Network key node identification method and system based on high-order adjacency information entropy

Abstract

The application relates to a network key node identification method and system based on high-order adjacent information entropy, which comprises the following steps: acquiring a high-order network to be identified, an interaction order weight, an interaction weight and a size normalization parameter; enumerating a high-order interaction inner node pair and constructing a high-order adjacent weight; calculating a node high-order strength and a high-order adjacent degree based on the high-order adjacent weight; constructing a high-order selection probability distribution in the node high-order neighborhood; calculating a high-order adjacent information entropy score according to the probability distribution, and outputting a network key node sorting sequence according to the score. The application can retain the interaction order, the interaction size and the neighbor high-order strength information in a high-order network such as a hypergraph or a simplex complex, reduce the influence of weakening the group interaction contribution of an ordinary projection network, and provide an interpretable, locally computable and fast screening result of key nodes.

Description

Technical Field

[0001] This invention relates to the fields of complex network analysis, systems engineering and data mining, and in particular to a method and system for identifying key network nodes based on high-order adjacency information entropy. Background Technology

[0002] Key node identification is used to discover nodes in complex systems such as transportation, power, communications, finance, social media, and public health that have a significant impact on network connectivity, propagation range, or operational efficiency. Traditional methods such as degree centrality, betweenness centrality, proximity centrality, PageRank, k-shell decomposition, LocalRank, and collective influence are mostly based on ordinary paired edge networks and can characterize the importance of nodes under edge adjacency relationships.

[0003] However, real-world systems such as scientific research collaborations, group interactions, label co-occurrence, metabolic responses, multi-agent collaboration, and online group interactions often involve group interactions with three or more nodes. If such group interactions are simply projected as ordinary edge networks, the interaction order, group size, closure structure, and multi-agent collaboration relationships are easily weakened or lost, thus affecting the selection results of key nodes.

[0004] Adjacency information entropy methods can characterize the uncertainty of node influence allocation using local adjacency probability distributions, offering interpretability and low computational complexity. However, existing adjacency information entropy methods typically rely on ordinary edge adjacency or projected networks, making it difficult to explicitly utilize higher-order structural information in hyperedges or simplexes. Therefore, a technical solution is needed that can preserve higher-order interaction contributions and output a ranking of key nodes. Summary of the Invention

[0005] This invention provides a method and system for identifying key network nodes based on high-order adjacency information entropy. It can comprehensively utilize interaction order, interaction scale, interaction weight, and neighbor high-order strength in high-order networks such as hypergraphs and simplex complexes, avoiding the weakening of group interaction contributions by ordinary projection networks.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0007] The first aspect is a network key node identification method based on high-order adjacency information entropy, which includes the following steps:

[0008] Acquire the high-order network data to be identified, including a set of nodes and a set of high-order interactions; obtain the interaction order weights, high-order interaction weights, and scale normalization parameters; enumerate node pairs for each high-order interaction in the high-order interaction set and construct the high-order adjacency weights for each node pair; calculate the high-order strength of each node based on the high-order adjacency weights of each node pair, which reflects the overall connection strength formed by the target node through high-order interactions; calculate the high-order adjacency degree of each node based on the high-order adjacency weights of each node and the high-order strengths of its neighboring nodes, which characterizes the quality of high-order neighbors;

[0009] Based on the higher-order adjacency weights and higher-order strengths of each node, a higher-order selection probability distribution for each node is formed.

[0010] Based on the higher-order selection probability distribution of each node, the higher-order adjacency information entropy of each node is calculated and used as the node importance score.

[0011] The nodes are sorted in descending order based on their importance scores to obtain the ranking sequence of key nodes in the network.

[0012] As a preferred option, the higher-order adjacency weights for each node pair include:

[0013] For any pair of nodes, iterate through all higher-order interactions that contain that pair of nodes;

[0014] For each higher-order interaction that includes the node pair, multiply the interaction weight of the higher-order interaction by the order weight of the corresponding order, and then divide by the scale normalization term corresponding to the scale of the higher-order interaction to obtain the adjacency weight contribution value of the higher-order interaction to the node pair; sum up the adjacency weight contribution values ​​of each higher-order interaction that includes the node pair to obtain the higher-order adjacency weight of the node pair.

[0015] As a preferred method, calculating higher-order adjacency includes:

[0016] For the target node, sum the higher-order adjacency weights of all its neighboring nodes in its higher-order neighborhood to obtain the higher-order strength of the target node.

[0017] For the target node, traverse all neighboring nodes in its higher-order neighborhood, multiply the higher-order adjacency weight between the target node and each neighboring node by the higher-order strength of the corresponding neighboring node to obtain the weighted strength value of each neighboring node, and sum the weighted strength values ​​of each neighboring node to obtain the higher-order adjacency degree of the target node.

