Figure 1.
An example of network disintegration processes under different ASR. Gray nodes indicate successful attacks, green nodes represent unsuccessful attacks, and blue nodes denote unattacked nodes. (a–f) represent scenarios where 5.9%, 20.6%, and 35.3% of network nodes are attacked, with ASR of 100% and 60% respectively.
Figure 1.
An example of network disintegration processes under different ASR. Gray nodes indicate successful attacks, green nodes represent unsuccessful attacks, and blue nodes denote unattacked nodes. (a–f) represent scenarios where 5.9%, 20.6%, and 35.3% of network nodes are attacked, with ASR of 100% and 60% respectively.
Figure 2.
An example of the MC integration method for approximating a definite integral over a one-dimensional unit interval. (a) illustrates the approximation of the integral by summing the areas of bars that correspond to the sampled points. Each bar’s height represents the value of at , and its width is , where K denotes the total number of samples. (b) demonstrates the sequential rearrangement of the bars to prevent overlapping on the -axis, ensuring a clear visualization of the areas.
Figure 2.
An example of the MC integration method for approximating a definite integral over a one-dimensional unit interval. (a) illustrates the approximation of the integral by summing the areas of bars that correspond to the sampled points. Each bar’s height represents the value of at , and its width is , where K denotes the total number of samples. (b) demonstrates the sequential rearrangement of the bars to prevent overlapping on the -axis, ensuring a clear visualization of the areas.
Figure 3.
A comparison of MC and QMC integration methods. (a,d) show the two-dimensional projections of a PRS and an LDS (a Sobol sequence), respectively. (b,c) depict the MC integration for approximating a definite integral over a one-dimensional unit interval, while (e,f) present the QMC integration for approximating a definite integral over a one-dimensional unit interval.
Figure 3.
A comparison of MC and QMC integration methods. (a,d) show the two-dimensional projections of a PRS and an LDS (a Sobol sequence), respectively. (b,c) depict the MC integration for approximating a definite integral over a one-dimensional unit interval, while (e,f) present the QMC integration for approximating a definite integral over a one-dimensional unit interval.
Figure 4.
An example to illustrate the division of the unit hypercube, where and . The unit hypercube is divided into 4 regions, namely , where each region corresponds to a state of , denoted by .
Figure 4.
An example to illustrate the division of the unit hypercube, where and . The unit hypercube is divided into 4 regions, namely , where each region corresponds to a state of , denoted by .
Figure 5.
An illustrative example of non-central nodes and comparison of BC and BCnns. In this figure, (a) highlights non-central nodes in red, (b) showcases node sizes based on BC, and (c) showcases node sizes based on BCnns.
Figure 5.
An illustrative example of non-central nodes and comparison of BC and BCnns. In this figure, (a) highlights non-central nodes in red, (b) showcases node sizes based on BC, and (c) showcases node sizes based on BCnns.
Figure 6.
The topologies of six real-world networks. The size of each node is proportional to its degree. (a) Karate, (b) Krebs, (c) Airport, (d) Crime, (e) Power, (f) Oregon1.
Figure 6.
The topologies of six real-world networks. The size of each node is proportional to its degree. (a) Karate, (b) Krebs, (c) Airport, (d) Crime, (e) Power, (f) Oregon1.
Figure 7.
Comparison of the convergence and error of the PRQMC, QMC, and MC methods in assessing robustness for two smaller-scale networks. Convergence and error curves of Karate (a,b), Krebs (c,d).
Figure 7.
Comparison of the convergence and error of the PRQMC, QMC, and MC methods in assessing robustness for two smaller-scale networks. Convergence and error curves of Karate (a,b), Krebs (c,d).
Figure 8.
Comparison of the convergence and standard deviation of the PRQMC, QMC, and MC methods in assessing robustness for four larger-scale networks. Convergence and standard deviation curves of Airport (a,e), Crime (b,f), Power (c,g), Oregon1 (d,h).
Figure 8.
Comparison of the convergence and standard deviation of the PRQMC, QMC, and MC methods in assessing robustness for four larger-scale networks. Convergence and standard deviation curves of Airport (a,e), Crime (b,f), Power (c,g), Oregon1 (d,h).
Figure 9.
The ANCw curves of networks under different attack strategies: (a) Karate, (b) Krebs, (c) Airport, (d) Crime, (e) Power, (f) Oregon1.
