1. Introduction

2. Preliminaries

3. Methods

3.3. The Proposed PRQMC Algorithm

To efficiently calculate the RASR for large-scale networks, the PRQMC algorithm is proposed, which leverages randomized QMC integration to approximate the RASR with a faster convergence rate and utilizes parallelization techniques to speed up the calculation. In the following, we first introduce the RASR calculation model based on QMC integration and then give the PRQMC algorithm.

3.3.1. RASR Calculation Model Based on QMC Integration

The RASR of a network G, as defined in Definition 5, can be expressed using Lebesgue integration based on the principle of MC integration (see Section 2), that is,

$$\begin{array}{c}\hfill RASR=E\left(ANCw(\mathit{Seq},\mathit{S})\right)={\int }_{\mathrm{\Omega }}ANCw(\mathit{Seq},\mathit{S})dP\left(\mathit{S}\right),\end{array}$$

(24)

where $\mathit{S}=(s_{v_1},s_{v_2},\dots ,s_{v_N})$ denotes a random variable representing the state of an attacking sequence $\mathit{Seq}$, $\mathrm{\Omega }$ is the sample space of $\mathit{S}$, $P\left(\mathit{S}\right)$ is the probability measure of $\mathit{S}$.

Let ${\mathit{P}}_{\mathit{v}}=(p_{v_1},p_{v_2},\dots ,p_{v_N})$ represent the ASR of each node corresponding to $\mathit{Seq}$, $\mathit{X}=(x_1,x_2,\dots x_N)$ is a uniformly distributed vector in ${[0,1]}^N$, where $x_i\in [0,1],i\in \{1,2,\dots ,N\}$. Then, $\mathit{S}=(s_{v_1},s_{v_2},\dots ,s_{v_N})$ can be represented as follows:

$$\mathit{S}=G\left(\mathit{X}\right),$$

(25)

where

$$s_{v_i}=G_i\left(x_i\right)=\left\{\begin{array}{cc}T,\hfill & \mathrm{if}{x}_i\le p_{v_i}\hfill \\ F,\hfill & \mathrm{otherwise}\hfill \end{array}\right.,i\in \{1,2,\dots ,N\}.$$

(26)

When the $\mathit{Seq}$ is determined, the $ANCw(\mathit{Seq},\mathit{S})$ can be represented as a function of $\mathit{X}$, that is,

$$F\left(\mathit{X}\right)=ANCw(\mathit{Seq},G(\mathit{X}\left)\right)=ANCw(\mathit{Seq},\mathit{S}).$$

(27)

By substituting Equation (27) into Equation (24) and transforming the integral space from $\mathrm{\Omega }$ to ${[0,1]}^N$, we obtain the following expression for RASR:

$$RASR=E\left[F\left(\mathit{X}\right)\right]={\int }_{{[0,1]}^N}F\left(\mathit{X}\right)dP\left(\mathit{S}\right).$$

(28)

This equation represents the integration of $F\left(\mathit{X}\right)$ with respect to the probability measure $P\left(\mathit{S}\right)$ over the N-dimensional unit hypercube ${[0,1]}^N$.

For the given network G, the sample space $\mathrm{\Omega }$ has a size of $2^N$. Let the state of $\mathit{Seq}$ be ${\mathit{S}}_i$, where $i\in \{1,2,3,\dots ,2^N\}$. Based on ${\mathit{P}}_{\mathit{v}}$, the unit hypercube ${[0,1]}^N$ can be divided into $2^N$ regions denoted by $Q_i$, where region $Q_i$ corresponds to state ${\mathit{S}}_i$, $i\in \{1,2,\dots 2^N\}$. Figure 4 illustrates this process for the case when $N=2$. Then, the integral in Equation (28) can be transformed into:

$${\int }_{{[0,1]}^N}F\left(\mathit{X}\right)dP\left(\mathit{S}\right)=\sum _{i=1}^{2^N}{\int }_{Q_i}F\left({\mathit{X}}^i\right)dP\left({\mathit{S}}_i\right),$$

(29)

where ${\mathit{X}}^{\left\{i\right\}}$ is a vector uniformly distributed within region $Q_i$.

