Graph convolutional networks-based robustness optimization for scale-free Internet of Things

Abstract

The Internet of Things (IoT) devices have limited resources and are vulnerable to attacks, so optimizing their network topology to resist random failures and malicious attacks has become a key issue. The scale-free network model has strong resistance to random attacks, but it is very vulnerable to malicious attacks. The existing studies mostly adopt heuristic algorithms to optimize the ability of scale-free networks to resist malicious attacks, but their high computational cost cannot meet the timeliness requirements of the real IoT. Therefore, this paper proposes an intelligent topology robustness optimization model based on a graph convolutional network (ROGCN). The model extracts the onion-like structural features of the highly robust network topology from the data set through supervised learning, and on this basis, different search strategies are designed to meet the needs of different IoT scenarios. The extensive experimental results demonstrate that ROGCN can more effectively improve the robustness of scale-free IoT networks against malicious attacks compared to two existing heuristic algorithms, with a lower computational cost.

Keywords

Robustness optimization internet of things graph convolutional network scale-free model malicious attacks

1. Introduction

As a multidisciplinary fusion system, the Internet of Things (IoT) integrates the fifth-generation (5G) ultra-dense cellular system [1], heterogeneous ad hoc networks [2], wireless sensor networks [3], hybrid mobile networks, wireless fidelity networks, and data center networks [4]. The IoT is widely used in social life, industrial production, national defense construction and other areas [5]. However, due to the limited storage, computing, communication capabilities and energy of IoT devices, the application environment is complex and diverse, making these devices vulnerable to man-made attacks, natural damage, energy depletion and other problems [6], which breaks the connectivity of the IoT. The connectivity between IoT devices is the basis of communication. If devices are regarded as nodes and communication between devices as connecting edges, the IoT can be abstracted as a complex network topology, and corresponding random failures and malicious attacks are regarded as destruction of the network topology. Therefore, efficiently constructing a robust network topology has become a research focus in the field of the IoT.

As a classic model in complex network theory, the scale-free model is widely used in modeling the homogeneous network topology of various real systems [7, 8]. This research is aimed at the homogeneous IoT; that is, the device nodes have similar communication ranges and bandwidths, so the scale-free network is used for network topology modeling of the IoT. The degree distribution of the scale-free network follows a power-law distribution, which makes it highly resistant to random attacks but very vulnerable to malicious attacks [9, 10]. Therefore, this paper aims to optimize the ability of the scale-free IoT to resist malicious attacks while ensuring strong robustness against random failures.

Robustness optimization of scale-free IoT topology under malicious attacks is an NP-hard problem [11]. Heuristic algorithms are widely used to solve this problem and have achieved good results, such as the hill climbing algorithm (HCA) [12], simulated annealing algorithm (SAA) [13], memetic algorithm (MA) [14] and genetic algorithm (GA) [15, 16]. The empirical results of heuristic algorithms confirm that the optimized highly robust network topology has an onion-like structure, which is characterized by nodes with a large degree located in the center of the topology surrounded by nodes with a gradually decreasing degree, and in which nodes with a similar degree tend to connect, showing an onion-like hierarchical structure. Tanizawa et al. [17] chose the onion-like structure of the interconnected random regular graph and proved its robustness against malicious attacks through rigorous mathematical theoretical analysis. The above theoretical and empirical studies show that there is a certain evolution pattern from the initial topology to a highly robust optimization topology. However, the evolution of heuristic algorithms from the initial topology to a highly robust optimization topology involves many edge selection, edge swapping, and robustness value calculation operations, which leads to a high time cost and cannot meet the needs of IoT topology optimization in low-latency scenarios. In addition, heuristic algorithms easily fall into the local optimum solution, their optimization effect decreases with increasing network size, and the scalability of the algorithm needs to be improved. Based on the above analysis, this paper introduces the deep learning method to learn the evolution pattern from the initial topology to the target topology to reduce the time cost and improve the scalability of the robust optimization method.

The IoT network topology, as a kind of graph structure data, is different from Euclidean structure data [18] such as images and voices and cannot be directly used as the input of classic deep learning models. A graph neural network is an end-to-end deep learning model designed specifically for graph structure data [19, 20], which is different from classic neural networks and has adaptability to graph structure data. Therefore, this paper transforms the topology robustness optimization problem into an evolution pattern problem of adding and deleting connected edges between the initial topology and the target topology with high robustness by learning the graph neural network. That is, according to the initial topology, we extract all node features and comprehensively calculate the edge features between nodes, obtain the probability of each edge in the target topology based on the edge features, and determine the target topology.

The introduction of graph neural networks to solve the topology robustness optimization problem faces the following difficulties: one is how to design the corresponding graph neural network input for the topology robustness optimization problem, which is crucial to the learning of evolutionary patterns. Considering that the onion-like structure of highly robust topology has a strong correlation with the degree difference between nodes, the traditional definition of degree difference is directly introduced into the model, which has a slow convergence speed. Therefore, a new degree difference representation method is proposed as the edge feature input of the graph neural network. The second difficulty is based on the constraint that the number of nodes and edges of the target topology should be consistent with the initial topology, and it concerns how to obtain the target topology according to the connected edge probability of the output of the graph neural network. Considering that different IoT scenarios have different requirements for the topology robustness optimization effect and timeliness, different search strategies are proposed to achieve the goal from the edge probability to the target topology.

The main contributions of this paper are as follows:

•
To the best of our knowledge, a graph convolutional network is applied to the robustness optimization problem of IoT topology for the first time.
•
For the topology robustness optimization problem, a new degree difference representation method is proposed to accelerate the convergence of the graph convolutional network model.
•
This paper proposes three different search strategies to suit the needs of different IoT scenarios.

