High-order network energy control robustness analysis method based on pure complex

Abstract

The invention provides a high-order network energy control robustness analysis method based on pure complex, and relates to the technical field of control and information. On the basis of three constructed simple complex network systems, an overall energy control robustness index after network nodes are attacked is obtained through theoretical derivation, and overall energy control robustness characteristics of the three simple complex network systems under random attacks and deliberate attacks are analyzed through simulation. And an influence mechanism of a network high-order structure and an attack mode on the overall energy control robustness is disclosed. And an energy control robustness joint evaluation mechanism is introduced, so that the response capability and stability of the network under external interference or structural disturbance can be comprehensively evaluated, and theoretical support is provided for control strategy design and fault-tolerant mechanism construction of an actual system. The provided analysis framework can adapt to simple complex structures of different dimensions and different scales, and can be widely applied to modeling and evaluation of various complex systems such as social networks, biological information networks, traffic systems, energy systems and the like.

Description

Technical Field

[0001] The present invention relates to the field of control and information technology, and in particular to a high-order network controllability robustness analysis method based on simplicial complexes. Background Art

[0002] Complex network models are widely used to simulate and analyze the operational mechanisms of real-world systems in fields such as information, energy, biology, and social systems. Controllability and robustness are key indicators of system performance, reflecting the system's responsiveness to external control signals and its stability in the face of structural disturbances or node failures, respectively.

[0003] Traditional complex network models are often constructed using graphs, where nodes represent the basic units of the system and edges represent the relationships between these units. However, graphs can only describe binary relationships and are unable to effectively depict higher-order interactions between multiple nodes. When faced with real-world systems with higher dimensions and complex coupling characteristics, the expressive and analytical capabilities of graph models are significantly limited.

[0004] In recent years, simplicial complexes (Simplicial Complexes) have gained increasing attention as a tool for modeling higher-order networks. Compared to traditional graph models, Simplicial Complexes (Simplicial Complexes) can represent the interactions between multiple nodes, providing richer structural information. They are suitable for describing high-order interaction structures such as group behavior in social networks, multiprotein complexes in biological networks, and multi-way intersections in transportation systems.

[0005] Although research has begun to focus on simplicial complex-based network modeling, the analysis of the controllability robustness of such networks is still in its infancy. Existing methods, in particular, struggle to effectively evaluate and analyze the overall structure of simplicial complex networks. Therefore, a systematic analysis method for simplicial complex networks is urgently needed. This method can incorporate high-order network structural characteristics to quantify the controllability robustness of the network and provide theoretical support for the optimization design, control strategy formulation, and stability assessment of practical systems. Summary of the Invention

[0006] In view of the shortcomings of the existing technology, the present invention provides a high-order network controllability robustness analysis method based on simplicial complex;

[0007] A method for analyzing the controllability robustness of high-order networks based on simplicial complexes includes the following steps:

[0008] Step 1: Construct three simplicial complex networks Determine the differential equations for the dynamics of simplicial and complex network systems;

[0009] Specifically: construct three simple complex networks respectively in, Indicates that the network center is a (k-1)-simplex, and it is a common face of the surrounding l k-simplices; Indicates that there are l k-simplices in the network, and adjacent simplices 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-simplex, there are x1 k-simplices adjacent to one face of the center simplex, there are x2 k-simplices adjacent to another face of the center simplex, and finally there are x k+1 k-simplexes are adjacent to the last remaining face of the central simplex, where x k+1 represents the number of k-simplexes;

[0010] The structural graphs of the three simple complex networks are all undirected connected graphs G = (V, E), where V = {v1, v2, ..., v N} represents a set of N nodes, v N is the Nth node, E represents the set of edges;

[0011] Adjacency tensor of q-simplicial complex represents N×N×…×N| q+1 -dimensional tensor, is the adjacency tensor elements, when (i1,i2,…,i q+1 )∈S (q) , otherwise Among them, S (q) represents a q-simplicial complex, i1,i2,…,i q+1 Indicates the node number, (i1,i2,…,i q+1 ) represents a q-simplex; the adjacency tensor Compressed into an N×N matrix A (q) , matrix A (q) Elements in The definition is as follows:

[0012]

[0013] in, is the adjacency tensor The elements have values of 0 or 1; the sum is traversed over all possible q-1 nodes i3,i4,…,i q+1 ; represents the total number of q-simplices in which nodes i1 and i2 participate together;

