Simple complex family network structure robustness analysis method

Abstract

The invention provides a simple complex family network structure robustness analysis method, and relates to the technical field of control and information. Different initial networks are constructed according to three simple complex network generation algorithms, structural robustness analytical expressions of the three simple complex networks are theoretically derived based on a structural robustness index calculation formula, and an influence mechanism of a network topology structure and an attack mode on the structural robustness of the system is disclosed. The simple complex family network is used for describing the structure of an actual complex system, the high-order interaction relation of an actual network can be effectively reflected, and the method is more suitable for modeling of a real network system; a node degree-based attack mode is adopted to simulate an actual deliberate attack suffered by a network system, and an influence mechanism of high-order interaction and the attack mode on structural robustness is disclosed, so that scientific theoretical support and technical implementation guidance can be provided for reasonably formulating a network attack defense strategy in actual application.

Description

Technical Field

[0001] The invention relates to the field of control and information technology, and in particular to a simple complex family network structure robustness analysis method. Background Art

[0002] The robustness of a network system quantifies the degree to which a node or edge in the system maintains structural integrity or functional feasibility after being attacked or failing. Complex network structural robustness technology is closely related to various fields of society and has an important impact on multiple fields such as transportation, energy, and social interaction. For example, in a transportation network, by studying structural robustness, key nodes and sections can be identified, and response strategies can be formulated in advance, such as adding backup lines or strengthening maintenance in sections prone to congestion or accidents, improving the network's ability to respond to emergencies, reducing the risk of traffic paralysis, and ensuring the smooth flow of personnel and materials. In the energy Internet, robustness research helps optimize the structure of energy networks, such as rationally arranging substations and transmission lines, enhancing the network's ability to resist interference such as natural disasters and man-made sabotage, ensuring the stability and reliability of energy supply, and reducing the probability of accidents such as large-scale power outages. Social networks can identify key users and information dissemination paths. When dealing with problems such as the spread of false information, they can prevent the large-scale spread of false information by controlling the information dissemination of key nodes, and maintain the normal order of social networks and the authenticity of information.

[0003] Existing research on the structural robustness of complex networks mainly focuses on low-order networks with binary interactions, and there is still little research on the structural robustness of high-order interaction networks. Most of the interaction units in real networks are often composed of three or more nodes. The network structure can no longer be described by simple binary node interaction relationships, but needs to be characterized by "high-order networks" with high-order interactions. Compared with low-order networks, high-order networks are more complex in structure and involve multi- and multi-layer relationships. Such network structures are closer to the interactions between real nodes and edges, and have a wider range of applications and value. As a typical representation of high-order networks, simplicial complexes are a natural and powerful tool for simulating group interactions in the real world. For example, when simulating cooperative networks, cooperation may involve three or more people, not just two people. The use of simplicial complex graphs can go beyond binary interaction patterns and capture group (multi-) interaction patterns. The mathematical complexity of simplicial complexes provides opportunities and challenges for studying robustness on simplicial complex graphs.

[0004] In addition, in terms of simulating attack methods, the existing technologies usually adopt random attacks or deliberate attacks, and the comparative study of network structure robustness under various attack methods is not comprehensive and in-depth. The present invention conducts in-depth comparisons from different angles for deliberate attacks, reveals the influence of network topology and attack methods on structural robustness, and provides scientific basis and technical guidance for the design of actual network systems and the improvement of anti-interference capabilities. Summary of the invention

[0005] In view of the deficiencies of the prior art, the present invention provides a method for analyzing the robustness of a simple complex family network structure;

[0006] A method for analyzing the robustness of a simple complex family network structure includes the following steps:

[0007] Step 1: Construct three simplicial complex networks t k (x1,x2,…,x k+1 );

[0008] Construct three simple complex networks respectively t k (x1,x2,…,x k+1 ), where k represents the dimension of the simplex, Indicates that the network center is a k-1 dimensional simplex, and it is a common surface of the surrounding l k-dimensional simplexes; It means that there are l k-dimensional simplexes in the network, and adjacent simplexes are connected to each other in the lower adjacent way; t k (x1,x2,…,x k+1 ) means that the network center is a k-dimensional simplex, there are x1 k-dimensional simplices adjacent to one face of the center simplex, there are x2 k-dimensional simplices adjacent to another face of the center simplex, and finally there are x k+1 The k-dimensional simplex is adjacent to the last remaining face of the central simplex; k+1 represents the number of k-dimensional simplexes;