[0018] As a preferred approach, the higher-order selection probability distribution of each node is formed based on the higher-order adjacency weights and the higher-order strengths of its neighbors, including:

[0019] Multiply the higher-order adjacency weights between the target node and each higher-order neighbor node by the higher-order strength of the corresponding higher-order neighbor node, and then divide by the higher-order adjacency degree of the target node.

[0020] As a preferred method, calculating the higher-order adjacency information entropy of each node includes:

[0021] For a target node with a non-empty higher-order neighborhood and a higher-order adjacency greater than zero, take the negative value of the base-2 logarithm of each higher-order selection probability in its higher-order selection probability distribution, multiply it by the corresponding higher-order selection probability, and sum the results of each product to obtain the higher-order adjacency information entropy of the target node; for a node with an empty higher-order neighborhood or a higher-order adjacency equal to zero, set its higher-order adjacency information entropy to zero.

[0022] Preferably, calculating the higher-order adjacency information entropy of each node also includes a second-order higher-order adjacency information entropy expansion step:

[0023] For the target node, its first-order higher-order adjacency and the higher-order adjacency of each of its higher-order neighbor nodes are weighted and fused by adjusting the weight parameters of direct and indirect higher-order adjacency to obtain the target node's second-order higher-order adjacency. Based on the higher-order adjacency weights, neighbor higher-order strengths, and neighbor higher-order adjacency of each node, a second-order higher-order selection probability distribution for each node is formed.

[0024] Preferably, the second-order higher-order adjacency information entropy expansion step includes: multiplying the first-order higher-order adjacency degree of the target node by the weight adjustment parameter to obtain the direct adjacency term; multiplying the higher-order adjacency weights between the target node and each higher-order neighbor node by the higher-order adjacency degree of the corresponding neighbor node, summing the results, and then multiplying by the complement value of the weight adjustment parameter to obtain the indirect adjacency term; and adding the direct adjacency term and the indirect adjacency term to obtain the second-order higher-order adjacency degree.

[0025] Secondly, a network key node identification system based on high-order adjacency information entropy is provided to implement the network key node identification method and its steps based on high-order adjacency information entropy as described in the first aspect, including:

[0026] The data acquisition module is used to acquire the high-order network data to be identified, including the node set and the high-order interaction set;

[0027] The weight construction module is used to obtain the interaction order weight, higher-order interaction weight and scale normalization parameter. For each higher-order interaction in the higher-order interaction set, it enumerates node pairs and constructs the higher-order adjacency weight of each node pair. Based on the higher-order adjacency weight of each node pair, it calculates the higher-order strength of each node, which reflects the overall connection strength formed by the target node through higher-order interactions.

[0028] The entropy calculation module is used to calculate the higher-order adjacency degree of each node based on the higher-order adjacency weights of each node and the higher-order strengths of its neighboring nodes, which is used to characterize the quality of higher-order neighbors; it forms the higher-order selection probability distribution of each node based on the higher-order adjacency weights of each node and the higher-order strengths of its neighbors; and it calculates the higher-order adjacency information entropy of each node based on the higher-order selection probability distribution of each node, which serves as the node importance score.

[0029] The node sorting module is used to sort the nodes in descending order according to their importance scores, thus obtaining the sorting sequence of key network nodes.

[0030] Compared with the prior art, the beneficial effects of the present invention are reflected in:

[0031] Compared with ordinary adjacency information entropy or node importance indicators based on projection networks, this invention explicitly preserves the order, scale, and weight information of nodes jointly participating in the hyperedge or simplex through high-order adjacency weights, which can reduce the problem of high-order interactions being weakened by ordinary projection.

[0032] This invention incorporates the higher-order strength of neighboring nodes into the local probability distribution by using higher-order adjacency and higher-order selection probability. This can characterize both the neighborhood size of the target node and the distribution pattern of influence among multiple higher-order neighbors.

[0033] This invention employs a sparse high-order adjacency weight table and a local traversal approach, avoiding the high computational overhead of global shortest path, spectral decomposition, or node-by-node propagation simulations. It is suitable as a tool for rapidly screening candidate key nodes in large-scale high-order networks. Attached Figure Description

[0034] Figure 1 This is a flowchart of the network key node identification method based on high-order adjacency information entropy of the present invention.

[0035] Figure 2 To verify the statistical graphs of the scale of first-order edges and second-order simplexes in the network, which are used to illustrate the differences in the scale of higher-order structures of different network samples.

[0036] Figure 3 A normalized scatter plot of HOAIE score and node propagation influence is used to illustrate the correlation between key node score and propagation influence.