[0014] A Q-dimensional simplicial complex network system with N nodes is represented by differential equations, where the dynamic differential equation of the i-th node is expressed as follows:

[0015]

[0016] in, represents the state of node i, Represents the mth input. When the i-th node receives input u m When the effect is im =1, otherwise b im =0; is an internal coupling function that satisfies the condition q=1,2,…,Q is the coupling strength, represents the set of real numbers, represents the set of positive real numbers, represents a set of (q+1)-dimensional real vectors; is the adjacency tensor Elements of j q Is the node label, indicating node j q ;

[0017] Let x=[x1,x2,…,x N ] T Represents the state vector of the node, u=[u1,u2,…,u M ] T represents the input vector, and the superscript T represents the transpose, then formula (2) can be written as

[0018]

[0019] in, is the input matrix, represents the set of N×M dimensional real matrices; σ q is the coupling strength; A (q) Represents the q-order adjacency tensor The compressed N×N matrix, α q is the matrix A (q) The correlation coefficient of the order, α q =q;

[0020] Step 2: Based on the simplicial complex network constructed in step 1, two different attack strategies are adopted on the network nodes based on the node degree ranking results. The network controllability robustness index values are calculated respectively, and then the controllability robustness of the network after the nodes are attacked is analyzed;

[0021] The attack strategy specifically includes a random attack strategy and a deliberate attack strategy;

[0022] The random attack strategy is specifically as follows: randomly selecting a node in the network to attack each time;

[0023] The deliberate attack strategy is specifically as follows: according to the simplicial complex network system constructed in step 1, the 1st degree, 2nd degree, ..., qth degree of all nodes in the network are calculated and sorted in descending order; when a node in the network is attacked, it is modeled as the node being deleted, that is, the node is removed from the network, and nodes with larger degrees are selected for attack according to the node q-order sorting results; when selecting nodes, if multiple nodes have the same q-order, the (q-1) degree of the nodes is compared, and the node with larger (q-1) degree is preferentially selected for deletion, and so on; when deleting node i, the edges associated with node i and the q-simplex in which node i participates are also deleted, wherein the q-order of node i is defined as follows:

[0024]

[0025] Where i = 1, 2, ..., N, is the adjacency tensor The elements of , the value is 0 or 1;

[0026] The controllable robustness index of node i in the network after being attacked is defined as follows:

[0027]

[0028] Among them, N represents the total number of initial nodes in the network, N D (i) represents the number of driver nodes required to ensure the network is controllable after node i is attacked; Ni is the number of remaining nodes in the network after node i is attacked;

[0029] The calculation formula for the number of driving nodes in the controllable robustness index is as follows:

[0030]

[0031] Where j = 1, 2, ..., N i , N i =Ni; is λ j The geometric multiplicity, λ j express The j-th eigenvalue of is the coupling strength, Represents the adjacency tensor Compressed N i ×N i matrix, is a matrix The correlation coefficient of the order, that is represents the unit matrix, the symbol rank(·) represents the rank of the matrix, and the symbol max represents the maximum value;

[0032] Step 3: Calculate the overall controllability robustness index of the network after the node is attacked

[0033] The overall controllability robustness of the network is defined as follows:

[0034]

[0035] in, is the controllable robustness index after the i-th node in the network is attacked, It is the overall controllability robustness index of the network; the smaller the overall controllability robustness index value is, the better the controllability robustness of the network system is.

[0036] The beneficial effects of adopting the above technical solution are:

[0037] The present invention provides a high-order network controllability robustness analysis method based on simplicial complexes, which can reveal the influence mechanism of the high-order interaction relationship of the network on the controllability robustness of the system and has stronger feasibility in application. Compared with the existing technology, the specific beneficial technical effects of the present invention are: (1) breaking through the limitations of traditional graph models: based on the simplicial complex modeling method, the present invention can express the high-order association relationship between multiple nodes, breaking through the expression bottleneck of the traditional binary edge structure in describing group interaction behavior, and is applicable to a wider range of practical application scenarios; (2) introducing a joint controllability robustness evaluation mechanism: combining controllability and robustness for quantitative analysis, it can comprehensively evaluate the response capability and stability of the network when subjected to external interference or structural disturbance, and provide theoretical support for the control strategy design and fault tolerance mechanism construction of the actual system; (3) having good scalability and versatility: the proposed analysis framework can adapt to simplicial complex structures of different dimensions and scales, has good algorithm scalability, and can be widely used in the modeling and evaluation of various complex systems such as social networks, bioinformatics networks, transportation systems, and energy systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 This is a flowchart of an implementation method for analyzing the controllability robustness of a high-order network based on a simplicial complex according to the present invention;

