A simplicial complex family network structure robustness analysis method

Abstract

The application provides a simplex family network structure robustness analysis method and relates to the technical field of control and information. According to three simplex network generation algorithms, different initial networks are constructed, and based on a structure robustness index calculation formula, analytical expressions of the structure robustness of the three simplex networks are derived theoretically, and the influence mechanism of network topology and attack mode on the system structure robustness is revealed. The simplex family network is used to describe the structure of an actual complex system, can effectively reflect the high-order interaction relationship of the actual network, and is more suitable for modeling of a real network system; the node degree-based attack mode is used to simulate the deliberate attack suffered by the actual network system, and the influence mechanism of the high-order interaction and the attack mode on the structure robustness is revealed, which can provide scientific theoretical support and technical implementation guidance for reasonably formulating a network attack defense strategy in practical applications.

Description

Technical Field

[0001] This invention relates to the field of control and information technology, and in particular to a robustness analysis method for simple complex family network structures. Background Technology

[0002] Robustness of a network system quantifies the degree to which nodes or connections within the system maintain structural integrity or functional availability after being attacked or experiencing failure. Robustness technology for complex network structures is closely linked to various sectors of society, having a significant impact on multiple fields such as transportation, energy, and social networks. For example, in transportation networks, studying structural robustness can identify key nodes and road segments, allowing for the development of proactive response strategies, such as adding backup lines or strengthening maintenance on congestion-prone or accident-prone sections, improving the network's ability to cope with emergencies, reducing the risk of traffic paralysis, and ensuring the smooth flow of people and goods. In the energy internet, robustness research helps optimize the energy network structure, such as the rational layout of substations and transmission lines, enhancing the network's ability to withstand interference from natural disasters and human-caused disruptions, ensuring a stable and reliable energy supply, and reducing the probability of large-scale power outages and other accidents. In social networks, robustness can identify key users and information dissemination paths. When dealing with issues such as the spread of misinformation, controlling the information dissemination at key nodes can prevent the large-scale spread of false information, maintaining the normal order and information authenticity of the social network.

[0003] Existing research on the robustness of complex network structures mainly focuses on low-order networks with binary interactions, while research on the structural robustness of high-order interactive networks is still scarce. Most real-world networks consist of three or more interacting units, and their structures can no longer be described by simple binary node interactions; instead, they require characterization through "high-order networks" with higher-order interactions. Compared to low-order networks, high-order networks are more complex, involving multiple and multi-layered relationships. Such network structures more closely resemble the interactions between real nodes and edges, and have a wider range of applications and value. Simplex graphs, as a typical representation of high-order networks, are a natural and powerful tool for simulating group interactions in the real world. For example, in simulating cooperative networks, cooperation may involve three or more people, not just two. Using simplex graphs can transcend binary interaction patterns and capture group (multi-layered) interaction patterns. The mathematical complexity of simplex graphs presents both opportunities and challenges for studying robustness on simplex graphs.

[0004] Furthermore, existing technologies for simulating attack methods typically employ random or deliberate attacks, lacking comprehensive and in-depth comparative studies of network structural robustness under various attack methods. This invention provides an in-depth comparison of deliberate attacks from different perspectives, revealing the impact of network topology and attack methods on structural robustness, and offering scientific basis and technical guidance for the design and improvement of anti-interference capabilities of practical network systems. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides a robustness analysis method for simple complex family network structures;

[0006] A robustness analysis method for simple complex family network structures includes the following steps:

[0007] Step 1: Construct three simple complex networks , , ;

[0008] Construct three simple complex networks respectively , , Where k represents the dimension of the simplex, This indicates that the center of the network is a k-1 dimensional simplex, and that it is a common face of the l surrounding k-dimensional simplexes; This indicates that there are l k-dimensional simplexes in the network, and adjacent simplexes are connected by a lower adjacency connection. This indicates that the center of the network is a k-dimensional simplex, there are x1 k-dimensional simplexes that are under-adjacent to one face of the central simplex, x2 k-dimensional simplexes that are under-adjacent to the other face of the central simplex, and finally x... k+1 The k-dimensional simplexes are under-adjacent to the last remaining face of the central simplex; x k+1 This represents the number of k-dimensional simplexes;

[0009] The structure diagrams of the three simple complex networks are all undirected connected graphs. ,in, Let N0 be the set of N0 nodes. Let N be the N0th node, and E represent the set of edges;