[0009] The structural graphs of the three simple complex networks are all undirected connected graphs G = (V, E), where represents a set of N0 nodes, is the N0th node, and E represents the set of edges;

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

[0011] According to the simple complex network generated in step 1, calculate the degree values ​​of all nodes in the network and sort them in descending order; the nodes in the network that are attacked by the network are modeled as being deleted, that is, the node is removed from the network, and the nodes are selected for attack in turn according to the node degree sorting results. When selecting nodes, if the node degree values ​​are the same, compare them according to the ratio Q(i) calculated after deleting the node. The nodes with a larger Q(i) ratio are deleted first, where Q(i) is defined as the ratio of the maximum number of connected branch nodes in the current network after the i-th network attack to the total number of nodes in the current network, expressed as follows:

[0012]

[0013] Where i = 1, 2, ..., N0, N0 represents the total number of initial network nodes, N represents the total number of current network nodes, N i represents the number of nodes contained in the largest connected branch in the current network, Q(i)∈[0,1], where there are three special cases: (1) after the i-th attack, the current network is a connected graph, and the corresponding ratio is Q(i)=1; (2) after the N0-1-th attack, the current network has only one isolated node, and the corresponding ratio is Q(N0-1)=1; (3) after the N0-th attack, there is no node in the current network, and the ratio is equal to 0, that is, Q(N0)=0;

[0014] The network attack adopts the following two different methods: (1) The first method is to sort the nodes in the initial network in descending order based on their degree values, and delete i nodes in the i-th attack; (2) The second method is to sort the nodes in the current network structure in descending order based on their degree values ​​before each network attack, and select the node with the largest degree value in the current network for deletion in each attack; the number of attacks from the beginning to the end of these two attack methods is equal to the total number of nodes in the initial network;

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

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

[0017]

[0018] Among them, Q(i) represents the ratio of the number of nodes contained in the largest connected branch in 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] According to the three simplex complex networks described in step 1, combined with formula (2), the theoretical analytical expressions of the structural robustness of the three simplex complex networks under the first attack mode are obtained respectively:

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

[0021] (2) When When the network's k=2 and the total number of initial network nodes N0 is an odd number, N0≥11, symbol Indicates rounding down to an integer; when When the network's k=2 and the total number of initial network nodes N0 is an even number, N0≥12;

[0022] (3) When t k (x1,x2,…,x k+1 ) When k=2,

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

[0024] The present invention provides a simple complex family network structural robustness analysis method, which can reveal the positive influence mechanism of the high-order interaction relationship of the network on the system structural robustness, and has stronger feasibility in application. Compared with the prior art, the specific beneficial technical effects of the present invention are: (1) using a simple complex network to characterize the high-order interaction relationship of the actual network system can achieve better modeling of the real system and has higher application value; (2) by analyzing the influence mechanism of network topology and attack mode on structural robustness, it is found that the third-order model structure in the high-order interaction is beneficial to improving the structural robustness of the system, which provides a theoretical basis and technical guidance for the reasonable formulation of attack and defense response strategies in practical applications.

[0025] The present 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 in the event of failure or attack, the network topology can be optimized to improve its anti-destruction ability and ensure the reliability and efficiency of data transmission. In power networks, robustness analysis can be used to assess the vulnerability of key nodes or connections and design a more robust power grid structure, thereby reducing the risk of power outages and responding to natural disasters or man-made damage. In transportation networks, robustness analysis can be used to optimize traffic flow distribution and improve emergency response capabilities to accidents or natural disasters, thereby ensuring the normal operation of the transportation system. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 It is a flow chart of the implementation of a simple complex family network structure robustness analysis method of the present invention;

[0027] Figure 2 Schematic diagram of three simple complex network topology structures in the implementation of the present invention;

[0028] Among them, (a) The network structure of (b) The network structure of (c)-t 2 (2,3,4) network structure;

[0029] Figure 3 In the implementation of the present invention When the network is at k=2, the first attack method is used, and the ratio Q(i) changes with the number of initial network nodes N0;

[0030] Figure 4 In the implementation of the present invention k (x1,x2,…,x k+1 ) When the network is k=2, the first attack method is used, and the ratio Q(i) changes with the number of initial network nodes N0;