[0037] Figure 4 A comparison of Kendall correlation coefficients between various centrality indicators and communication influence under the main communication parameters.

[0038] Figure 5 These are inaccurate function curves for different centrality indices, used to compare the error levels of different sorting methods in identifying the first K nodes.

[0039] Figure 6This is a heatmap of Top-K accuracy for different metrics, used to demonstrate the recognition accuracy of different networks under different K values.

[0040] Figure 7 The curves showing the excess destruction rate of higher-order closures under different centrality sortings are used to illustrate the destructive effect of removing key nodes on the structure of higher-order closures.

[0041] Figure 8 The curves showing the change in the proportion of the remaining second-order simplex under different centrality orders are used to reflect the degree of preservation of higher-order ternary interaction structures.

[0042] Figure 9 This is a heatmap of the Kendall correlation coefficients between centrality and propagation influence at different first-order propagation rates, used to illustrate the impact of changes in propagation parameters on ordination correlation.

[0043] Figure 10 This is a heatmap of the Kendall correlation coefficients between centrality and propagation influence under different second-order propagation rates, used to illustrate the impact of changes in second-order propagation parameters on ranking correlation.

[0044] Figure 11 The ablation experiment results for the components of HOAIE are used to illustrate the contribution of components such as direct adjacency, higher-order intensity, and neighbor intensity to the recognition performance. Detailed Implementation

[0045] To make the technical means, inventive features, objectives, and effects of the invention readily understandable, the invention is further described below with reference to specific illustrations. However, the invention is not limited to the embodiments described below.

[0046] It should be noted that the structures, proportions, sizes, etc., illustrated in the accompanying drawings of this specification are only used to complement the content disclosed in the specification for those skilled in the art to understand and read, and are not intended to limit the conditions under which the present invention can be implemented. Therefore, they have no substantial technical significance. Any modifications to the structure, changes in the proportions, or adjustments to the size, without affecting the effects and objectives that the present invention can produce, should still fall within the scope of the technical content disclosed in the present invention.

[0047] The technical problem to be solved by this invention is to provide a network key node identification method based on high-order adjacency information entropy, which can comprehensively utilize interaction order, interaction scale, interaction weight and neighbor high-order strength in high-order networks such as hypergraphs and simplex complexes, and avoid ordinary projection networks weakening the group interaction contribution.

[0048] To address the aforementioned technical problems, this invention provides a method for identifying key network nodes based on higher-order adjacency information entropy, comprising: acquiring the higher-order network to be identified and related parameters; enumerating pairs of nodes within higher-order interactions and constructing higher-order adjacency weights; calculating higher-order strength and higher-order adjacency degree; forming a higher-order selection probability distribution based on the higher-order adjacency weights and the higher-order strength of neighbors; calculating the higher-order adjacency information entropy score; and outputting a sorted sequence of key network nodes in descending order of score.

[0049] Example 1:

[0050] In this embodiment, the high-order network to be identified is the urban traffic signal cooperative control network. This network uses intersection signal controllers, key detection sections, or traffic control units as nodes, and records the combined effects of three or more nodes in the same traffic phase, continuous road segment congestion propagation chain, green wave cooperative control scheme, or multi-intersection joint release event as high-order interactions; the interaction weight can be determined by traffic flow, queue length, average delay, green wave ratio cooperative intensity, or historical congestion propagation frequency.

[0051] Through the modeling method described above, this invention does not deal with abstract numbers themselves, but rather with high-order correlations formed by natural laws such as intersection nodes, traffic phases, vehicle queuing, and congestion propagation during traffic flow. The output key node ranking can be used to prioritize signal timing adjustments, deploy detection equipment, investigate abnormal signal control units, or formulate congestion mitigation plans, thereby solving the technical problems of inaccurate identification of key nodes in traffic networks and the weakening of group collaborative effects by ordinary side networks.

[0052] like Figure 1 As shown, the high-order network to be identified is first obtained. .in, For a set of nodes, This is a set of higher-order interactions. For a hypergraph, Indicates the superedge; for simple complexes, Represents a maximal simplex or a simplex of a specified order. When At that time, Called Meta-interaction, whose higher order is For unauthorized high-order networks, it can be made For high-level networks with authority, It is given by the corresponding business records, interaction frequency, or confidence level.

[0053] Figure 1 The overall processing flow of the present invention is demonstrated: first, traffic signal cooperative control network or other high-order network data are input; then, a high-order adjacency weight table is constructed; subsequently, high-order strength, high-order adjacency degree, high-order selection probability and high-order adjacency information entropy score are calculated; finally, the key node ranking results that can be used for engineering disposal are output.