[0010] Step 2: Based on the simple complex network generated in Step 1, calculate the ratio Q(i) of the nodes after being attacked by the network;

[0011] Based on the simplex complex network generated in step 1, the degree values ​​of all nodes in the network are calculated and sorted in descending order. A node being attacked is modeled as being deleted, i.e., the node is removed from the network. Nodes are selected for attack in sequence according to the degree value sorting results. When selecting nodes, if nodes with the same degree value are encountered, the ratio Q(i) after deleting the node is compared. The node with the larger Q(i) ratio is deleted first, where Q(i) is defined as the ratio of the number of nodes in the largest connected component in the current network after the i-th network attack to the total number of nodes in the current network, expressed as follows:

[0012] (1);

[0013] in, N0 represents the initial total number of network nodes, and N represents the current total number of network nodes. This represents the number of nodes contained in the largest connected component in the current network. Among them, there are three special cases: (1) After the i-th attack, the current network is a connected graph, then the corresponding ratio ;(2) No. After this attack, the network now has only one isolated node, corresponding to the ratio. ;(3) No. After the attack, there are no nodes left in the current network, and the ratio equals 0. ;

[0014] The network attack takes the following two different approaches: (1) The first approach is to sort the nodes in descending order based on the initial network node degree values ​​and delete i nodes in the i-th attack; (2) The second approach is to sort the nodes in descending order according to the current network structure node degree values ​​before each network attack and delete the node with the largest degree value in the current network in each attack. The number of attacks in both approaches from start to finish is equal to the total number of nodes in the initial network.

[0015] Step 3: Calculate the network structure robustness index after a node is attacked;

[0016] The robustness index of the network structure is defined as follows:

[0017] (2);

[0018] Where Q(i) represents the ratio of the number of nodes in the largest connected component of the current network after the i-th attack to the total number of nodes in the current network. Indicates the robustness of the network structure;

[0019] Based on the three simple complex networks described in step 1, and combined with formula (2), the theoretical analytical expressions for the structural robustness of the three simple complex networks under the first attack method are obtained respectively:

[0020] (1) When When k=2 in the network, ;

[0021] (2) When When k=2 and the initial total number of network nodes N0 is odd, , ,symbol Indicates rounding down to the nearest integer; when When k=2 and the initial total number of network nodes N0 is even, , ;

[0022] (3) When When k=2 in the network, .

[0023] The beneficial effects of adopting the above technical solution are as follows:

[0024] This invention provides a robustness analysis method for simple complex family network structures, which can reveal the positive influence mechanism of higher-order network interactions on system structural robustness, and is more feasible in application. Compared with the prior art, the specific beneficial technical effects of this invention are: (1) Using simple complex networks to characterize the higher-order interaction relationships of actual network systems can achieve better modeling of real systems and has higher application value; (2) By analyzing the influence mechanism of network topology and attack methods on structural robustness, it is found that the third-order motif structure in higher-order interactions is beneficial to improving the structural robustness of the system, providing a theoretical basis and technical guidance for rationally formulating attack and defense response strategies in practical applications.

[0025] This invention can be applied to fields such as communication networks, power networks, transportation networks, biological networks, and social networks. For example, in communication networks, by analyzing the robustness of network nodes or connections under fault or attack conditions, network topology can be optimized to improve its resilience and ensure the reliability and efficiency of data transmission. In power networks, robustness analysis can be used to assess the vulnerability of critical nodes or connections, design more robust power grid structures, thereby reducing the risk of power outages and responding to natural disasters or human-caused damage. In transportation networks, robustness analysis can be used to optimize traffic flow distribution, improve emergency response capabilities to accidents or natural disasters, and thus ensure the normal operation of the transportation system. Attached Figure Description

[0026] Figure 1 This is a flowchart illustrating the implementation of a robustness analysis method for a simple complex family network structure according to the present invention.

[0027] Figure 2These are schematic diagrams of three simple complex network topologies used in this invention.

[0028] Among them, (a)- Network structure, (b)- Network structure, (c)- The network structure;

[0029] Figure 3 In the implementation of this invention The Internet When using the first attack method, the ratio With the initial number of network nodes The change graph;

[0030] Figure 4 In the implementation of this invention The Internet When using the first attack method, the ratio With the initial number of network nodes The change graph;

[0031] Figure 5 In the implementation of this invention The Internet When using the first attack method, the ratio With the initial number of network nodes The change graph;

[0032] Figure 6 In this invention, all three simple complex networks employ the first attack method, and the structural robustness index... With the initial number of network nodes The change graph;

[0033] Figure 7 In the implementation of this invention The Internet A comparison of the structural robustness index as a function of the initial number of nodes under two different structures, both using the first attack method;

[0034] Figure 8 In the implementation of this invention The Internet Two different attack methods were used, and the trend of structural robustness index changing with the initial number of nodes was compared in the image. Among them, attack1 is the first attack method, which is based on the initial network node degree value sorted in descending order, and the i-th attack deletes i nodes; attack2 is the second attack method, which is based on the current network node degree value sorted in descending order, and deletes one node in each attack.