[0039] Figure 2 Schematic diagrams of three simple complex network topologies in the implementation of the present invention;

[0040] Among them, (a)- The network structure of (b)- The network structure of (c)-t 2 (2,3,4) network structure;

[0041] Figure 3 In the implementation of the present invention The network adopts random attack mode, the index value and Graph showing changes with the number of node deletions i;

[0042] Figure 4 In the implementation of the present invention The network uses deliberate attack methods, the index value and Graph showing changes with the number of node deletions i;

[0043] Figure 5 For the three simple complex networks in the present invention, when k=2, the random attack mode is used, and the index value is Comparison chart of changes with the number of node deletions i;

[0044] Figure 6 For the three simple complex networks in the present invention, when k=2, the random attack mode is used, and the index value is Comparison chart of changes with the number of node deletions i;

[0045] Figure 7 For the three simple complex networks in the present invention, when k=2, the index value is Comparison chart of changes with the number of node deletions i;

[0046] Figure 8 For the three simple complex networks in the present invention, when k=2, the index value is Comparison chart of changes with the number of node deletions i. DETAILED DESCRIPTION

[0047] The following embodiments of the present invention are described in further detail with reference to 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.

[0048] A method for analyzing the robustness of controllable high-order networks based on simplicial complexes, such as Figure 1 As shown, the following steps are included:

[0049] Step 1: Construct three simplicial complex networks Determine the differential equations for the dynamics of simplicial and complex network systems;

[0050] Specifically: construct three simple complex networks respectively in, Indicates that the network center is a (k-1)-simplex, and it is a common face of the surrounding l k-simplices; Indicates that there are l k-simplices in the network, and adjacent simplices 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-simplex, there are x1 k-simplices adjacent to one face of the center simplex, there are x2 k-simplices adjacent to another face of the center simplex, and finally there are x k+1 k-simplexes are adjacent to the last remaining face of the central simplex, where x k+1 represents the number of k-simplexes;

[0051] The structural graphs of the three simple complex networks are all undirected connected graphs G = (V, E), where V = {v1, v2, ..., v N} represents a set of N nodes, v N is the Nth node, E represents the set of edges;

[0052] Adjacency tensor of q-simplicial complex represents N×N×…×N| q+1 -dimensional tensor, is the adjacency tensor elements, when (i1,i2,…,i q+1 )∈S (q) , otherwise Among them, S (q) represents a q-simplicial complex, i1,i2,…,i q+1 Indicates the node number, (i1,i2,…,i q+1 ) represents a q-simplex; the adjacency tensor Compressed into an N×N matrix A (q) , matrix A (q) Elements in The definition is as follows:

[0053]

[0054] in, is the adjacency tensor The elements have values of 0 or 1; the sum is traversed over all possible q-1 nodes i3,i4,…,i q+1 ; represents the total number of q-simplices in which nodes i1 and i2 participate together;

[0055] A Q-dimensional simplicial complex network system with N nodes is represented by differential equations, where the dynamic differential equation of the i-th node is expressed as follows:

[0056]

[0057] in, represents the state of node i, Represents the mth input. When the i-th node receives input u m When the effect is im =1, otherwise b im =0; is an internal coupling function that satisfies the condition is the coupling strength, represents the set of real numbers, represents the set of positive real numbers, represents a set of (q+1)-dimensional real vectors; is the adjacency tensor Elements of j q Is the node label, indicating node j q ;

[0058] Let x=[x1,x2,…,x N ] T Represents the state vector of the node, u=[u1,u2,…,u M ] T represents the input vector, and the superscript T represents the transpose, then formula (2) can be written as

[0059]

[0060] in, is the input matrix, represents the set of N×M dimensional real matrices; σ q is the coupling strength; A (q) Represents the q-order adjacency tensor The compressed N×N matrix, α q is the matrix A (q) The correlation coefficient of the order, α q =q;

[0061] 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), where 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) represents t 2 The network structure of (2,3,4) has a total number of nodes N0 = 12; these three simple complex networks are all undirected connected graphs;