Figure 9.
The ANCw curves of networks under different attack strategies: (a) Karate, (b) Krebs, (c) Airport, (d) Crime, (e) Power, (f) Oregon1.
Table 1.
Basic information for six real-world networks. N and M represent the number of nodes and edges, respectively; <k> and denote the network’s average degree and maximal degree, respectively; and C is the average clustering coefficient.
Table 1.
Basic information for six real-world networks. N and M represent the number of nodes and edges, respectively; <k> and denote the network’s average degree and maximal degree, respectively; and C is the average clustering coefficient.
Network | N | M | <k> | | C |
---|
Karate [41] | 34 | 78 | 4.59 | 17 | 0.571 |
Krebs [10] | 62 | 159 | 5.13 | 22 | 0.591 |
Airport [42] | 332 | 2126 | 12.81 | 139 | 0.625 |
Crime [42] | 829 | 1473 | 3.55 | 25 | 0.008 |
Power [42] | 4941 | 6594 | 2.67 | 19 | 0.107 |
Oregon1 [43] | 10,670 | 22,002 | 4.12 | 2312 | 0.456 |
Table 2.
Computational time comparison of PRQMC, QMC, and MC methods (s). Smaller values are better (best in bold).
Table 2.
Computational time comparison of PRQMC, QMC, and MC methods (s). Smaller values are better (best in bold).
Network | MC | QMC | PRQMC |
---|
Karate | 1.8 | 1.7 | 0.4 |
Krebs | 4.5 | 4.3 | 0.6 |
Airport | 106.7 | 104.9 | 3.0 |
Crime | 518.2 | 520.6 | 7.4 |
Power | 20,525.7 | 20,529.1 | 343.7 |
Oregon1 | 213,748.2 | 213,758.1 | 4262.4 |
Table 3.
The sample numbers ( and ) for different networks used in HBnnsAGP.
Table 3.
The sample numbers ( and ) for different networks used in HBnnsAGP.
Network | | |
---|
Karate | 16 | 8 |
Krebs | 30 | 16 |
Airport | 100 | 60 |
Crime | 120 | 80 |
Power | 1300 | 80 |
Oregon1 | 2300 | 80 |
Table 4.
The robustness of networks under different ASR. All values are multiplied by 100. Smaller values represent better attack destructiveness for attack strategies (best in bold).
Table 4.
The robustness of networks under different ASR. All values are multiplied by 100. Smaller values represent better attack destructiveness for attack strategies (best in bold).
Scenario | Network | HBnnsAGP | FINDER | HBA | HDA | RF |
---|
1. ASR = 100% | Karate | 12.77 | 14.12 | 15.04 | 15.04 | 42.86 |
Krebs | 12.26 | 16.26 | 14.21 | 17.23 | 42.96 |
Airport | 7.53 | 10.25 | 7.93 | 11.10 | 43.19 |
Crime | 9.90 | 11.04 | 10.14 | 11.54 | 39.57 |
Power | 0.91 | 5.02 | 1.01 | 5.23 | 20.29 |
Oregon1 | 0.68 | 1.06 | 0.73 | 1.01 | 36.47 |
Avg score | 7.34 | 9.63 | 8.18 | 10.19 | 37.56 |
2. ASR = 90% | Karate | 22.26 | 23.92 | 23.79 | 24.81 | 47.25 |
Krebs | 20.70 | 23.59 | 21.49 | 24.40 | 47.09 |
Airport | 20.94 | 23.35 | 22.23 | 23.74 | 47.74 |
Crime | 16.44 | 17.06 | 17.23 | 16.95 | 43.52 |
Power | 2.41 | 6.06 | 2.37 | 6.39 | 22.51 |