[0035] Figure 9 In the implementation of this invention The Internet A comparison chart showing how the structural robustness index changes with the initial number of nodes under two different structures, both using the first attack method. Detailed Implementation

[0036] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and are not intended to limit the scope of the invention.

[0037] A robustness analysis method for simple complex family network structures, such as Figure 1 As shown, it includes the following steps:

[0038] Step 1: Construct three simple complex networks , , ;

[0039] Construct three simple complex networks respectively , , Where k represents the dimension of the simplex, This indicates that the center of the network is a k-1 dimensional simplex, and that it is a common face of the l surrounding k-dimensional simplexes; This indicates that there are l k-dimensional simplexes in the network, and adjacent simplexes are connected by a lower adjacency connection. This indicates that the center of the network is a k-dimensional simplex, there are x1 k-dimensional simplexes that are under-adjacent to one face of the central simplex, x2 k-dimensional simplexes that are under-adjacent to the other face of the central simplex, and finally x... k+1 The k-dimensional simplexes are under-adjacent to the last remaining face of the central simplex; x k+1 This represents the number of k-dimensional simplexes;

[0040] The structure diagrams of the three simple complex networks are all undirected connected graphs. ,in, express A set of nodes, For the first There are nodes, and E represents the set of edges;

[0041] In this embodiment, three simple complex networks are constructed. , , The structures are as follows: Figure 2 of (a), Figure 2 (b) and Figure 2 As shown in (c), Figure 2 (a) indicates The network structure has a total of nodes. ; Figure 2 (b) indicates The network structure has a total of nodes. ; Figure 2 (c) represents The network structure has a total of nodes. In the three simple complex networks constructed, k is always 2, where... The 2D simplex is a triangle used to represent higher-order interactions between three nodes. These three simplex networks are all undirected connected graphs and can be used to model real-world social networks, brain networks, intelligent transportation systems, etc. For example, a 2D simplex in a social network can be used to represent a paper co-authored by three authors.

[0042] Step 2: Based on the simplex complex network generated in Step 1, calculate the ratio of nodes after a network attack. ;

[0043] Based on the simplex complex network generated in step 1, calculate the degree values ​​of all nodes in the network and sort them in descending order. A node being attacked is modeled as being deleted, i.e., removed from the network. Attacks are performed on nodes sequentially based on their degree values. If nodes with the same degree value are encountered during node selection, the ratio is calculated based on the value after deleting that node. Comparison, Nodes with larger ratios are deleted first. Defined as the ratio of the number of nodes in the largest connected branch in the current network after the i-th network attack to the total number of nodes in the current network, expressed as follows:

[0044] (1);

[0045] in, N0 represents the initial total number of network nodes, and N represents the current total number of network nodes. This represents the number of nodes contained in the largest connected component in the current network. Among them, there are three special cases: (1) After the i-th attack, the current network is a connected graph, then the corresponding ratio ;(2) No. After this attack, the network now has only one isolated node, corresponding to the ratio. ;(3) No. After the attack, there are no nodes left in the current network, and the ratio equals 0. ;

[0046] The network attack takes the following two different methods: (1) The first method is to sort the initial network node degree values ​​in descending order, and use the i-th attack to delete i nodes, that is, the first attack selects the node with the largest degree value from the initial network node sorting result and deletes it, and calculates the... The second attack involves deleting the node with the second-largest degree value from the initial network node ranking results, and calculating... ; (2) The second method is to sort the nodes in the current network structure in descending order before each network attack, and then select the node with the largest degree value in the current network for deletion in each attack. That is, the first attack selects the node with the largest degree value from the current network node sorting results for deletion, and calculates... The second attack involves sorting the nodes according to the current network structure, selecting the node with the highest degree value, and deleting it. The calculation... ; Both of these attack methods involve the same number of attacks from start to finish, equal to the initial total number of network nodes.