[0031] Figure 5 In the implementation of the present invention When the network is at k=2, the first attack method is used, and the ratio Q(i) changes with the number of initial network nodes N0;

[0032] Figure 6 In the implementation of the present invention, the three simple complex networks all adopt the first attack mode, and the structural robustness index is The graph changes with the number of initial network nodes N0;

[0033] Figure 7 In the implementation of the present invention The network adopts the first attack method under two different structures of k=1 and 2, and the comparison chart of the structural robustness index changes with the initial number of nodes;

[0034] Figure 8 In the implementation of the present invention When the network is k=2, two different attack methods are used, and the trend of the structural robustness index changing with the initial number of nodes is compared. Among them, attack1 is the first attack method, that is, based on the descending order of the initial network node degree value, the i-th attack deletes i nodes; attack2 is the second attack method, that is, based on the descending order of the current network node degree value, one node is deleted each time the attack is made.

[0035] Fig. 9 In the implementation of the present invention The network uses the first attack method under two different structures of k=1 and 2, and the comparison chart shows the change of structural robustness index with the initial number of nodes. DETAILED DESCRIPTION

[0036] The specific implementation of the present invention is further described in detail below in conjunction with the accompanying drawings and examples. The following examples are used to illustrate the present invention, but are not intended to limit the scope of the present invention.

[0037] A method for analyzing the robustness of simple complex family network structures, such as Figure 1 As shown, the following steps are included:

[0038] Step 1: Construct three simplicial complex networks t k (x1,x2,…,x k+1 );

[0039] Construct three simple complex networks respectively t k (x1,x2,…,x k+1 ), where k represents the dimension of the simplex, Indicates that the network center is a k-1 dimensional simplex, and it is a common surface of the surrounding l k-dimensional simplexes; It means that there are l k-dimensional simplexes in the network, and adjacent simplexes are connected to each other in the lower adjacent way; t k (x1,x2,…,x k+1 ) means that the network center is a k-dimensional simplex, there are x1 k-dimensional simplices adjacent to one face of the center simplex, there are x2 k-dimensional simplices adjacent to another face of the center simplex, and finally there are x k+1 The k-dimensional simplex is adjacent to the last remaining face of the central simplex; k+1 represents the number of k-dimensional simplexes;

[0040] The structural graphs of the three simple complex networks are all undirected connected graphs G = (V, E), where represents a set of N0 nodes, is the N0th node, and E represents the set of edges;

[0041] In this embodiment, three simple complex networks are constructed t 2 (2,3,4), the structures are as follows Figure 2 (a) Figure 2 (b) and Figure 2 (c) Figure 2 (a) indicates The network structure has a total number of nodes N0 = 10; Figure 2 (b) indicates The network structure has a total number of nodes N0 = 12; Figure 2 (c) indicates t 2(2,3,4) network structure, the total number of nodes is N0 = 12; k in the three constructed simplicial complex networks is 2, where k = 2 represents a 2-dimensional simplex, a 2-dimensional simplex is a triangle, used to represent the high-order interactions between the three nodes. These three simplicial complex networks are all undirected connected graphs, which can be used to model social networks, brain networks, intelligent transportation systems, etc. in the real world. For example, a 2-dimensional simplex in a social network can be used to represent three authors cooperating to publish a paper;

[0042] Step 2: Based on the simplex complex network generated in step 1, calculate the ratio Q(i) of the node after being attacked by the network;

[0043] According to the simple complex network generated in step 1, calculate the degree values ​​of all nodes in the network and sort them in descending order; the nodes in the network that are attacked by the network are modeled as being deleted, that is, the node is removed from the network, and the nodes are selected for attack in turn according to the node degree sorting results. When selecting nodes, if the node degree values ​​are the same, compare them according to the ratio Q(i) calculated after deleting the node. The nodes with a larger Q(i) ratio are deleted first, where Q(i) is defined as the ratio of the maximum number of connected branch nodes in the current network after the i-th network attack to the total number of nodes in the current network, expressed as follows:

[0044]