The rest of this paper is organized as follows: In Section 2, we review the background and related work. In Section 3, we present the new degree difference representation method and propose our algorithm. In Section 4, we present the performance evaluation results. Finally, Section 5 concludes this paper.
2. Background and related work

2.1 Scale-free IoT topology generation

The scale-free property means that a small number of nodes in the network have a large number of connections, while most nodes have only a few connections, which is shown in the power-law distribution of node degrees [21, 22]. To explain the mechanism of this phenomenon, Barabási and Albert [23] proposed the classic Barabási-Albert (BA) model for constructing a scale-free network topology. However, due to the limited communication range and energy of IoT devices, nodes can only connect to neighboring nodes, and the maximum number of connections that can be connected is limited [24]. Based on this constraint, Qiu et al. [25] improved the traditional BA model and proposed a scale-free IoT topology construction method. The process is as follows: first, all nodes are placed in the communication area, and then, beginning from the starting node of the topology center, connections are established between the nodes in sequence, where a newly added node selects neighbors according to the degrees of the other nodes in the communication range, and tend to connect the nodes with larger degrees and not reaching the maximum degree limit. The process is implemented through a roulette mechanism.

2.2 Types of cyber-attacks

The main dangers facing IoT topology can be divided into random attacks and malicious attacks. Random attacks include random equipment failure, natural disasters and equal probability attacks, while malicious attacks include intrusion attacks, terrorist attacks and energy depletion [26]. Corresponding to the attack method in complex network theory, a random attack randomly selects any node in the network to attack, and all nodes are attacked with the same probability; a malicious attack refers to the targeted selection of the most important node in the network to attack. Its purpose is to cause the greatest damage to the network with the least attack cost. In this paper, a high-degree adaptive (HDA) attack strategy [27] is used to simulate real malicious attacks on the IoT topology. Each time the node with the largest degree of the current network is selected for attack, the node and its connected edges are deleted.

2.3 Robustness metric

Based on percolation theory [28] and statistical methods, Schneider et al. [29] proposed a network topology robustness metric $R$ , which considers the size of the maximum connected subgraph under all possible malicious attacks. The measurement formula is given as Eq. (1).

$\displaystyle R=\frac{1}{{N+1}}\sum\limits_{i={\rm{0}}}^{N}{\frac{{\textit{MCS% }_{\textit{size}}^{i}}}{N}}$ (1)

where $N$ represents the total number of nodes in the network. $\textit{MCS}_{\textit{size}}^{i}$ represents the number of nodes contained in the largest connected subgraph of the network after the $i$ -th attack, and the normalized factor $1/({N+1})$ enables the robustness comparison of networks of different sizes and edge densities. The fully connected network topology has the strongest ability to resist malicious attacks and the highest robustness value, with a corresponding $R$ value of 0.5. The star network topology has the weakest ability to resist malicious attacks and the lowest robustness value, and the corresponding $R$ value is $2/({N+1})$ . Therefore, the robustness metric $R$ ranges in $[2/(N+1),0.5]$ . The larger the $R$ value is, the stronger the robustness of the network. The metric $R$ and its deformation have been widely used by researchers in the study of network robustness.

2.4 Research on the robustness optimization of scale-free networks

In recent years, significant progress has been made in the research of robustness enhancement strategies for scale-free networks under malicious attacks. Herrmann et al. [12] proposed an HCA based on a random swapping edge strategy. The algorithm is simple and effective, but it easily falls into a local optimal solution. Buesser et al. [13] proposed a probabilistic edge swapping strategy based on an SAA, which solves the problem of the HCA easily falling into a local optimum but performs redundant edge swapping operations and R value calculations in a larger solution space, resulting in higher calculation costs. Zhou et al. [14] proposed an MA by combining the global search capability of the population and the local heuristic search capability of the individual. Rong et al. [30] analyzed the types of edges that affect the size of the largest connected subgraph and proposed a heuristic algorithm based on edge classification, which to some extent solves the problem of a poor optimization effect when the network size is large. However, the MA and the edge classification algorithm do not consider the energy and communication range limitations of nodes, so they are not suitable for a scale-free IoT. Aiming at the problem that traditional genetic algorithms tend to fall into premature convergence [31], Qiu et al. [16, 32] proposed multiple-population genetic algorithms (MPGA) and adaptive robustness evolution algorithms (AREA), which can effectively enhance the robustness of scale-free IoT topology. However, with increasing network size, the computational efficiency of genetic algorithms decreases, and they cannot meet the low-latency requirements of the IoT. To avoid the above problems, this paper adopts another research idea, based on deep learning technology, to transform the topology robustness optimization problem into an evolution pattern problem from learning the initial topology to the highly robust target topology. Compared with traditional heuristic algorithms, this method has a lower computational cost and better scalability, and is suitable for different network sizes.

3. ROGCN model

Figure 1.

Overview of ROGCN.

An overview of the ROGCN model proposed is shown in Fig. 1. The main process is as follows:

(1)

Given the initial scale-free IoT topology, based on the onion-like structural features of the evolution from the initial topology to the highly robust topology, the design new degree difference matrix as a priori knowledge to extract the onion-like evolution pattern of the network. The new degree difference matrix and adjacency matrix are used as the initial features of the network into the graph convolutional network.

(2)

The feature representations of all nodes and edges are extracted through the designed multi-layer graph convolutional network layer, the final feature representations of all edges are input to the multi-layer perceptron layer, and the probability matrix of connected and unconnected edges in the predicted topology is output.

(3)

The loss of the adjacency matrix between the prediction matrix and the label topology is calculated by using the weighted binary cross-entropy loss function. Then, the loss is minimized through gradient descent, and repeated iterations are used to complete the training of the model parameters.

(4)

In the testing phase, considering the prerequisite constraints that the number of nodes and edges of the final predicted topology need to be consistent with the initial topology, and different IoT scenarios have different requirements for optimization effects and calculation costs, this paper designs different search strategies to convert the edge prediction probability matrix obtained by the model into the adjacency matrix of the final predicted topology.

The ROGCN is trained based on supervised learning, using the state-of-the-art AREA in topology robustness optimization as the label data.