[0062] Step 2: Based on the simplicial complex network constructed in step 1, two different attack strategies are adopted on the network nodes based on the node degree ranking results. The network controllability robustness index values are calculated respectively, and then the controllability robustness of the network after the nodes are attacked is analyzed;

[0063] The attack strategy specifically includes a random attack strategy and a deliberate attack strategy;

[0064] The random attack strategy is specifically as follows: randomly selecting a node in the network to attack each time;

[0065] The deliberate attack strategy is specifically as follows: according to the simplicial complex network system constructed in step 1, the 1st degree, 2nd degree, ..., qth degree of all nodes in the network are calculated and sorted in descending order; when a node in the network is attacked, it is modeled as the node being deleted, that is, the node is removed from the network, and nodes with larger degrees are selected for attack according to the node q-order sorting results; when selecting nodes, if multiple nodes have the same q-order, the (q-1) degree of the nodes is compared, and the node with larger (q-1) degree is preferentially selected for deletion, and so on; when deleting node i, the edges associated with node i and the q-simplex in which node i participates are also deleted, wherein the q-order of node i is defined as follows:

[0066]

[0067] Where i = 1, 2, ..., N, is the adjacency tensor The elements of , the value is 0 or 1;

[0068] The controllable robustness index of node i in the network after being attacked is defined as follows:

[0069]

[0070] Among them, N represents the total number of initial nodes in the network, N D (i) represents the number of driver nodes required to ensure the network is controllable after node i is attacked; Ni is the number of remaining nodes in the network after node i is attacked;

[0071] The calculation formula for the number of driving nodes in the controllable robustness index is as follows:

[0072]

[0073] Where j = 1, 2, ..., N i , N i =Ni; is λ j The geometric multiplicity, λ j express The j-th eigenvalue of is the coupling strength, Represents the adjacency tensor Compressed N i ×N i matrix, is a matrix The correlation coefficient of the order, that is represents the unit matrix, the symbol rank(·) represents the rank of the matrix, and the symbol max represents the maximum value;

[0074] This example uses the following two attack methods:

[0075] 1. The first random attack method: For example, the network structure is as follows Figure 2 (a)

[0076] (1) First attack, delete v5, indicator value

[0077] (2) Second attack, delete v7, indicator value

[0078] (3) The third attack, deleting v8, indicator value

[0079] (4) The fourth attack, deleting v3, indicator value

[0080] (5) The fifth attack, deleting v4, indicator value

[0081] (6) The sixth attack, delete v 10 , indicator value

[0082] (7) The seventh attack, delete v1, indicator value

[0083] (8) The eighth attack, delete v9, indicator value

[0084] (9) The ninth attack, deleting v6, indicator value

[0085] 2. The second deliberate attack method: still with For example, the network structure is as follows Figure 2 As shown in (a); according to the current network structure, the 1st degree value and 2nd degree value of all nodes are calculated, and the node degree values are sorted in descending order as follows:

[0086]

[0087] in Represents node v i The second-order value of Represents node v i The first-order value of

[0088] (1) First attack, delete v2, indicator value

[0089] After the first attack, based on the current network structure, the node degree values are sorted in descending order as follows:

[0090]

[0091] (2) Second attack, delete v3, indicator value

[0092] After the second attack, based on the current network structure, the node degree values are sorted in descending order as follows:

[0093]

[0094] (3) The third attack, deleting v1, indicator value

[0095] After the third attack, based on the current network structure, the node degree values are sorted in descending order as follows:

[0096]

[0097] (4) The fourth attack, delete v4, indicator value

[0098] After the fourth attack, based on the current network structure, the node degree values are sorted in descending order as follows:

[0099]

[0100] (5) The fifth attack, delete v5, indicator value

[0101] After the fifth attack, based on the current network structure, the node degree values are sorted in descending order as follows:

[0102]

[0103] (6) The sixth attack, deleting v6, all nodes in the current network are isolated, the index value

[0104] After the sixth attack, based on the current network structure, the node degree values are sorted in descending order as follows:

[0105]

[0106] (7) The seventh attack, delete v7, indicator value

[0107] After the seventh attack, based on the current network structure, the node degree values are sorted in descending order as follows:

[0108]

[0109] (8) The eighth attack, deleting v8, indicator value

[0110] After the eighth attack, based on the current network structure, the node degree values are sorted in descending order as follows:

[0111]

[0112] (9) The ninth attack, delete v9, indicator value

[0113] Step 3: Calculate the overall controllability robustness index of the network after the node is attacked

[0114] The overall controllability robustness of the network is defined as follows:

[0115]

[0116] in, is the controllable robustness index after the i-th node in the network is attacked, It is the overall controllability robustness index of the network; the smaller the overall controllability robustness index value is, the better the controllability robustness of the network system is.