[0045] Where i = 1, 2, ..., N0, N0 represents the total number of initial network nodes, N represents the total number of current network nodes, N i represents the number of nodes contained in the largest connected branch in the current network, Q(i)∈[0,1], where there are three special cases: (1) after the i-th attack, the current network is a connected graph, and the corresponding ratio is Q(i)=1; (2) after the N0-1-th attack, the current network has only one isolated node, and the corresponding ratio is Q(N0-1)=1; (3) after the N0-th attack, there is no node in the current network, and the ratio is equal to 0, that is, Q(N0)=0;

[0046] The network attack adopts the following two different methods: (1) The first method is to sort the initial network node in descending order based on the degree value, 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 to delete, and calculates Q(1); the second attack selects the node with the second largest degree value from the initial network node sorting result to delete, and calculates Q(2); .... (2) The second method is to sort the nodes in the current network structure in descending order before each network attack, and select the node with the largest degree value in the current network to delete each time, that is, the first attack selects the node with the largest degree value from the current network node sorting result to delete, and calculates Q(1); the second attack, sorts the nodes according to the current network structure, selects the node with the largest degree value to delete, and calculates Q(2); .... The number of attacks from the beginning to the end of these two attack methods is equal to the total number of initial network nodes;

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

[0048] 1. The first attack method: For example, the network structure is as follows Figure 2 (b) As shown. According to the initial network structure, all node degrees are calculated and the node degrees are sorted in descending order 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 node v i The degree value.

[0049] (1) In the first attack, v5 is deleted. The current network is a connected graph, and the ratio Q(1) = 1;

[0050] (2) The second attack, deleting v5 and v7, the current network is a connected graph, and the ratio Q(2) = 1;

[0051] (3) The third attack, delete v5, v7, and v2. The current network is a connected graph, and the ratio Q(3) = 1;

[0052] (4) The fourth attack, delete v5, v7, v2, v 10 , the current network is a connected graph, the ratio Q(4) = 1;

[0053] (5) The fifth attack, delete v5, v7, v9, v 10 , v8, the maximum number of nodes in the current network is 5, the total number of nodes in the current network is 7, the ratio

[0054] (6) The sixth attack, delete v5, v7, v2, v 10, v9, v8, the maximum number of nodes in the current network is 4, the total number of remaining network nodes is 6, the ratio

[0055] (7) The seventh attack, delete v5, v7, v2, v4, v 10 , v9, v8, the maximum number of nodes in the current network is 2, the total number of nodes in the current network is 5, the ratio

[0056] (8) The eighth attack, delete v5, v7, v2, v4, v 10 , v9, v8, v3, the maximum number of nodes in the current network is 2, the total number of nodes in the current network is 4, the ratio

[0057] (9) The ninth attack, delete v5, v7, v2, v4, v 10 , v9, v8, v3, v6, the maximum number of nodes in the current network is 2, the total number of nodes in the current network is 3, the ratio

[0058] (10) The tenth attack, delete v5, v7, v2, v4, v 10 , v9, v8, v3, v6, v 11 , the maximum number of nodes in the current network connected branch is 1, the total number of nodes in the current network is 2, the ratio

[0059] (11) The eleventh attack, delete v5, v7, v2, v4, v 10 , v9, v8, v3, v6, v 11 , v1, the current network contains only one isolated node, the ratio Q(11) = 1;

[0060] (12) The twelfth attack, delete all nodes in the network. The current network does not have a node, and the ratio Q(12) = 0.

[0061] It should be noted that in the sixth attack, according to the initial network structure node degree value sorting result, d(2) = d(4). If v5, v7, v2, v 10 , v9, v8, the maximum number of nodes in the current network is 4, the total number of nodes in the current network is 6, the ratio If you delete v5, v7, v4, v 10 , v9, v8, the maximum number of nodes in the current network is 3, the total number of nodes in the current network is 6, the ratio Comparison ratio It can be concluded that v5, v7, v2, and v10 , v9, and v8 to delete.

[0062] 2. The second attack method: still with For example, the network structure is as follows Figure 2 (b) as shown.

[0063] (1) In the first attack, all node degrees are calculated according to the current network structure and the node degrees 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). v5 is deleted. The current network is a connected graph and the ratio Q(1)=1.