The Lebesgue measure of region $Q_i$ in ${[0,1]}^N$, denoted by ${\lambda }_N\left(Q_i\right)$, is equivalent to the probability measure of ${\mathit{S}}_{\mathit{i}}$, denoted as $P\left({\mathit{S}}_i\right)$. Based on the principle of MC integration, we have:

$$\begin{array}{c}\hfill \sum _{i=1}^{2^N}{\int }_{Q_i}F\left({\mathit{X}}^{\left\{i\right\}}\right)dP\left({\mathit{S}}_i\right)=\sum _{i=1}^{2^N}{\int }_{Q_i}F\left({\mathit{X}}^{\left\{i\right\}}\right)d{\mathit{X}}^{\left\{i\right\}}={\int }_{{[0,1]}^N}F\left(\mathit{X}\right)d\mathit{X}.\end{array}$$

(30)

Combining Equation (28), Equation (29), and Equation (30), we obtain:

$$RASR=E\left[F\left(\mathit{X}\right)\right]={\int }_{{[0,1]}^N}F\left(\mathit{X}\right)d\mathit{X}.$$

(31)

By referencing Equation (14) and Equation (31), the RASR of a network can be approximated using the QMC integration method. The approximation of RASR, denoted by $\widehat{R}$, is defined as follows.

Definition 6.

Consider a network $G=(V,E)$ with N nodes. Suppose a sequence of nodes $\mathit{Seq}=(v_1,v_2,\dots ,v_N)$ is targeted for attack, ${\mathit{P}}_{\mathit{v}}=(p_{v_1},p_{v_2},\dots ,p_{v_N})$ signifies the ASR of each node. The RASR of the network G can be approximated by $\widehat{R}$, which is defined as:

$$\begin{array}{c}\hfill \widehat{R}=\frac{1}{K}\sum _{i=1}^{K}F\left({\mathit{Y}}_i\right)\approx RASR.\end{array}$$

(32)

Here, $\{{\mathit{Y}}_{\mathbf{1}},{\mathit{Y}}_{\mathbf{2}},\dots ,{\mathit{Y}}_{\mathit{K}}\}$, as specified in Equation (14), represents a set of points obtained from an N-dimensional LDS. K is the total number of samples. The function $F\left(\mathit{X}\right)$ is defined in Equation (27).

The error bound of the QMC integral is determined by the star discrepancy of the chosen LDS, making the selection of LDSs important for improving the accuracy of approximations. Two frequently used LDSs are the Halton sequence and the Sobol sequence [40]. In this research, the Sobol sequence is adopted, as it demonstrates better performance in higher dimensions compared to the Halton sequence [41].

3.3.2. Parallel Randomized QMC (PRQMC) Algorithm

Despite the faster convergence rate of the QMC integration method compared to MC integration, it still necessitates a large number of samples to calculate the average value. Furthermore, the calculation of function $ANCw(\mathit{Seq},\mathit{S})$, typically done through attack simulations, demands considerable computational resources, especially for large-scale networks [42]. Consequently, the computational process of obtaining $\widehat{R}$ for large-scale networks remains time-consuming. Additionally, due to the deterministic nature of the LDS, the QMC integration method can be seen as a deterministic algorithm, thus presenting challenges in assessing the reliability of numerical integration results and potentially leading to being stuck in local optima. In light of these issues, the PRQMC algorithm capitalizes on the benefits of the Randomized QMC method and parallelization.

The PRQMC algorithm improves computational efficiency through parallelization. This is because the computational cost of sampling the attack sequence’s state $\mathit{S}$ is significantly lower than that of computing the function $ANCw(\mathit{Seq},\mathit{S})$. Therefore, by initially sampling the attack sequence’s state $\mathit{S}$ and obtaining a sufficient number of samples, it is possible to calculate the $\widehat{R}$ by parallelizing the computation of the function $ANCw(\mathit{Seq},\mathit{S})$ with various samples. This approach effectively accelerates the calculation process by distributing the task across multiple processors or computing nodes.

Additionally, the PRQMC algorithm enhances randomness by randomly sampling points from the LDS, providing unbiased estimation and improved variance reduction capabilities. This is particularly advantageous in high-dimensional problems, where RQMC often outperforms QMC in terms of accuracy and efficiency [43].

The procedure of the PRQMC algorithm is presented in Algorithm 1, which consists of two main steps: “sampling state" and “paralleling stage". In the sampling stage, we first randomly sample K points $\{{\mathit{Y}}_{\mathbf{1}},{\mathit{Y}}_{\mathbf{2}},\dots ,{\mathit{Y}}_{\mathit{K}}\}$ from an N-dimensional Sobol sequence, then determine K states of the attack sequence, $\{{\mathit{S}}_1,{\mathit{S}}_2,\dots ,{\mathit{S}}_K\}$, by comparing the values of each dimension of the sampled points with the ASR of each node. In the paralleling stag, we parallelize the computation of the function $ANCw(\mathit{Seq},{\mathit{S}}_i)$, then obtain $\widehat{R}$ by calculating the average value of $ANCw(\mathit{Seq},{\mathit{S}}_i)$.