[0047] This embodiment employs the following two attack methods:

[0048] 1. First type of attack: using For example, the network structure is as follows Figure 2 As shown in (b). Based on the initial network structure, the degree values ​​of all nodes are calculated and the results of the node degree values ​​in descending order are as follows: d(5)>d(7)=d(8)=d(9)=d(2)=d(4)=d(10)>d(3)=d(11)=d(6)>d(1)=d(12), where d(i) represents the node The degree value.

[0049] (1) First attack, delete The current network is a connected graph, and the ratio is... ;

[0050] (2) Second attack: delete v5 and v7. The current network is a connected graph. Ratio ;

[0051] (3) The third attack deletes v5, v7, and v2. The current network is a connected graph. The ratio ;

[0052] (4) The fourth attack deletes v5, v7, v2, and v 10 The current network is a connected graph, and the ratio is... ;

[0053] (5) The fifth attack will delete v5, v7, v9, and v 10 v8, the current network has a maximum connected component containing 5 nodes, and a total of 7 nodes. The ratio is... ;

[0054] (6) The sixth attack deleted v5, v7, v2, and v 10v9, v8, the current network has a maximum connected component containing 4 nodes, and the remaining network has a total of 6 nodes. The ratio ;

[0055] (7) The seventh attack deletes v5, v7, v2, v4, and v 10 v9, v8, the current network has a maximum connected component containing 2 nodes, and the current network has a total of 5 nodes. The ratio is... ;

[0056] (8) The eighth attack deleted v5, v7, v2, v4, and v 10 v9, v8, v3, the current network has a maximum connected component containing 2 nodes, and the current network has a total of 4 nodes. The ratio is... ;

[0057] (9) Ninth attack, delete v5, v7, v2, v4, v 10 v9, v8, v3, v6, the current network has a maximum connected component containing 2 nodes, and the current network has a total of 3 nodes. The ratio ;

[0058] (10) The tenth attack deletes v5, v7, v2, v4, and v 10 v9, v8, v3, v6, v 11 The largest connected component in the current network contains 1 node, and the total number of nodes in the current network is 2. The ratio is... ;

[0059] (11) The eleventh attack deleted v5, v7, v2, v4, and v 10 v9, v8, v3, v6, v 11 v1, the current network contains only one isolated node, the ratio ;

[0060] (12) The twelfth attack deletes all nodes in the network, leaving the network with no nodes. The ratio .

[0061] It should be noted that during the sixth attack, based on the initial network structure node degree values, d(2) = d(4). If v5, v7, v2, and v... 10 v9, v8, the current maximum connected component in the network contains 4 nodes, the current total number of nodes in the network is 6, the ratio If you delete v5, v7, v4, v 10 v9, v8, the current network has a maximum connected component containing 3 nodes, and the current network has a total of 6 nodes. The ratio Comparison ratio Therefore, we can conclude that v5, v7, v2, and v are the preferred choices. 10 Delete v9 and v8.

[0062] 2. Second attack method: still using For example, the network structure is as follows Figure 2 As shown in (b).

[0063] (1) In the first attack, the degree values ​​of all nodes are calculated based on the current network structure, and the node degree values ​​are sorted in descending order: d(5)>d(7)=d(8)=d(9)=d(2)=d(4)=d(10)>d(3)=d(11)=d(6)>d(1)=d(12). Delete The current network is a connected graph, and the ratio is... ;

[0064] (2) The second attack, based on the first attack, calculates the node degree values ​​in descending order according to the current network structure: d(9)=d(10)>d(7)=d(8)=d(11)=d(2)=d(4)=d(3)>=d(6)=d(1)=d(12), and deletes them. The current network is a connected graph, and the ratio is... ;

[0065] (3) The third attack, based on the second attack, calculates the node degree values ​​in descending order according to the current network structure: d(8)=d(9)=d(2)=d(4)=d(3)>=d(6)=d(1)=d(7)=d(11)>d(12), and deletes them. The current network is a connected graph, and the ratio is... ;

[0066] (4) The fourth attack, based on the third attack, calculates the node degree values ​​in descending order according to the current network structure: d(8)=d(9)>d(4)=d(3)=d(6)=d(7)=d(11)>d(1)=d(12), and deletes them. The largest connected component in the current network contains 6 nodes, and the total number of nodes in the current network is 8. The ratio is... ;