3.1 New node degree difference representation

The main feature of the onion-like structure is that nodes with similar degrees are connected together as much as possible; that is, the degree difference between the two nodes of each edge is as small as possible. In this way, when a node in the network fails, its connected nodes can replace its original function to the greatest extent, weakening the damage caused by malicious attacks [33]. Therefore, the onion-like structure has a strong correlation with the degree difference between nodes, and the degree difference can be used as the input of the graph convolutional network to learn the evolution pattern from the initial topology to the highly robust target topology. However, by using the traditional definition of degree difference as the input of the graph convolutional network, it is found that the convergence speed of the model is slow. To better represent the degree difference information between nodes and improve the convergence speed of the model, this paper proposes a new representation method for the degree difference between nodes, as shown in Eq. (2).

$\displaystyle{D_{ij}}=\left\{{\begin{array}[]{ll}{1-\frac{{2\Delta{d_{ij}}}}{{% \max\{{\Delta{d_{i}}}\}+\max\{{\Delta{d_{j}}}\}+1}}},&{i\neq j}\\ 0,&{i=j}\end{array}}\right.$ (2)

where ${D_{ij}}$ represents the new degree difference between nodes $i$ and $j$ , $\Delta{d_{ij}}$ represents the degree difference between nodes $i$ and $j$ , and ${{\max}}\{{\Delta{d_{i}}}\}$ represents the maximum degree difference between node $i$ and all connected nodes. When the degrees of nodes $i$ and $j$ are the same, the ${D_{ij}}$ value is the largest, and its value is 1. When $\Delta{d_{ij}}={{\max}}\{{\Delta{d_{i}}}\}={{\max}}\{{\Delta{d_{j}}}\}$ , the ${D_{ij}}$ value is the smallest, and its value is $1/({{{\max}}\{{\Delta{d_{i}}}\}+{{\max}}\{{\Delta{d_{j}}}\}+1})$ . Therefore, the value range of ${D_{ij}}$ is $({0,1}]$ .

3.2 Graph convolutional network

Input layer: The input layer contains node features and edge features. In this article, the adjacency matrix of the initial topology is embedded into the $h$ -dimensional node features, as given in Eq. (3).

$\displaystyle{\alpha_{i}}={W_{1}}{a_{i}}+{b_{1}}$ (3)

where ${W_{1}}\in{\mathbb{R}^{h\times n}}$ represents the weight matrix and ${a_{i}}\in{\{0,1\}^{n\times 1}}$ represents the $i$ -th column of the adjacency matrix. $h$ represents the feature dimensions of nodes and edges in the graph convolutional layer. The new degree difference representation is embedded into an $h/2$ dimensions feature vector. The communication range index ${\delta_{ij}}$ is introduced, when nodes $i$ and $j$ are within the communication range of each other, their value is equal to 1; otherwise, their value is 0. ${\delta_{ij}}$ is also embedded into the feature vector of $h/2$ dimensions. The edge input feature is defined as in Eq. (4).

$\displaystyle{\beta_{ij}}=({{W_{2}}{d_{ij}}+{b_{2}}})||{W_{3}}{\delta_{ij}}$ (4)

where ${W_{2}}\in{\mathbb{R}^{h/2\times 1}}$ and ${W_{2}}\in{\mathbb{R}^{h/2\times 1}}$ , ${d_{ij}}$ represent the new degree difference between nodes $i$ and $j$ , $\cdot{{||}}\cdot$ represents the concatenation operation, and ${\delta_{ij}}$ narrows the search space and accelerates the learning process because nodes in the label topology can only be connected to nodes within the communication range. The initial values of ${W_{1}}$ , ${W_{2}}$ and ${W_{3}}$ are determined by random initialization. With continuous training iterations, their values are updated by gradient descent, and finally converge to a certain value.

Graph convolution layer: In a convolutional network, the concept of a neighborhood is determined by the Euclidean distance [34]. In graph convolutional networks, the concept of a neighborhood is determined by the graph topology. When there are connected edges between two nodes, they are considered neighborhoods. Let $x_{i}^{l}$ denote the feature vector of node $i$ in the $l$ -th layer. By nonlinear transformation and activation on the feature vector of all nodes $j$ in the neighborhood of node $i$ , the feature vector $x_{i}^{l+1}$ of node $i$ in the next layer is obtained. Therefore, the general form of the feature vector $x_{i}^{l+1}$ of the graph convolutional network at node $i$ can be obtained as in Eq. (5).

$\displaystyle x_{i}^{l+1}=f(x_{i}^{l},\{x_{j}^{l}:j\in{\theta_{i}}\})$ (5)

where ${\theta_{i}}$ represents the set of neighboring nodes centered on node $i$ . Different definitions of the mapping $f$ determine different types of graph convolutional networks, such as those of Derr et al. [35], Rossi et al. [36] and Bresson et al. [37].

In this work, we leverage the graph convolutional network framework proposed by Bresson et al. [37, 38], as shown in Fig. 2. Let $x_{i}^{l+1}$ and $e_{ij}^{l+1}$ denote the node feature vector and edge feature vector of node $i$ and edge $i j$ at layer $l+1$ , respectively, which are defined as follows:

$\displaystyle x_{i}^{l+1}=x_{i}^{l}+\text{ReLU}\left(\text{BN}\left(W_{1}^{l}x% _{i}^{l}+\sum\limits_{j\in{\theta_{i}}}{\eta_{ij}^{l}\odot}W_{2}^{l}x_{j}^{l}% \right)\right)\;\text{with}\;\eta_{ij}^{l}=\frac{{\sigma({\text{e}}_{ij}^{l})}% }{{\sum\nolimits_{j^{\prime}\in{\theta_{i}}}{\sigma({\text{e}}_{ij^{\prime}}^{% l})+\varepsilon}}}$ (6) $\displaystyle e_{ij}^{l+1}=e_{ij}^{l}+\text{ReLU}(\text{BN}(W_{3}^{l}e_{ij}^{l% }+W_{4}^{l}x_{i}^{l}+W_{5}^{l}x_{j}^{l}))$ (7)

where ${W_{1-5}}\in{\mathbb{R}^{h\times h}}$ , $\sigma$ is the sigmoid function, $\varepsilon$ is a small value, ReLU is the rectified linear unit, and BN stands for batch normalization. The 0th layer of the graph convolutional network is the input layer, namely, $x_{i}^{l=0}={\alpha_{i}}$ and $e_{ij}^{l=0}={\beta_{ij}}$ . Since the graph topology has no specific direction (up, down, left, or right), the diffusion process on the graph structure should be isotropic. In fact, this is not the case. Different neighbors of the node contain different information and weights. Therefore, the graph convolutional network framework makes the diffusion process anisotropic by point-wise multiplication operations with learnable normalized edge gates [39]. In addition, the residual connection mechanism [40] effectively weakens the effect of the vanishing gradient problem on backpropagation.

Figure 2.

Residual gated graph convolutional network.

Output layer: The output layer implements binary classification through a multi-layer perceptron (MLP) [41]. The edge feature vector $e_{ij}^{l}$ of the last layer of the graph convolution is used as the input of the MLP, and output the probability $P\in{[{0,1}]^{n\times n\times 2}}$ of unconnected and connected edges in the prediction topology. Specific definitions are given in Eq. (8).

$\displaystyle{p_{ij}}=\textit{MLP}(e_{ij}^{l})$ (8)

Loss function: Given the adjacency matrix $Y\in{\{{0,1}\}^{n\times n}}$ of the label topology and the edge probability matrix $P\in{[{0,1}]^{n\times n\times 2}}$ of the output layer, due to the large difference in the number of 0 and 1 values in $Y$ , the classification task becomes highly unbalanced toward 0. Therefore, appropriate class weights are required to counterbalance this effect. The class weights are defined in Eq. (9).

$\displaystyle\textit{weight}=\{{w_{0}},{w_{1}}\}=\left\{\frac{{{n^{2}}}}{{({n^% {2}}-2nm)\times c}},\frac{{{n^{2}}}}{{(2nm)\times c}}\right\}$ (9)

where ${w_{0}},{w_{1}}$ are the weights of labels 0 and 1, $n$ is the number of nodes, $m$ is the edge density of the scale-free network, and $c$ is the number of classes; here, $c=$ 2. In addition, this paper uses the weighted binary cross-entropy loss function [42] to calculate the loss value of the prediction matrix $P$ and label topology adjacency matrix $Y$ . The loss is defined in Eq. (10).

$\displaystyle\textit{Loss}(Y,P)=\frac{{{1}}}{{{n^{2}}}}\sum\limits_{i=0}^{n-1}% {\sum\limits_{j=0}^{n-1}{\textit{weight}[{Y_{ij}}]\left(-{P_{ij}}[{Y_{ij}}]+% \log\left(\sum\limits_{k=0}^{c-1}{\exp({P_{ij}}[k])}\right)\right)}}$ (10)

3.3 Search strategy

As seen from the above section, the output of the graph convolutional network model is the probability ${p_{ij}}\in{[{{{0,1}}}]^{{2}}}$ of connected and unconnected edges in the predicted topology; ${p_{ij}}[0]$ represents the probability that no edges exist between nodes $i$ and $j$ , and ${p_{ij}}[1]$ represents the probability that edges exist between nodes $i$ and $j$ . If the adjacency matrix of the predicted topology is directly obtained by the argmax function, redundant connected edges will usually be generated compared with the initial topology. The goal of this paper is to enhance the robustness of the network by adjusting the connection relationship between nodes under the condition that the number of nodes and connected edges remains unchanged. Therefore, three search strategies are proposed to keep the number of edges of the predicted topology consistent with that of the initial topology.

TOP-K strategy: Assuming that the number of edges in the initial topology is $K$ , the top $2K$ maximum values in the edge existence probability matrix ${p_{ij}}[1]$ are directly selected, and the corresponding adjacency matrix positions are assigned as 1 and the rest as 0; then, the adjacency matrix of the predicted topology can be obtained.

Degree search strategy: This search strategy aims at obtaining a node degree of the predicted topology that is consistent with that of the initial topology by continuously selecting the edges that meet the conditions to add to the predicted topology until the number of edges of the predicted topology equals that of the initial topology. Algorithm 1 describes the detailed process of the degree search strategy. The variables used in the algorithm are as follows:

[b] Degree search $N$ , $K$ , ${P_{\textit{pred}}}$ , ${D_{\textit{init}}}$ ${A_{\textit{pred}}}$ ${A_{\textit{pred}}}\leftarrow$ Initialize to 0 ${P_{\textit{index}}}={\mathop{\rm argmax}\nolimits}({P_{\textit{pred}}},% \textit{dim}=2)$ ${P_{\textit{value}}}\leftarrow({{P_{\textit{pred}}},{P_{\textit{index}}}})$ i = 1 to N $P_{\textit{index}}^{i}.\textit{sum}()==D_{\textit{init}}^{i}$ j = 1 to N $P_{\textit{index}}^{ij}==1$ $[A_{\textit{pred}}^{ij},A_{\textit{pred}}^{ji}]=1$ $[D_{\textit{init}}^{i},D_{\textit{init}}^{j}]=[D_{\textit{init}}^{i},D_{% \textit{init}}^{j}]-1$ ${A_{\textit{pred}}}.\textit{sum}()$ $\neq$ $2K$ ${P_{\textit{value}}}\leftarrow{\mathop{\rm sort}\nolimits}\_{\rm descending}({% {P_{\textit{value}}}\odot({P_{\textit{index}}}-{A_{\textit{pred}}})})$ ${p_{\textit{temp}}}$ in ${P_{\textit{value}}}$ $[m,n]\leftarrow{p_{\textit{temp}}}$ (Convert to the corresponding index) $D_{\textit{init}}^{m}>0$ and $D_{\textit{init}}^{n}>0$ $[A_{\textit{pred}}^{mn},A_{\textit{pred}}^{nm}]=1$ $[D_{\textit{init}}^{m},D_{\textit{init}}^{n}]=(D_{\textit{init}}^{m},D_{% \textit{init}}^{n})-1$ ${A_{\textit{pred}}}.\textit{sum}()==2K$ break