Oregon1 | 7.30 | 9.01 | 8.18 | 8.76 | 40.89 |
Avg score | 15.01 | 17.17 | 15.88 | 17.51 | 41.50 |
3. ASR = 80% | Karate | 31.52 | 33.00 | 32.93 | 34.09 | 52.25 |
Krebs | 28.75 | 30.82 | 29.12 | 31.37 | 51.82 |
Airport | 31.87 | 33.86 | 33.59 | 34.17 | 52.77 |
Crime | 23.81 | 26.00 | 25.72 | 25.43 | 48.37 |
Power | 4.09 | 7.44 | 4.04 | 7.77 | 25.44 |
Oregon1 | 16.71 | 19.26 | 18.41 | 18.91 | 46.12 |
Avg score | 22.79 | 25.06 | 23.97 | 25.29 | 46.13 |
4. ASR = 70% | Karate | 40.89 | 42.1 | 42.29 | 43.3 | 57.75 |
Krebs | 37.7 | 39.2 | 37.8 | 39.44 | 57.13 |
Airport | 41.65 | 43.25 | 43.23 | 43.51 | 58.10 |
Crime | 34.1 | 37.76 | 37.11 | 37.3 | 54.47 |
Power | 6.55 | 9.65 | 6.57 | 9.95 | 29.24 |
Oregon1 | 27.37 | 30.02 | 29.36 | 29.68 | 51.95 |
Avg score | 31.38 | 33.66 | 32.73 | 33.86 | 51.44 |
5. ASR = 60% | Karate | 50.39 | 51.33 | 51.70 | 52.42 | 63.70 |
Krebs | 47.39 | 48.49 | 47.52 | 48.64 | 62.99 |
Airport | 50.73 | 51.98 | 52.03 | 52.23 | 63.66 |
Crime | 46.48 | 49.33 | 49.08 | 48.91 | 61.13 |
Power | 10.33 | 13.35 | 10.58 | 13.61 | 34.34 |
Oregon1 | 38.50 | 40.87 | 40.41 | 40.56 | 58.25 |
Avg score | 40.64 | 42.56 | 41.89 | 42.73 | 57.35 |
6. ASR = 50% | Karate | 59.90 | 60.55 | 60.88 | 61.41 | 69.90 |
Krebs | 57.61 | 58.38 | 57.74 | 58.33 | 62.99 |
Airport | 59.45 | 60.32 | 60.39 | 60.52 | 63.66 |
Crime | 57.59 | 59.50 | 59.50 | 59.08 | 61.13 |
Power | 16.54 | 19.73 | 17.31 | 19.93 | 34.34 |
Oregon1 | 49.68 | 51.58 | 51.28 | 51.33 | 58.25 |
Avg score | 50.13 | 51.69 | 51.18 | 51.77 | 57.35 |
7. ASR = 50% for the first 30% of nodes | Karate | 48.61 | 50.38 | 49.76 | 50.57 | 50.79 |
Krebs | 45.51 | 47.16 | 45.78 | 47.38 | 53.00 |
Airport | 48.78 | 50.68 | 51.00 | 50.47 | 49.76 |
Crime | 41.84 | 48.17 | 46.90 | 47.12 | 50.06 |
Power | 14.87 | 17.79 | 16.32 | 17.86 | 27.04 |
Oregon1 | 41.26 | 42.91 | 42.83 | 43.14 | 48.92 |
Avg score | 40.15 | 42.91 | 42.10 | 42.76 | 46.60 |
8. Random ASR | Karate | 35.12 | 36.29 | 36.50 | 37.57 | 54.11 |
Krebs | 30.39 | 32.99 | 31.02 | 33.36 | 53.91 |
Airport | 36.95 | 38.58 | 38.67 | 38.99 | 55.24 |
Crime | 27.90 | 30.96 | 30.48 | 30.42 | 51.06 |
Power | 5.17 | 8.46 | 5.18 | 8.80 | 27.40 |
Oregon1 | 21.61 | 24.23 | 23.52 | 23.86 | 48.79 |
Avg score | 26.19 | 28.56 | 27.55 | 28.79 | 48.42 |
Table 5.
The computation time of different attack strategies (ms). Smaller values are better (best in bold).
Table 5.
The computation time of different attack strategies (ms). Smaller values are better (best in bold).
Network | HBnnsAGP | FINDER | HBA | HDA |
---|
Karate | 1.6 | 16.3 | 1.9 | 0.5 |
Krebs | 3.6 | 36.6 | 4.6 | 2.3 |
Airport | 82.3 | 218.3 | 211.0 | 11.1 |
Crime | 552.1 | 369.3 | 4434.6 | 49.1 |
Power | 6760.7 | 1397.9 | 78,119.9 | 1796.8 |
Oregon1 | 15,799.1 | 8641.5 | 477,802.8 | 2065.9 |