[0067] (5) The fifth attack, based on the fourth attack, calculates the node degree values ​​in descending order according to the current network structure: d(8)=d(4)=d(3)=d(6)>d(7)=d(11)=d(1)=d(12), and deletes them. The current maximum connected component in the network contains 4 nodes, and the current total number of nodes in the network is 7. The ratio is... ;

[0068] (6) The sixth attack, based on the fifth attack, calculates the node degree values ​​in descending order according to the current network structure: d(4)=d(3)>d(6)=d(11)=d(1)=d(12)>d(7), and deletes them. The current maximum connected component in the network contains 2 nodes, and the current total number of nodes in the network is 6. The ratio is... ;

[0069] (7) The seventh attack, based on the sixth attack, calculates the node degree values ​​in descending order according to the current network structure: d(3)=d(11)=d(1)=d(12)>d(7)=d(6), and deletes them. The current maximum connected component in the network contains 2 nodes, and the current total number of nodes in the network is 5. The ratio is... ;

[0070] (8) The eighth attack, based on the seventh attack, calculates the node degree values ​​in descending order according to the current network structure: d(11)=d(12)>d(1)=d(6)=d(7), and deletes them. The largest connected component in the current network contains 1 node, and the total number of nodes in the current network is 4. The ratio is... ;

[0071] (9) The ninth attack, based on the eighth attack, leaves only isolated nodes in the current network, all with the same degree value. These nodes are then deleted. The largest connected component in the current network contains 1 node, and the total number of nodes in the current network is 3. The ratio is... ;

[0072] (10) Tenth attack, delete The largest connected component in the current network contains 1 node, and the total number of nodes in the current network is 2. The ratio is... ;

[0073] (11) Eleventh attack, deletion The current network contains only one isolated node, and the ratio ;

[0074] (12) Twelfth attack, deletion Currently, there are no nodes in the network, and the ratio is... ;

[0075] Step 3: Calculate the network structure robustness index after a node is attacked;

[0076] The robustness index of the network structure is defined as follows:

[0077] (2);

[0078] Where Q(i) represents the ratio of the number of nodes in the largest connected component of the current network after the i-th attack to the total number of nodes in the current network. This indicates the robustness of the network structure; the larger the structural robustness index value, the better the robustness of the network system.

[0079] Based on the three simple complex networks described in step 1, and combined with formula (2), through theoretical derivation, the theoretical analytical expressions for the structural robustness of the three simple complex networks under the first attack method can be obtained respectively:

[0080] (1) When When k=2 in the network, ;

[0081] (2) When When k=2 and the initial total number of network nodes N0 is odd, , ,symbol Indicates rounding down to the nearest integer; when When k=2 and the initial total number of network nodes N0 is even, , ;

[0082] (3) When When k=2 in the network, ;

[0083] In summary, when k=2, with the same initial total number of nodes and attack methods, the structural robustness relationship of the three simple complex networks is as follows: ,in, The number of 2D simplex, i.e., third-order modalities in the network, is greater than The Internet, and The number of 2D simplex, i.e., third-order modalities in the network, is greater than This indicates that the 2D simplex, or third-order motif, in high-order interactive networks plays an important role in structural robustness.

[0084] In this embodiment, For example (e.g.) Figure 2 The first attack method shown in (b) ultimately yields... .Depend on Figures 3 to 9 It can be seen Figure 3 In the implementation of this invention The Internet When using the first attack method, the ratio With the initial number of network nodes The change graph; Figure 4 for The Internet When using the first attack method, the ratio With the initial number of network nodes The change graph; Figure 5 for When the network has k=2, using the first attack method, the ratio With the initial number of network nodes The change graph; by Figures 3 to 5 It can be seen that the theoretical and simulation results of the structural robustness indices of the three simple complex networks considered in this invention are consistent. Figure 6 For all three simple complex networks, the first attack method is used, and the structural robustness index is... With the initial number of network nodes The change graph; Figure 6 The structural robustness relationship of the three simple complex networks was verified as follows: . Figure 7 for The Internet A comparison of the structural robustness index as a function of the initial number of nodes under two different structures, both using the first attack method; Figure 8 for When the network has k=2, two different attack methods are used, and the trend of the structural robustness index changes with the initial number of nodes is compared in the image. Among them, attack1 is the first attack method, which is based on the initial network node degree value sorted in descending order, and the i-th attack deletes i nodes; attack2 is the second attack method, which is based on the current network node degree value sorted in descending order, and deletes one node in each attack. Figure 9 for The Internet A comparison of the structural robustness index as a function of the initial number of nodes under two different structures, both using the first attack method; Figures 7 to 9 This demonstrates that attack methods and network topology significantly impact structural robustness, and that 2D simplexes (i.e., third-order modalities) in high-order interactive networks are beneficial for improving structural robustness. This indicates that the structural robustness analysis method proposed in this invention is reasonable and effective, providing a theoretical basis and technical guidance for the rational formulation of attack and defense response strategies in practical applications.