[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), delete v 10 , the current network is a connected graph, the ratio Q(2) = 1;

[0065] (3) The third attack, based on the second attack, calculate the descending order of node degree values ​​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), delete v2, the current network is a connected graph, and the ratio Q(3)=1;

[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), delete v9, the number of nodes in the largest connected branch in the current network is 6, the total number of nodes in the current network is 8, and 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), delete v8, the maximum number of nodes in the current network is 4, the total number of nodes in the current network is 7, and the ratio is

[0068] (6) The sixth attack, based on the fifth attack, is to calculate the descending order of node degree values ​​according to the current network structure: d(4) = d(3) > d(6) = d(11) = d(1) = d(12) > d(7). Delete v4. The maximum number of nodes in the current network is 2. The total number of nodes in the current network is 6. The ratio

[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), delete v3, the maximum number of nodes in the current network is 2, the total number of nodes in the current network is 5, and 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), delete v 11 , the maximum number of nodes in the current network connected branch is 1, the total number of nodes in the current network is 4, the ratio

[0071] (9) The ninth attack: Based on the eighth attack, all the remaining nodes in the current network are isolated nodes with the same node degree. Delete v1. The maximum number of nodes in the current network is 1. The total number of nodes in the current network is 3. The ratio

[0072] (10) The tenth attack, deleting v6, the number of nodes in the largest connected branch in the current network is 1, the total number of nodes in the current network is 2, and the ratio is

[0073] (11) The eleventh attack, deleting v7, the current network contains only one isolated node, the ratio Q(11) = 1;

[0074] (12) Twelfth attack, delete v 12 , the current network does not have a node, the ratio Q(12) = 0;

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

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

[0077]

[0078] Among them, Q(i) represents the ratio of the number of nodes contained in the largest connected branch in 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; the larger the value of the structural robustness index, the better the robustness of the network system;

[0079] According to the three simplex complex networks described in step 1, combined with formula (2), through theoretical derivation and proof, the theoretical analytical expressions of the structural robustness of the three simplex complex networks under the first attack mode can be obtained respectively:

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

[0081] (2) When When the network's k=2 and the total number of initial network nodes N0 is an odd number, N0≥11, symbol Indicates rounding down to an integer; when When the network's k=2 and the total number of initial network nodes N0 is an even number, N0≥12;

[0082] (3) When t k (x1,x2,…,x k+1 ) When k=2 of the network,

[0083] In summary, when k=2, under the same total number of initial nodes and attack methods, the structural robustness relationship of the three simple complex networks is: in, The number of 2-dimensional simplex, i.e., third-order motifs of the network is greater than t k (x1,x2,…,x k+1 ) network, and t k (x1,x2,…,x k+1 ) The number of 2-dimensional simplex, i.e., third-order motifs of the network is greater than This indicates that the 2-dimensional simplex, i.e., the third-order motif, in high-order interaction networks plays an important role in structural robustness.

[0084] In this embodiment, For example (such as Figure 2 The first attack method shown in (b) can finally obtain Depend on Figures 3 to 9 It can be seen Figure 3 In the implementation of the present invention When the network is at k=2, the first attack method is used, and the ratio Q(i) changes with the number of initial network nodes N0; Figure 4 t k (x1,x2,…,x k+1) When the network is k=2, the first attack method is used, and the ratio Q(i) changes with the number of initial network nodes N0; Figure 5 for When the network is at k=2, the first attack method is used, and the ratio Q(i) changes with the number of initial network nodes N0; Figures 3 to 5 It can be seen that the theoretical results and simulation results of the structural robustness indicators of the three simple complex networks considered in the present invention are consistent. Figure 6 The first attack method is used for all three simple complex networks, and the structural robustness index The graph changes with the number of initial network nodes N0; Figure 6 It is verified that the structural robustness relationship of the three simple complex networks is Figure 7 for The network adopts the first attack method under two different structures of k=1 and 2, and the comparison chart of the structural robustness index changes with the initial number of nodes; Figure 8 for When the network is k=2, two different attack methods are used, and the trend of the structural robustness index changing with the initial number of nodes is compared. Among them, attack1 is the first attack method, that is, based on the descending order of the initial network node degree value, the i-th attack deletes i nodes; attack2 is the second attack method, that is, based on the descending order of the current network node degree value, one node is deleted each time the attack is made. Fig. 9 for The network adopts the first attack method under two different structures of k=1 and 2, and the comparison chart of the structural robustness index changes with the initial number of nodes; Figures 7 to 9 It shows that the attack mode and network topology have a significant impact on structural robustness, and the 2-dimensional simplex, i.e., the third-order motif, in the high-order interactive network is conducive to improving structural robustness. This shows that the structural robustness analysis method proposed in the present invention is reasonable and effective, and provides a theoretical basis and technical guidance for the rational formulation of attack and defense response strategies in practical applications.