•
$N$ : the total number of nodes in the initial network topology.
•
$K$ : the total number of edges in the initial network topology.
•
${P_{\textit{pred}}}$ : the predicted probability of connected and unconnected edges in the network topology.
•
${D_{\textit{init}}}$ : the node degree of the initial topology.
•
${A_{\textit{pred}}}$ : the final adjacency matrix of the predicted topology.
•
${P_{\textit{index}}}$ : the conversion of ${P_{\textit{pred}}}$ to the corresponding adjacency matrix.
•
${P_{\textit{value}}}$ : the existence probability of the corresponding edge when the ${P_{\textit{index}}}$ value is 1.
•
${p_{\textit{temp}}}$ : a temporarily stored element in ${P_{\textit{value}}}$ .

The algorithm is implemented in two stages. The first stage is to select nodes whose predicted topology is consistent with the initial topology degree. The connected edges of such nodes will be the final output; the corresponding position of the adjacency matrix is assigned as 1, and the corresponding degree is also modified (lines 4–13). The second stage selects edges according to the probability of edge existence in descending order. When the degrees of the nodes at both ends of the selected edge are greater than 0, the edge is determined as the final output, and the adjacency matrix and node degree are modified until the total number of edges in the predicted adjacency matrix is equal to the total number of edges in the initial topology when the search stops (lines 15–25).

Robustness search strategy: The search strategy aims at maximizing the robustness value. It continuously selects the edges that meet the conditions and deletes them from the predicted topology until the number of edges in the predicted topology is equal to that of the initial topology. Algorithm 2 describes the detailed process of the robustness search strategy. The variables used in the algorithm are as follows (the variables used in Algorithm 1 are not described again):

•
${A_{\textit{temp}}}$ : temporarily store the adjacency matrix of the predicted topology after deleting a pair of edges

[b] Robustness search $N$ , $K$ , ${P_{\textit{pred}}}$ , ${D_{\textit{init}}}$ ${A_{\textit{pred}}}$ ${P_{\textit{index}}}={\mathop{\rm argmax}\nolimits}({P_{\textit{pred}}},% \textit{dim}=2)$ , ${A_{\textit{pred}}}={P_{\textit{index}}}$ ${P_{\textit{value}}}\leftarrow({{P_{\textit{pred}}},{P_{\textit{index}}}})$ i = 1 to N $P_{\textit{index}}^{i}.\textit{sum}()==D_{\textit{init}}^{i}$ j = 1 to N $P_{\textit{index}}^{ij}==1$ ${P_{\textit{value}}}\leftarrow{P_{\textit{value}}}$ (Remove the probability of the existence of ${e_{ij}},{e_{ji}}$ ) ${P_{\textit{value}}}\leftarrow{\mathop{\rm sort}\nolimits}\_{\rm ascending}({{% P_{\textit{value}}}})$ ${p_{\textit{temp}}}$ in ${P_{\textit{value}}}$ $[m,n]\leftarrow{p_{\textit{temp}}}$ (Convert to the corresponding index) ${A_{\textit{temp}}}\leftarrow{A_{\textit{pred}}}$ (Remove ${e_{mn}},{e_{nm}}$ ) NetworkFullConnected ( ${A_{\textit{temp}}}$ ) and $R({{A_{\textit{temp}}}})>R({{A_{\textit{pred}}}})$ ${A_{\textit{pred}}}={A_{\textit{temp}}}$ ${P_{\textit{value}}}\leftarrow{P_{\textit{value}}}$ (Remove the probability of the existence of ${e_{mn}},{e_{nm}}$ ) ${A_{\textit{pred}}}.\textit{sum}()==2K$ break ${A_{\textit{pred}}}.\textit{sum}()>2K$ Execute lines 13–23, of which 16 lines $R({{A_{\textit{temp}}}})>R({{A_{\textit{pred}}}})$ is changed to $R({{A_{\textit{temp}}}})\geqslant R({{A_{\textit{pred}}}})$

The algorithm is implemented in three stages. In the first stage, the nodes whose predicted topology is consistent with the initial topology degree are selected. The connecting edges of such nodes are retained and do not participate in subsequent edge deleting operations (lines 3–11). In the second stage, the edges are selected in ascending order according to the probability of edge existence. After the selected edges are deleted, if the network remains fully connected and the robustness value increases, the deletion of the edges is accepted; otherwise, it is rejected. This process repeats until the remaining connected edges in the predicted topology are equal to the number of initial topological edges (lines 12–23). If the number of predicted topological edges is still greater than that in the initial topology after the second stage, the third stage is executed. The difference between the third stage and the second stage is that the condition of edge deletion is relaxed. When the network remains fully connected and robustness does not decrease after an edge is deleted, edge deletion is accepted until the number of remaining connected edges in the predicted topology is equal to the number of edges in the initial topology (lines 24–26).
4. Experimental evaluation

This section first introduces the data set, experimental environment and parameter settings. Then, we evaluate the convergence of the loss values of the model under the traditional and new degree difference representation, and we compare the effects, advantages and disadvantages of different search strategies and the topology changes before and after optimization. Second, we compare the connectivity of the scale-free topology optimized by the ROGCN, SAA and AREA algorithms under random attacks and malicious attacks, and we observe the robustness optimization effects of various algorithms under different numbers of nodes and edge densities. Finally, the optimization efficiency of the different algorithms is evaluated.

In addition, the experimental results in this section are the average of $c({c\geqslant 20})$ independent repeated experiments carried out by the model on the test set, unless otherwise specified.

