_{0} = 8$ , $n = 10$ . It can be seen from the figure that except for a few nodes whose degree values exceed 10, the degree values of other nodes are mostly concentrated between 3 and 8, which also fully shows that the generated multi-cooperative communication network has scale-free characteristics. The degree distribution after network evolution is shown in Figure 3. It can be seen from the figure that when $k \in {3, 4, 5}$ , the proportion of network nodes is the highest. However, with the increase in $k$ , the degree distribution of network nodes finally evolves into the power-law distribution. According to the analysis of the network, the network diameter $(d)$ , average path length $(L)$ , average degree $(< k >)$ , and average clustering coefficient $(C)$ can be obtained

\begin{matrix} d_{\max} & = 7 \\ L & = \frac{2}{N (N - 1)} \sum_{i \geq j} d_{ij} \approx 3.4 \\ < k > & = \frac{\sum_{j} (1 - p) k_{j} + p}{N} \approx 5.68 \\ C & = \frac{1}{N} \sum_{i}^{N} C_{i} \approx 0.34 \end{matrix}

Figure 2.

The degree value corresponding to the node after the evolution of multi-domain cooperative communication network.

Figure 3.

Degree distribution of multi-domain cooperative communication networks.

Robustness analysis

Based on the multi-domain cooperative communication network generated above, the robustness of the network is analyzed using random attack, intentional attack, and hybrid attack strategies. An intentional attack means attacking a node with a higher degree of probability with a certain probability. A random attack is a random attack on nodes in the network with a certain probability. The hybrid attack is to attack nodes in the network with two attack strategies at the same time.

The influence of the two attack methods on the degree distribution of cooperative communication networks under different attack probabilities is shown in Figures 4 and 5. Here, the number of attack nodes is 0, 40, and 80, respectively, and the corresponding attack probability is 0, 0.2, and 0.4, respectively. Other parameters $p_{1} = 0.2$ , $p_{2} = 0.25$ , $p_{3} = 0.3$ , $p_{4} = 0.05$ , $p_{5} = 0.2$ , $p = 0.5$ , $m_{0} = 5$ , $m_{1} = 3$ , $m_{2} = 3$ , $e_{0} = 8$ , $n = 10$ , $N = 200$ are selected here. From Figure 4, with the increase in attack probability, although the tail will arrive in advance with the increase in $k$ , the overall network degree distribution will follow the power-law distribution. The impact of the random attack on the network degree distribution is not significant, while in Figure 5, the network degree distribution changes significantly with the increase in attack probability. When the attack probability is 0.2 and 0.4, the degree distribution of the network has become the exponential distribution. This is because the deliberate attack is on the “key nodes” with large medium values in the network. These nodes play a vital role in the cooperative communication network. Once removed, the communication efficiency of the whole network will decline sharply. Previous studies have shown that the network generated by random connection is not very sensitive to intentional attacks, so the value can be adjusted to make the connection mode of nodes in the cooperative communication network more random so that the generated network model has better resistance to intentional attacks. When the attack probability is 0.2 and 0.4, the impact of the mixed attack method on the degree distribution of the cooperative communication network is shown in Figures 6 and 7, where $α$ represents the ratio of deliberate attack to random attack in mixed attack, where $α = 1 / 3, 1, 3$ . It can be seen from Figure 6 that when $η$ is small and the proportion of deliberate attacks in mixed attacks is not particularly large, the degree distribution of the network will not change significantly. It is found in Figure 7 that when $η = 0.4$ , the degree distribution of the network presents the characteristics of exponential distribution regardless of the proportion of deliberate attack and random attack in the hybrid attack. This also shows that as long as there is a deliberate attack in the hybrid attack, the degree distribution of the cooperative communication network will be affected to some extent. Figures 8 and 9 are the network evolution topology after the random attack, and the corresponding attack probabilities are 0.3 and 0.6, respectively. Due to the obvious scale-free characteristics of the generated network, most of the nodes under random attack are nodes with small degree values, so as the attack probability increases, the network topology does not change much after the random attack. Figures 10 and 11 are the network evolution topology after the hybrid attack; the corresponding attack probabilities are 0.3 and 0.6, respectively; and the ratio of the deliberate attack to the random attack in the hybrid attack is $α = 1$ . Figures 12 and 13 are the network evolution topology after the deliberate attack, and the corresponding attack probabilities are 0.3 and 0.6, respectively. Compared with random attacks, both hybrid attacks and deliberate attacks deliberately attack nodes with large degree values, so that the degree values of network nodes are relatively small after the attack, and the contrast of topology graphs is also quite obvious.

Figure 4.

Influence of random attack on degree distribution of multi-domain cooperative communication network.

Figure 5.

Influence of intentional attack on degree distribution of multi-domain cooperative communication network.

Figure 6.

Influence of hybrid attack with attack probability of 0.2 on degree distribution of multi-domain cooperative communication network.

Figure 7.

Influence of hybrid attack with attack probability of 0.4 on degree distribution of multi-domain cooperative communication network.

Figure 8.

The network evolution topology of random attack with attack probability of 0.3.

Figure 9.

The network evolution topology of random attack with attack probability of 0.6.