[0085] The above description is only a preferred embodiment of the present disclosure and an explanation of the technical principles used. Those skilled in the art should understand that the scope of the invention involved in the embodiments of the present disclosure is not limited to the technical solutions formed by a specific combination of the above-mentioned technical features, but should also cover other technical solutions formed by any combination of the above-mentioned technical features or their equivalent features without departing from the above-mentioned inventive concept. For example, the above-mentioned features are replaced with the technical features with similar functions disclosed in the embodiments of the present disclosure (but not limited to) to form a technical solution.

Claims

1. A method for analyzing the robustness of simple complex family network structures, characterized in that: The following steps are involved: Step 1: Construct three simplicial complex networks t k (x1,x2,…,x k+1 ); Construct three simple complex networks respectively t k (x1,x2,…,x k+1 ), where k represents the dimension of the simplex, Indicates that the network center is a k-1 dimensional simplex, and it is a common surface of the surrounding l k-dimensional simplexes; It means that there are l k-dimensional simplexes in the network, and adjacent simplexes are connected to each other in the lower adjacent way; t k (x1,x2,…,x k+1 ) means that the network center is a k-dimensional simplex, there are x1 k-dimensional simplices adjacent to one face of the center simplex, there are x2 k-dimensional simplices adjacent to another face of the center simplex, and finally there are x k+1 The k-dimensional simplex is adjacent to the last remaining face of the central simplex; k+1 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 node after being attacked by the network; Step 3: Calculate the network structure robustness index after the node is attacked.

2. The method for analyzing the robustness of a simple complex family network structure according to claim 1, characterized in that: The structural graphs of the three simple complex networks described in step 1 are all undirected connected graphs G = (V, E), where represents a set of N0 nodes, is the N0th node, and E represents the set of edges.

3. The method for analyzing the robustness of a simple complex family network structure according to claim 1, characterized in that: The step 2 is specifically as follows: according to the simple complex network generated in step 1, all node degrees in the network are calculated and sorted in descending order; nodes in the network that are attacked by the network are modeled as being deleted, that is, the node is removed from the network, and nodes are selected for attack in sequence according to the node degree sorting results. When selecting nodes, if the node degrees are the same, the ratio Q(i) calculated after deleting the node is compared, and the nodes with a larger Q(i) ratio are deleted first, where Q(i) is defined as the ratio of the maximum number of connected branch nodes in the current network after the i-th network attack to the total number of nodes in the current network, expressed as follows: Where i = 1, 2, ..., N0, N0 represents the total number of initial network nodes, N represents the total number of current network nodes, N i represents the number of nodes contained in the largest connected branch in the current network, Q(i)∈[0,1], among which there are three special cases: (1) after the i-th attack, the current network is a connected graph, and the corresponding ratio is Q(i)=1; (2) after the N0-1-th attack, the current network has only one isolated node, and the corresponding ratio is Q(N0-1)=1; (3) after the N0-th attack, there is no node in the current network, and the ratio is equal to 0, that is, Q(N0)=0.

4. The method for analyzing the robustness of a simple complex family network structure according to claim 3, characterized in that: The network attack adopts the following two different methods: (1) The first method 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 method is to sort the nodes in the current network structure in descending order before each network attack, and select the node with the largest degree value in the current network for deletion in each attack; the number of attacks from the beginning to the end of these two attack methods is equal to the total number of initial network nodes.

5. The method for analyzing the robustness of a simple complex family network structure according to claim 1, characterized in that: The network structure robustness index described in step 3 is defined as follows: Among them, Q(i) represents the ratio of the number of nodes contained in the largest connected branch in 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; According to the three simplex complex networks described in step 1, combined with formula (2), the theoretical analytical expressions of the structural robustness of the three simplex complex networks under the first attack mode are obtained respectively: (1) When When the network k = 2, (2) When When the network's k=2 and the total number of initial network nodes N0 is an odd number, N0≥11, symbol Indicates rounding down to an integer; when When the network's k=2 and the total number of initial network nodes N0 is an even number, N0≥12; (3) When t k (x1,x2,…,x k+1 ) When k=2,

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