4.1 Data sets

In this paper, the initial scale-free IoT topological data set is generated based on the method in [25], AREA is used to optimize the robustness of the initial topology data set, and the obtained optimized topology of the onion-like structure is used as the label data set. Specifically, the data set is ${\text{D}}=\{{X,Y}\}=\{{({x_{1}},{y_{1}}),\cdots,({x_{N}},{y_{N}})}\}$ , where $N$ represents the number of samples, ${x_{i}}$ represents the adjacency matrix of the $i$ -th initial topology, and ${y_{i}}$ represents the adjacency matrix of the $i$ -th label topology. To ensure that the network topology has strong resistance to random attacks, the AREA optimizes the robustness of the scale-free IoT under the premise that the initial degree distribution remains unchanged, and it simultaneously follows the constraints of the scale-free IoT on the communication range and the maximum number of connections. The data set is divided into training samples, verification samples and test samples, which account for 60%, 20%, and 20% of the samples, respectively. The model extracts the rules of evolution from the initial topology to the label topology through the training samples and then adjusts the hyper-parameters in the graph convolutional network with the verification samples. Finally, the accuracy of the model is tested in the test samples.

4.2 Parameter settings

This experiment is based on PyTorch 1.8.1 and Python 3.7. The initial scale-free IoT topology is constructed by randomly deploying $N$ nodes with a communication range of 200 $m$ in an area of dimensions 500 $\times$ 500 ${m^{2}}$ . The parameters involved in the experiment are shown in Table 1.

$L\times W$ is the area where the nodes are deployed. $r$ is the communication range of the device node. Both parameters belong to experimental environment settings, and their values refer to related work [32] to facilitate comparison with the experimental results. $N$ and $M$ are the number of deployed device nodes and the edge density, respectively, and the edge density refers to the number of edges when a new node joins the scale-free network. ${D_{\max}}$ is the maximum number of neighbors to which the device node can connect. The three parameters take different values to obtain different data sets, and then these different data sets are used for training and testing in the proposed model to prove the effectiveness and scalability of the model.

${L_{\textit{conv}}}$ is the number of layers of the graph convolution layer, ${L_{\textit{mlp}}}$ is the number of layers of multi-layer perceptron classification, and the node features and edge features of each layer are represented by $h$ dimensions. The settings of ${L_{\textit{conv}}}$ and $h$ refer to the graph convolutional neural network framework [37] cited in this paper. Through experiments on the robustness optimization effects ${L_{\textit{conv}}}$ of $h$ and under different values, it is found that the data sets of different network sizes require different parameter settings to achieve the best results, but increasing the experimental parameters will not affect the experimental results. For the consistency of analysis across network sizes, this experiment uses the same model hyper-parameter settings on different data sets, that is, the maximum model capacity. The results show that the experimental effect is the best when ${L_{\textit{conv}}}=$ 20 and $h=$ 200. Increasing the values of both cannot improve the effect, but it will increase the cost of training time. Similarly, the multi-layer perceptron layer mainly classifies edge feature information, and the number of layers is usually set to 2 or 3. This paper compares experiments and finds that the effect is best when ${L_{\textit{mlp}}}=$ 3, which is suitable for a variety of different data sets.

${B_{\textit{size}}}$ is the size of a batch. $l r$ and $d r$ are the learning rate and the decay rate of the learning rate, respectively. ${T_{\max}}$ is the maximum number of iterations, and the model is trained by the adaptive moment estimation (Adam) optimizer. Their specific values are obtained from previous experience and a large number of experimental results.

Table 1
Parameter settings of ROGCN

Parameter	Value	Parameter	Value
$L\times W$	500 $\times$ 500 ${m^{2}}$	${L_{\textit{mlp}}}$	3
$r$	200 $m$	$h$	200
$N$	100 $\sim$ 300	${B_{\textit{size}}}$	20
$M$	1 $\sim$ 5	$l r$	0.001
${D_{\max}}$	20 $\sim$ 45	$d r$	1.01
${L_{\textit{conv}}}$	20	${T_{\max}}$	1000

4.3 Convergence of ROGCN

This section verifies the convergence ability of the ROGCN model under the traditional and new node degree difference representation based on the scale-free network topology with the number of nodes $N=$ 100 and edge density $M=$ 2. Figure 3 shows the relationship between the loss value and the number of iterations during the training process. The ROGCN based on the traditional node degree difference (TNDD) definition gradually converges from the 175 iterations, while the ROGCN based on the new node degree difference (NNDD) representation method gradually converges from the 80 iterations. The new degree difference representation method significantly improves the convergence speed of the model. Subsequent experiments are based on the new node degree difference representation method. Additionally, with the gradual convergence of the ROGCN model, the robustness of the scale-free IoT topology is close to the optimal value of the current model, which proves the effectiveness of the model.

Figure 3.

The convergence of the ROGCN loss.

Figure 4.

Comparison of the robustness optimization effects of three search strategies on scale-free IoT topologies with different network sizes.

4.4 Comparison of different search strategies

This section compares the robustness of the three search strategies against malicious attacks under different network sizes. The number of scale-free IoT nodes $N$ is set to 100, 150, 200, 250, and 300, and the edge density $M$ is set to 2. The experimental results are shown in Fig. 4. Compared with the initial scale-free IoT topology, the three search strategies all significantly improve the robustness of the network topology; the optimization effects of ${\text{ROGC}}{{\text{N}}_{{\text{TOP-K}}}}$ , degree search ${\text{ROGC}}{{\text{N}}_{{\text{DS}}}}$ and robustness search ${\text{ROGC}}{{\text{N}}_{{\text{RS}}}}$ increase in turn. The optimization effect of ${\text{ROGC}}{{\text{N}}_{{\text{TOP-K}}}}$ is obviously weaker than that of the other two search strategies and will not be used in subsequent experiments. The time complexity comparison of the three search strategies is given in the last section.