Figure 10.

The network evolution topology of hybrid attack with attack probability of 0.3.

Figure 11.

The network evolution topology of hybrid attack with attack probability of 0.6.

Figure 12.

The network evolution topology of intentional attack with attack probability of 0.3.

Figure 13.

The network evolution topology of intentional attack with attack probability of 0.6.

From the above, we know that the average degree of network is 5.68, the average path length is 3.4, and the average cluster coefficient is 0.34. Other parameters $p_{1} = 0.2$ , $p_{2} = 0.25$ , $p_{3} = 0.3$ , $p_{4} = 0.05$ , $p_{5} = 0.2$ , $p = 0.5$ , $m_{0} = 5$ , $m_{1} = 3$ , $m_{2} = 3$ , $e_{0} = 8$ , $n = 10$ , $N = 200$ are selected here. Analyze the impact of the two attack modes on the average degree, average path length and average cluster coefficient. The number of attack nodes is 0, 20, 40, 60, 80, 100, 120, 140, 160, 180, and 200, respectively, and the corresponding attack probability is 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1. The corresponding 11 sets of scatter plot coordinates can be obtained through calculation, as shown in Figures 14 and 15. It can be seen from Figure 14 that the average degree and average path length are quite sensitive to deliberate attacks. When the attack probability $η = 0.1$ , compared with random attacks, the average degree decreases sharply in the face of deliberate attacks. This is because the node group with a large medium value in the network is removed at once, and the average degree of the whole network is bound to fluctuate significantly. When the attack probability increases gradually, the average path length of the network will also increase. When the attack probability increases to $η = 0.2$ , more and more isolated nodes will appear in the network, resulting in the gradual reduction of the average path length. As can be seen from Figure 15, as the proportion of deliberate attacks in mixed attacks increases, the average degree decreases rapidly to 0 with the increase in the attack probability, the average path length rises sharply to a certain extent, and then rapidly decreases to 0 with the increase in the attack probability. From Figure 16, when the attack probability $η < 0.5$ , compared with the random attack, the average clustering coefficient of the network gradually decreases to 0 when faced with deliberate attacks and mixed attacks, where $α = 3$ . When the attack probability $η > 0.6$ , the random attack will have a significant impact on the average clustering coefficient of the network. However, as long as there is an element of a deliberate attack in the hybrid attack, the average clustering coefficient of the network will accelerate the downward trend with the increase in the attack probability. In the mixed attack with $α = 3$ , the clustering coefficient of more nodes with smaller degree values decreases due to the random attack component, which makes the global clustering coefficient decrease faster. Therefore, the impact of the mixed attack with a significantly dominant proportion of deliberate attacks on the average clustering coefficient of the network is slightly stronger than that of the deliberate attacks.

Figure 14.

The effect of random attack and intentional attack on average degree and average path length of multi-domain cooperative communication network under different attack probabilities.

Figure 15.

The effect of hybrid attack on average degree and average path length of multi-domain cooperative communication network under different attack probabilities.

Figure 16.

The effect of three attack modes on average clustering coefficient of multi-domain cooperative communication network under different attack probabilities.

Relay selection strategy of the combined maximum degree and minimum clustering coefficient

In the future, wireless communication networks will carry more high-speed real-time communication services. Using relays can not only expand cell coverage and reduce communication blind spots but also save terminal transmit power, effectively achieve optimal resource allocation, and provide users with more reasonable high-speed communication services. Therefore, it is of great practical significance to study the relay node selection technology in cooperative communication networks. Combined with the transmission characteristics of the wireless communication system, it is necessary to formulate a relay selection scheme to determine the reliability of the relay location to improve the accuracy of communication. The selection of relay nodes is the key to realizing communication in a cooperative relay system. How to select the most suitable relay node to optimize the communication performance of the sender is the primary issue we consider, because an appropriate relay selection strategy can not only select the node that can effectively help the source node transmit information, improve the transmission quality, but also reduce the power waste caused by the relay forwarding signals with poor channel conditions, enabling energy-saving and efficient communication.

Based on the multi-domain cooperative communication network, the relay node selection strategy is studied and analyzed. From the above research and analysis, a multi-domain cooperative communication network has both scale-free characteristics and small-world characteristics. Most of the selected source nodes $s$ and target nodes $t$ are in different node focus areas. Therefore, the minimum clustering coefficient relay selection strategy that can quickly transfer regions will make the relay selection time relatively short to a certain extent and effectively improve the selection efficiency. However, the effect of this strategy is not as efficient as the maximum relay selection strategy in the network with obvious scale-free characteristics, and considering that the maximum relay selection strategy will cause the communication congestion of some nodes with large degree value, a relay node selection strategy combining maximum degree and minimum clustering coefficient is proposed, and the average selection steps and average selection time are used as the effectiveness indexes. The strategy sets a boundary value $k_{0}$ , and the maximum relay node selection strategy is continued for the nodes whose degree value is less than the boundary value, if it is greater than the boundary value, the relay node selection strategy with minimum clustering coefficient is used, which not only ensures the efficiency of the maximum relay node selection strategy but also greatly reduces the selection time and improves the selection efficiency.