[0117] In this embodiment, For example (such as Figure 2 The first random attack method shown in (a) can finally get The second deliberate attack method can eventually obtain Depend on Figures 3 to 8 It can be seen that Figure 3 In the implementation of the present invention When the network is at k=2, random attack is adopted, and the index value and Graph showing changes with the number of node deletions i; Figure 4 In the implementation of the present invention When the network is at k=2, the index value is and The graph of changes with the number of node deletions i; Figures 3 and 4 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 5 In the implementation of the present invention, the three simple complex networks all use random attack mode when k=2, and the index value is Comparison chart of changes with the number of node deletions i. Figure 6 For three simple complex networks, when k=2, all use random attack mode, and the index value is Comparison chart of changes with the number of node deletions i. Figure 7 For three simple complex networks, when k=2, all use deliberate attack mode, and the index value is Comparison chart of changes with the number of node deletions i. Figure 8 For three simple complex networks, when k=2, all use deliberate attack mode, and the index value is Comparison chart of changes with the number of node deletions i; Figure 6 and Figure 8 It is verified that the overall controllability robustness relationship of the three simple complex networks is Figure 6 and Figure 8 This shows that the attack method and network topology have a significant impact on the overall controllability robustness. The two-dimensional simplex, i.e., the third-order motif, in the high-order interaction network is conducive to improving the overall controllability robustness of the network.

[0118] In summary, we can draw the following conclusions:

[0119] (1) Simple complex network Among them, the one with the best overall controllability robustness is

[0120] (2) Simple complex network In the whole controllable robustness The sorting results are:

[0121] (3) In the case of a deliberate attack: Network, after deleting 1 node, the indicator value Start equal to 1; Network, delete After nodes, the index value Start equal to 1; t 2 (x1,x2,x3) network, after deleting 3 nodes, the indicator value Start equal to 1;

[0122] (4) Index value The change range is large, and the indicator value The change is small. This is because The network contains more k-simplexes, forming a denser high-order coupling network with strong redundancy. When a small number of nodes are deleted at the beginning, the redundant structure can maintain controllability. After the key nodes are removed in the later stage, the high-order structure suddenly collapses, causing the curve to change suddenly. The number of k-simplices in the network is extremely small, and controllability is maintained by relying on a few key nodes. The initial attack on the first key node will lead to the loss of most controllability. The remaining nodes in the later stage have limited contribution to controllability, and the room for increase is limited, which makes the curve tend to be flat.

[0123] (5) The overall controllability robustness index values of the network show an increasing trend, indicating that as nodes are removed, the proportion of driving nodes required to maintain controllability gradually increases, and the controllability of the network gradually deteriorates.

[0124] The examples illustrate that the controllable robustness analysis method proposed in the present invention is reasonable and effective, and provides a theoretical basis and technical guidance for rationally formulating attack and defense response strategies in practical applications.

[0125] The above description is merely a preferred embodiment of the present 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 the present disclosure is not limited to the technical solutions formed by a specific combination of the above-mentioned technical features, but should also encompass other technical solutions formed by any combination of the above-mentioned technical features or their equivalents without departing from the above-mentioned inventive concept. For example, a technical solution formed by mutually replacing the above-mentioned features with (but not limited to) technical features with similar functions disclosed in the embodiments of the present disclosure.

Claims

1. A high-order network controllability robustness analysis method based on simplicial complexes, characterized by: The following steps are involved: Step 1: Construct three simplicial complex networks t k (x1,x2,…,x k+1 ), determine the differential equations of the dynamics of the simplex complex network system; Step 2: Based on the simplicial complex network constructed in step 1, two different attack strategies are adopted on the network nodes based on the node degree ranking results. The network controllability robustness index values are calculated respectively, and then the controllability robustness of the network after the nodes are attacked is analyzed; Step 3: Calculate the overall controllability robustness index of the network after the node is attacked 2. The method for analyzing the controllability robustness of a high-order network based on a simplicial complex according to claim 1, characterized in that: Step 1 is as follows: construct three simple complex networks respectively t k (x1,x2,…,x k+1 ),in, Indicates that the network center is a (k-1)-simplex, and it is a common face of the surrounding l k-simplices; Indicates that there are l k-simplices in the network, and adjacent simplices 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-simplex, there are x1 k-simplices adjacent to one face of the center simplex, there are x2 k-simplices adjacent to another face of the center simplex, and finally there are x k+1 k-simplexes are adjacent to the last remaining face of the central simplex, x k+1 represents the number of k-simplices.