4.5 Comparison between the initial topology and optimized topology

This section compares the changes in the network topology before and after ROGCN optimization. The basic topological features of the data sets are shown in Table 2. |V| and |E| are the total numbers of nodes and edges, respectively. $M$ represents the edge density, and $<k>$ represents the average degree of nodes. $C$ and $D$ are the average clustering coefficient and network diameter, respectively. $L$ and $A$ are the average shortest path length and assortativity coefficient, respectively, and $R$ represents the network robustness. By comparing the data features of the initial topology and the optimized topology using ${\text{ROGC}}{{\text{N}}_{{\text{RS}}}}$ , it can be found that as the robustness of the network increases, the network diameter and the average shortest path length become larger, which means that the network transmission efficiency decreases, indicating that the network robustness optimization is achieved at the cost of a certain transmission efficiency. The network assortativity coefficient is used to evaluate whether nodes with similar degree values tend to be connected with each other. The larger the network assortativity coefficient is, the stronger the assortativity. The optimized network topology assortativity coefficient increased significantly, indicating that the optimization process tends to connect nodes to similar degrees, which is consistent with the onion-like structure features. This also reflects that the optimized topology tends to form an onion-like structure, which is consistent with the research conclusions of the classical methods of network robustness optimization [12, 13], and verifies the rationality and effectiveness of the model proposed in this paper.

Table 2
Basic topological features of the data sets

Data sets				Initial topology					Optimized topology
\|V\|	M	\|E\|	$<k>$	C	D	L	A	R	C	D	L	A	R
100	2	200	4	0.146	6.015	3.145	$-$ 0.097	0.143	0.110	9.626	3.802	0.165	0.234
150	2	300	4	0.107	6.251	3.342	$-$ 0.091	0.133	0.079	10.734	4.128	0.198	0.231
200	2	400	4	0.088	6.518	3.481	$-$ 0.084	0.128	0.062	11.117	4.271	0.207	0.227
250	2	500	4	0.074	6.815	3.587	$-$ 0.081	0.125	0.052	11.183	4.358	0.214	0.224
300	2	600	4	0.065	6.95	3.670	$-$ 0.076	0.123	0.045	11.207	4.415	0.207	0.222
100	1	100	2	0.002	10.609	4.782	$-$ 0.201	0.029	0.001	10.329	4.643	$-$ 0.162	0.065
100	3	300	6	0.185	5.043	2.715	$-$ 0.040	0.231	0.157	6.045	2.922	0.168	0.337
100	4	400	8	0.226	4.712	2.492	$-$ 0.003	0.292	0.209	5.029	2.580	0.142	0.390
100	5	500	10	0.267	4.281	2.351	0.027	0.335	0.257	4.644	2.399	0.129	0.418

To compare the changes in the network topology before and after the algorithm optimization more vividly, a scale-free IoT topology with $N=$ 300 nodes and edge density $M=$ 2 is adopted. The experimental results are shown in Fig. 5, where a larger node diameter represents a higher degree. Figure 5a shows the initial topology, and Fig. 5b shows the topology optimized by ${\text{ROGC}}{{\text{N}}_{{\text{RS}}}}$ , in which the node positions correspond one-to-one. Figure 5c and d are generated by the Fruchterman-Reingold layout in the complex network analysis tool Gephi, corresponding to another representation of Fig. 5a and b, respectively. The robustness of the initial topology is $R=$ 0.1288, and the robustness after ${\text{ROGC}}{{\text{N}}_{{\text{RS}}}}$ optimization is $R=$ 0.2356, which is improved by 82.9%. In addition, by observing the network topology before and after optimization, it is obvious that the nodes with similar degrees in the optimized topology are connected with each other, presenting a hierarchical structure, which verifies that the topology optimized by the proposed algorithm has the characteristics of an onion-like structure.

Figure 5.

Comparison of the topology before and after optimization.

Figure 6.

Comparison of the connectivity of scale-free IoT topologies optimized by different algorithms under random attacks.

4.6 Comparison of the connectivity of scale-free IoT topologies optimized by different algorithms under attacks

This section compares the connectivity of network topologies optimized by different algorithms under random and malicious attacks. The experiment is based on a scale-free IoT topology with the number of nodes $N=$ 100 and edge density $M=$ 2. The connectivity of the network is measured by the number of nodes in the most connected subgraph after deleting the attack nodes. Figure 6 shows the variation trend of the maximum connected subgraph size of the scale-free IoT topology under a random attack. The results show that the initial scale-free IoT topology has strong resistance to random attacks, and the connectivity deviation between the topology optimized by various algorithms and the initial topology under random attacks is small, indicating that the optimized topology retains the ability of the scale-free network to resist random attacks well.

Figure 7 shows the variation trend of the maximum connected subgraph size of the scale-free IoT topology under malicious attacks. With the progression of the malicious attack, the number of nodes in the maximum connected subgraph of the initial scale-free IoT topology decreases rapidly, indicating that it is vulnerable to malicious attacks. The topology optimized by the various algorithms for resisting malicious attacks has been improved, among which ${\text{ROGC}}{{\text{N}}_{{\text{RS}}}}$ has the best effect and significantly improves the robustness of the scale-free IoT topology against malicious attacks.

Figure 7.

Comparison of the connectivity of scale-free IoT topologies optimized by different algorithms under malicious attacks.