The pseudo-code of relay node selection strategy combining maximum degree and minimum clustering coefficient is shown in Algorithm 1:

Algorithm 1.

The relay node selection strategy combining maximum degree and minimum clustering coefficient.

Input: starting node

s

; destination node

t

; maximum steps of relay selection

step

Output:

step

; the set of nodes passed through

Pat h_{-} List

a \leftarrow s

;
2: While

step

> 0 do
3: if

Neigh s_{-} a ¹ t

then
4:

t \in Pat h_{-} List

;
5:

step \leftarrow step - 1

;
6: break
7: else
8: if

k_{a} \leq k_{0}

then
9:

p \leftarrow i

and

k_{i} = \max k_{j \in Neigh s_{-} a}

;
10: else
11:

p \leftarrow i

and

c_{i} = \min c_{j \in Neigh s_{-} a}

;
12:

p \in Pat h_{-} List

;
13:

step \leftarrow step - 1

;
14:

a \leftarrow p

;
15: end if
16: end if
17: end while
18: return

step

;

Pat h_{-} List

Note. In the process of relay selection, the same node can be queried multiple times, but the same link can only be accessed once.

The basic parameter values of the generated network are shown in Table 2.

Table 2.

Basic parameters of the generated multi-domain cooperative communication network data set.

Nodes	Links	Average degree	Average clustering coefficient
200	568	5.68	0.34

By analyzing the network performance generated by the relay node selection strategy of maximum degree and minimum clustering coefficient, the average selection steps and average selection time are obtained, as shown in Table 3.

Table 3.

Comparison of relay node selection strategies with maximum degree and minimum clustering coefficients.

Selection strategy	Average selection step	Average selection time (s)
Max-degree	3.23	0.0066
Min-clustering coefficient	3.50	0.0073

The impact of different boundary values $k_{0}$ on the relay node selection performance of the relay node selection strategy combining maximum degree and minimum clustering coefficient is shown in Table 4.

Table 4.

The effect of improved relay node selection with different threshold value $k_{0}$ .

$k_{0}$	Average selection step	Average selection time (s)
6	3.43	0.0068
8	3.12	0.0063
9	2.94	0.0059
10	2.80	0.0055
12	2.76	0.0044
13	2.93	0.0060
15	3.17	0.0062

It can be seen from Tables 3 and 4 that the relay node selection strategy combining the maximum degree and the minimum clustering coefficient is generally better than the maximum degree relay node selection strategy and the minimum clustering coefficient relay node selection strategy. When $k_{0} = 12$ , the relay node selection strategy combining maximum degree and minimum clustering coefficient achieves the best effect, that is, the average selection steps and average selection time are the shortest.

Conclusion

In the context of the IoT, according to the actual characteristics of the cooperative communication network, a multi-domain cooperative communication network evolution model with preferential attachment and random attachment was constructed in this article, which improves the network transmission capacity and alleviates the problem of network congestion. The evolution model was proposed in five steps with five different probabilities. Using the rate equation and mean-field method, the topological properties such as the degree distribution of the network were studied and then the robustness of the network model was analyzed. Compared with random attacks, as the proportion of deliberate attacks in mixed attacks increases, the impact on network characteristics gradually increased. It could be seen that deliberate attacks had a greater impact on the degree distribution, average degree, average shortest path length, and average clustering coefficient of the network, but the attachment mode of nodes in the network can be more random by adjusting the $p$ value, which can effectively solve the problem of communication congestion. Aiming at the problem of cooperative communication relay node selection, considering that the maximum relay selection strategy will cause the communication congestion of some nodes with large degree values, a relay node selection strategy combining the maximum degree and the minimum clustering coefficient is proposed. Through simulation experiments, the advantages and disadvantages of the three relay node selection strategies combining the maximum degree and the minimum clustering coefficient and the maximum degree and the minimum clustering coefficient were compared. The simulation results showed that the relay node selection strategy combining maximum degree and minimum clustering coefficient was obviously superior in the selection steps and selection time, and the best selection of relay nodes was obtained, which greatly improved the performance of relay selection in multi-domain cooperative communication networks.

Footnotes

Handling Editor: Yanjiao Chen

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Key Science Research and Development Program of China (Grant No. 2018YF2000701), China Postdoctoral Science Foundation (Grant No. 2021M692400), Research and Development Plan of Science and Technology of China National Railway Group Co., Ltd (Grant No. P2021S005 and No. N2021X007), Research Project of China Academy of Railway Sciences Group Co., Ltd (Grant No. 2021YJ136), and National Natural Science Foundation of China (Grant No. 61973201 and No. 62006169).

ORCID iD

Wei Bai

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Evolutionary modeling and robustness analysis of multi-domain cooperative communication network under the environment of Internet of things

Abstract

Keywords

Introduction

Related work

Evolutionary modeling of multi-domain cooperative communication network

Initial state

Step 1

Step 2

Step 3

Step 4

Step 5

Analysis of network characteristics

Degree distribution calculation

Numerical simulation

Robustness analysis

Relay selection strategy of the combined maximum degree and minimum clustering coefficient

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References