[0085] The above description is merely a preferred embodiment of this disclosure and an explanation of the technical principles employed. Those skilled in the art should understand that the scope of the invention involved in the embodiments of this disclosure is not limited to technical solutions formed by specific combinations of the above-described technical features, but should also cover other technical solutions formed by arbitrary combinations of the above-described technical features or their equivalents without departing from the above-described inventive concept. For example, technical solutions formed by substituting the above-described features with (but not limited to) technical features with similar functions disclosed in the embodiments of this disclosure.

Claims

1. A robustness analysis method for simple complex family network structures, characterized in that, Includes the following steps: Step 1: Construct three simple complex networks , , ; Construct three simple complex networks respectively , , Where k represents the dimension of the simplex, This indicates that the center of the network is a k-1 dimensional simplex, and that it is a common face of the l surrounding k-dimensional simplexes; This indicates that there are l k-dimensional simplexes in the network, and adjacent simplexes are connected by a lower adjacency connection. This indicates that the center of the network is a k-dimensional simplex, there are x1 k-dimensional simplexes that are under-adjacent to one face of the central simplex, x2 k-dimensional simplexes that are under-adjacent to the other face of the central simplex, and finally x... k+1 The k-dimensional simplexes are under-adjacent to the last remaining face of the central simplex; x k+1 This represents the number of k-dimensional simplexes; Step 2: Based on the simplex complex network generated in Step 1, calculate the ratio Q(i) of the number of nodes in the largest connected component in the current network after the i-th network attack to the total number of nodes in the current network. Step 3: Calculate the network structure robustness index after a node is attacked; The robustness index of the network structure is defined as follows: （2）； Where Q(i) represents the ratio of the number of nodes in the largest connected component of the current network after the i-th attack to the total number of nodes in the current network. This indicates the robustness of the network structure, where N0 represents the initial total number of network nodes; Based on the three simple complex networks described in step 1, and combined with formula (2), the theoretical analytical expressions for the structural robustness of the three simple complex networks under the first attack method are obtained respectively: (1) When When k=2 in the network, ; (2) When When k=2 and the initial total number of network nodes N0 is odd, , ,symbol Indicates rounding down to the nearest integer; when When k=2 and the initial total number of network nodes N0 is even, , ; (3) When When k=2 in the network, .

2. The robustness analysis method for simple complex family network structures according to claim 1, characterized in that, The structure diagrams of the three simple complex networks described in step 1 are all undirected connected graphs. ,in, Let N0 be the set of N0 nodes. Let N be the N0th node, and E represent the set of edges.

3. The robustness analysis method for simple complex family network structures according to claim 1, characterized in that, Step 2 specifically involves: calculating the degree values ​​of all nodes in the simplex complex network generated in Step 1 and sorting them in descending order; a node being attacked is modeled as being deleted, i.e., the node is removed from the network. Nodes are selected for attack sequentially based on their degree value sorting. When selecting nodes, if nodes have the same degree value, their ratios Q(i) after deletion are compared. Nodes with larger Q(i) ratios are deleted first, where Q(i) is defined as the ratio of the number of nodes in the largest connected component in the current network after the i-th network attack to the total number of nodes in the current network, expressed as follows: （1）； in, N0 represents the initial total number of network nodes, and N represents the current total number of network nodes. This represents the number of nodes contained in the largest connected component in the current network. Among them, there are three special cases: (1) After the i-th attack, the current network is a connected graph, then the corresponding ratio ;(2) No. After this attack, the network now has only one isolated node, corresponding to the ratio. ;(3) No. After the attack, there are no nodes left in the current network, and the ratio equals 0. .

4. The robustness analysis method for simple complex family network structures according to claim 3, characterized in that, The network attack takes two different approaches: (1) The first approach is to sort the nodes in descending order based on the initial network node degree values ​​and delete i nodes in the i-th attack; (2) The second approach is to sort the nodes in descending order according to the current network structure node degree values ​​before each network attack and delete the node with the largest degree value in the current network in each attack. The number of attacks in both approaches is equal to the total number of nodes in the initial network from start to finish.

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