4.7 Comparison between ROGCN and other algorithms on scale-free IoT topologies with different numbers of nodes

This section compares the robustness optimization effects of different algorithms against malicious attacks in different network sizes. The number of scale-free IoT nodes $N$ is set to 100, 150, 200, 250, and 300, and the edge density $M$ is set to 2. The experimental results are shown in Fig. 8. The four algorithms all significantly improve the robustness of the initial topology. With the increase in the number of nodes, the topology robustness of the SAA and ${\text{ROGC}}{{\text{N}}_{{\text{DS}}}}$ algorithm optimization shows a downward trend, while the topology robustness of the AREA and ${\text{ROGC}}{{\text{N}}_{{\text{RS}}}}$ algorithm optimization has no significant change. In addition, the ${\text{ROGC}}{{\text{N}}_{{\text{RS}}}}$ algorithm performs best in networks with different numbers of nodes. Compared with the AREA algorithm, the robustness of the five network sizes is increased by 2.5%, 2.1%, 1.6%, 1.2%, and 1.2%, with an average increase of 1.7%. This is because the ${\text{ROGC}}{{\text{N}}_{{\text{RS}}}}$ algorithm extracts the features of enhanced robustness, and the predicted adjacency matrix expands the optimal solution search space. On this basis, the robustness search strategy is more likely to obtain a better solution than the AREA algorithm.

Figure 8.

Comparison of the robustness optimization effects of ROGCN and other algorithms on scale-free IoT topologies with different network sizes.

Figure 9.

Comparison of the robustness optimization effects of ROGCN and other algorithms on scale-free IoT topologies with different edge densities.

4.8 Comparison between ROGCN and other algorithms on scale-free IoT topologies with different edge densities

This section compares the robustness optimization effects of the ROGCN, SAA and AREA algorithms against malicious attacks under different edge densities. The number of scale-free IoT nodes $N$ is set to 100, and the edge density $M$ is set to 1, 2, 3, 4, and 5. The experimental results are shown in Fig. 9. The robustness of the network topology increases with increasing edge density. The four algorithms significantly improve the $R$ value compared with that of the initial network topology, which means that the ability of the network topology to resist malicious attacks is gradually enhanced. In addition, in network topologies with different edge densities, the optimization effect of ${\text{ROGC}}{{\text{N}}_{{\text{RS}}}}$ is always better than that of the other algorithms. Compared with the AREA algorithm, the robustness under the five different edge densities is improved by 5.8%, 2.5%, 1.9%, 1.6%, and 1.5%, with an average increase of 2.7%.

Table 3
Time cost of different algorithms for optimizing a single IoT topology in different network sizes (unit: s)

N	SAA	AREA	${\text{ROGC}}{{\text{N}}_{{\text{TOP-K}}}}$	${\text{ROGC}}{{\text{N}}_{{\text{DS}}}}$	${\text{ROGC}}{{\text{N}}_{{\text{RS}}}}$
100	383.238	134.146	1.578	2.336	16.054
150	1031.158	522.533	1.875	3.185	91.374
200	2264.841	1552.905	2.132	4.726	370.268
250	4476.473	3557.283	3.252	7.355	992.649
300	8238.532	6867.424	4.665	12.519	1702.932

4.9 Running time of different algorithms

This section gives the statistics of the running time of various algorithms for different network sizes. The number of scale-free IoT nodes $N$ is set to 100, 150, 200, 250, and 300, and the edge density $M$ is set to 2. The results are shown in Table 3, which shows the robustness optimization time of a single topology and the test time of ROGCN on a single topology.

In the ROGCN model based on three search strategies, the computational cost of ${\text{ROGC}}{{\text{N}}_{{\text{RS}}}}$ is much higher than that of the other two strategies. This is because this strategy involves more robustness value (Eq. (1)) calculations and has higher complexity, but the optimization effect of ${\text{ROGC}}{{\text{N}}_{{\text{RS}}}}$ is the best, and it is suitable for scenarios with high requirements for topology robustness and general requirements for timeliness. The topology robustness optimization effect using ${\text{ROGC}}{{\text{N}}_{{\text{DS}}}}$ is general, but the computational cost is low, which is suitable for scenarios with high timeliness requirements. Considering that the training time of the ROGCN model is longer, the trained model has a significant advantage in the test time. Therefore, the ROGCN model can be trained offline, and then the trained model can be used to optimize the IoT topology online to achieve the goal of reducing the time cost.

5. Conclusion

Considering the limited communication range, connection degree and computing power of device nodes in the real IoT and the vulnerability to malicious attacks, this paper proposes a scale-free IoT topology robustness optimization method based on graph convolutional networks. Referring to the onion-like structural characteristics of highly robust topology, a new degree difference representation method is proposed to meet the feature extraction requirements of graph convolutional networks and accelerate the convergence speed. Then, the graph convolutional network with residual gating is used to extract the onion-like structural features from the initial topology to the label topology evolution process, and on this basis, different search strategies are designed to suit the needs of different IoT scenarios and effectively improve the ability of the IoT topology to resist malicious attacks. Finally, the experiments show that the proposed algorithm is superior to the SAA and AREA algorithms in terms of topology robustness optimization performance and computational cost.

In future research, transfer learning and reinforcement learning will be considered to achieve larger-scale network topology optimization and better optimization performance. In addition, future endeavors will explore a robustness enhancement mechanism that combines a small-world model and heterogeneous IoT. With the coming era of the Internet of Everything, topology optimization of the IoT will be challenging and meaningful research work.

Footnotes

Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 61803384). We would like to thank Ning Chen for his kind helps. We also would like to thank the anonymous reviewers for helpful comments and suggestions that certainly contribute to improve this paper.

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Graph convolutional networks-based robustness optimization for scale-free Internet of Things

Abstract

Keywords

1. Introduction

2.1 Scale-free IoT topology generation

2.2 Types of cyber-attacks

2.3 Robustness metric

3. ROGCN model

4.1 Data sets

4.2 Parameter settings

Table 1 Parameter settings of ROGCN

4.5 Comparison between the initial topology and optimized topology

Table 2 Basic topological features of the data sets

Table 3 Time cost of different algorithms for optimizing a single IoT topology in different network sizes (unit: s)

5. Conclusion

Footnotes

Acknowledgments

References

Table 1
Parameter settings of ROGCN

Table 2
Basic topological features of the data sets

Table 3
Time cost of different algorithms for optimizing a single IoT topology in different network sizes (unit: s)