Algorithm 1 PRQMC($G,\mathit{Seq},\mathit{P},K$)

3.4. The Proposed HBnnsAGP Attack Strategy

To assess the lower bound of network RASR, a new attack strategy called the High BCnns Adaptive GCC-Priority (HBnnsAGP) is presented. In HBnnsAGP, a novel centrality measure called BCnns is proposed to quantify the significance of a node, and GCC-priority attack strategy is utilized to improve attack effectiveness. Algorithm 2 describes the procedure of HBnnsAGP, which contains two steps: “obtaining the first part of $\mathit{Seq}$" and “obtaining the second part of $\mathit{Seq}$". In the first step, the algorithm obtains the first part of the attack sequence by iteratively removing the node with the highest BCnns in GCC and recalculating BCnns for the remaining nodes until only isolated nodes remain in the residual network. In the second step, the algorithm arranges these isolated nodes in descending order according to their DC values in the initial network to obtain the second part of the attack sequence. This procedure is aimed at improving the effectiveness of attacks when the ASR is below 100%. It is important to note that isolated nodes when the ASR is 100% may no longer remain isolated, as depicted in, as shown in Figure 1. Additionally, previous research has shown that there is minimal difference in destructiveness between simultaneous attacks and sequential attacks based on DC [9]. Therefore, by sorting these isolated nodes in descending order based on their DC values from the initial network (similar to the approach used in simultaneous attacks), the second step further improves the effectiveness of attacks when the ASR is less than 100%.

In the following, we first introduce the BCnns and then give the GCC-priority attack strategy.

Algorithm 2 HBnnsAGP($G,N_1,N_2$)

3.4.1. Non-central Nodes Sampling Betweenness Centrality (BCnns)

Contrasted with BC (see Definition 2), which evaluates a node’s role as a mediator in the network based on the count of shortest paths it traverses for all node pairs. BCnns quantifies the importance of nodes acting as bridge nodes between different network communities by counting the number of shortest paths that pass through a node for specific pairs of non-central nodes (nodes located in the periphery of the network and with less importance). These bridge nodes typically serve as mediators for non-central nodes across different communities. The BCnns is defined as follows.

Definition 7.

For a network $G=(V,E)$ with N nodes, the BCnns of node v in network G is:

$$\begin{array}{c}\hfill B{C}_{\mathit{nns}}\left(v\right)=\sum _{\begin{array}{c}s\in S,t\in T\end{array}}\frac{\sigma (s,t\mid v)}{\sigma (s,t)},\end{array}$$

(33)

where $S,T\subset V_{nns}$, $V_{nns}$ is the set of non-central nodes sampled from V, $V_{nns}\subset V$, $S\cap T=\varnothing$. The $\sigma (s,t)$ and $\sigma (s,t\mid v)$ have the same meaning as defined in Definition 2.

By selecting the appropriate pairs of non-central nodes, BCnns can more effectively measure the significance of nodes as bridges between different communities in a network. While these bridge nodes may not have the highest BC value, they are crucial for maintaining overall network connectivity and could potentially have the highest BCnns value.

The Definition of BCnns highlights the importance of selecting suitable nodes for sets S and T. Thus, we proposed an algorithm called Selection$ST$ for node selection. Algorithm 3 describes the procedure of Selection$ST$. Initially, the nodes are sorted in ascending order based on their DC values, and the first $N_1$ nodes with lower DC values are selected to create the non-central nodes set $V_{nns}$. This is because nodes with lower DC values typically have lower centrality and are considered non-central nodes. Next, in order to achieve a more balanced sampling, $V_{nns}$ is divided into two subsets: $V_{nns}^{odd}$ containing nodes at odd indices and $V_{nns}^{even}$ containing nodes at even indices. Lastly, $N_2$ nodes are randomly sampled from $V_{nns}^{odd}$ to create set S, and $N_2$ nodes are similarly sampled from $V_{nns}^{even}$ to form set T.

Algorithm 3 Selection$ST$($G_c,N_1,N_2$)

The $N_1$ and $N_2$ are chosen based on the size of the network and the node degree distribution. Typically, both $N_1$ and $N_2$ are much smaller compared to the total number of nodes N. Therefore, BCnns have higher computational efficiency compared to BC, especially for large-scale networks.

Figure 5 demonstrates the differences between BC and BCnns. Specifically, Figure 5a identifies the non-central nodes in red, Figure 5b showcases node sizes based on BC values, and Figure 5c adjusts node sizes based on their BCnns values. Notably, node 14 plays a critical bridging role between two communities, a role that BCnns captures more accurately than BC.

3.4.2. GCC-Priority Attack Strategy

As the attack progresses, the network fragments into connected components of varying sizes. The importance of these components varies within the residual network. The GCC refers to the largest connected component containing the most nodes. The destruction of the GCC accelerates the collapse of the network. The GCC-priority attack strategy enhances the attack’s effectiveness by targeting nodes within the GCC at each stage of the attack process.

4. Experimental Studies

5. Conclusion

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