3. The method for analyzing the controllability robustness of a high-order network based on a simplicial complex according to claim 2, 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 V = {v1, v2, ..., v N } represents a set of N nodes, v N is the Nth node, E represents the set of edges; Adjacency tensor of q-simplicial complex represents N×N×…×N| q+1 -dimensional tensor, is the adjacency tensor elements, when (i1,i2,…,i q+1 )∈S (q) , otherwise Among them, S (q) represents a q-simplicial complex, i1,i2,…,i q+1 Indicates the node number, (i1,i2,…,i q+1 ) represents a q-simplex; the adjacency tensor Compressed into an N×N matrix A (q) , matrix A (q) Elements in The definition is as follows: in, is the adjacency tensor The elements have values of 0 or 1; the sum is traversed over all possible q-1 nodes i3,i4,…,i q+1 ; represents the total number of q-simplices in which nodes i1 and i2 participate together; A Q-dimensional simplicial complex network system with N nodes is represented by differential equations, where the dynamic differential equation of the i-th node is expressed as follows: in, represents the state of node i, Represents the mth input. When the i-th node receives input u m When the effect is im =1, otherwise b im =0; is an internal coupling function that satisfies the condition q=1,2,…,Q is the coupling strength, represents the set of real numbers, represents the set of positive real numbers, represents a set of (q+1)-dimensional real vectors; q=1,2,…,Q is the adjacency tensor Elements of j q Is the node label, indicating node j q ; Let x=[x1,x2,…,x N ] T Represents the state vector of the node, u=[u1,u2,…,u M ] T represents the input vector, and the superscript T represents the transpose, then formula (2) can be written as in, is the input matrix, represents the set of N×M dimensional real matrices; σ q is the coupling strength; A (q) Represents the q-order adjacency tensor The compressed N×N matrix, α q is the matrix A (q) The correlation coefficient of the order, α q =q.

4. The method for analyzing the controllability robustness of a high-order network based on a simplicial complex according to claim 1, characterized in that: The attack strategy described in step 2 specifically includes a random attack strategy and a deliberate attack strategy; The random attack strategy is specifically as follows: randomly selecting a node in the network to attack each time; The deliberate attack strategy is specifically as follows: according to the simplicial complex network system constructed in step 1, the 1st degree, 2nd degree, ..., qth degree of all nodes in the network are calculated and sorted in descending order; when a node in the network is attacked, it is modeled as the node being deleted, that is, the node is removed from the network, and nodes with larger degrees are selected for attack according to the node q-order sorting results; when selecting nodes, if multiple nodes have the same q-order, the (q-1) degree of the nodes is compared, and the node with larger (q-1) degree is preferentially selected for deletion, and so on; when deleting node i, the edges associated with node i and the q-simplex in which node i participates are also deleted, wherein the q-order of node i is defined as follows: Where i = 1, 2, ..., N, is the adjacency tensor The elements of , the value is 0 or 1; The controllable robustness index of node i in the network after being attacked is defined as follows: Among them, N represents the total number of initial nodes in the network, N D (i) represents the number of driver nodes required to ensure the network is controllable after node i is attacked; Ni is the number of remaining nodes in the network after node i is attacked; The calculation formula for the number of driving nodes in the controllable robustness index is as follows: Where j = 1, 2, ..., N i , N i =Ni; is λ j The geometric multiplicity, λ j express The j-th eigenvalue of q i =1,2,…,Q i is the coupling strength, Represents the adjacency tensor Compressed N i ×N i matrix, is a matrix The correlation coefficient of the order, that is q i =1,2,…,Q i ; represents the unit matrix, the symbol rank(·) represents the rank of the matrix, and the symbol max represents the maximum value.

5. The method for analyzing the controllability robustness of a high-order network based on a simplicial complex according to claim 1, characterized in that: The overall controllable robustness index of the network described in step 3 The definition is as follows: in, is the controllable robustness index after the i-th node in the network is attacked, It is the overall controllability robustness index of the network; the smaller the overall controllability robustness index value is, the better the controllability robustness of the network system is.

Images