[0054] Subsequently, for each higher-order interaction Enumerate the node pairs Calculate and accumulate the higher-order adjacency weights of node pairs. :

[0055]

[0056] in, Represents a node With nodes Higher-order adjacency weights between them Indicates the interaction order weight. This represents the weight of the higher-order interaction e. This represents the scale normalization parameter. This formula sums the contributions of the same node pair in higher-order interactions of different orders and scales.

[0057] Higher-order adjacency weights are simultaneously influenced by the interaction order weight η and the interaction weights. and scale normalization parameters The influence of this interaction allows us to distinguish between the contributions of ordinary binary contact and triadic or higher group interactions.

[0058] After obtaining the higher-order adjacency weight table, calculate the higher-order strength of the nodes. :

[0059]

[0060]

[0061] in, Represents a node The higher-order neighborhood of , where V represents the set of nodes, E represents the set of higher-order interactions, and e represents the hyperedge or simplex. Used to identify candidate neighbors that have participated in at least one higher-order interaction with the target node; Represents a node The higher-order strength is used to characterize the overall connection strength formed by nodes through higher-order interactions.

[0062] Calculate higher-order adjacency :

[0063]

[0064] in, Represents a node The higher-order adjacency, Representing neighboring nodes The higher-order strength is determined by combining adjacency weight with neighbor strength to reflect the quality of higher-order neighbors.

[0065] Higher-order strength reflects the overall connection strength formed by the target node through higher-order interactions. Higher-order adjacency further introduces the higher-order strength of neighboring nodes to characterize the quality of higher-order neighbors.

[0066] For each target node vi, if its higher-order neighborhood is non-empty and its higher-order adjacency is... If the value is greater than 0, then a higher-order selection probability distribution is formed in its higher-order neighborhood:

[0067]

[0068] in, Represents a node To the neighbors The higher-order selection probability. This probability normalizes the common higher-order interaction strength of node pairs and the higher-order strength of neighbors into a local probability distribution.

[0069] compute nodes Higher-order adjacency information entropy :

[0070]

[0071] in, Represents a node The higher-order adjacency information entropy. This formula is used to transform the uncertainty of the allocation of local higher-order adjacency influence into a node importance score. All nodes are then ranked according to their higher-order adjacency information entropy. Sort the nodes from largest to smallest to obtain the sorted sequence of key nodes in the network. If the higher-order neighborhood is empty or the higher-order adjacency is... If the value is 0, then the higher-order adjacency information entropy score of that node is set to 0.

[0072] When all interactions in a high-order network are binary edges, and take , and When the node pairs higher-order adjacency weights degenerate into ordinary edge adjacencies, the higher-order adjacency information entropy degenerates into the adjacency information entropy form in ordinary networks; when there is group interaction, the weight terms can retain the higher-order structural differences that are easily lost in ordinary projections.

[0073] Example 2: Second-order higher-order adjacency information entropy expansion

[0074] In some application scenarios with significant indirect diffusion or two-hop cascading effects, higher-order adjacency capabilities of neighboring nodes can be further introduced. Specifically, a second-order higher-order adjacency degree can be constructed. :

[0075]

[0076] in, This represents the second-order higher-order adjacency degree. ∈[0,1] represents the weight adjustment parameter for direct higher-order adjacency and indirect higher-order adjacency. This formula is used to control the relative contributions of direct and indirect higher-order adjacencies. It is used to construct an optional second-order higher-order adjacency information entropy expansion.

[0077] Taking traffic signal cooperative control network as an example, the congestion status of adjacent intersections or upstream key road sections will have an indirect impact through traffic queuing and phase coordination. By introducing second-order higher adjacency through formula (7), the higher-order connectivity of the target intersection itself and the higher-order connectivity of adjacent intersections can be included in the identification of key nodes, which is more suitable for technical scenarios with indirect transmission characteristics such as traffic congestion cascading propagation, communication failure propagation or public health contact transmission.

[0078] when When, the second-order form degenerates into a first-order higher-order adjacency information entropy; when In practice, the ranking primarily relies on the higher-order adjacency capabilities of neighbors. In real-world applications, ranking can be selected based on historical propagation records, node removal records, expert annotation results, or ranking stability tests. , and Calculate the second-order higher-order selection probability based on the second-order higher-order adjacency:

[0079]

[0080] in, This represents the second-order higher-order selection probability, with the denominator normalized to the candidate neighbors within the higher-order neighborhood of the target node. Substituting this second-order selection probability into the definition of higher-order adjacency information entropy yields the second-order higher-order adjacency information entropy score.

[0081] Example 3: Implementation of electronic devices and storage media

[0082] The method of this invention can be executed by an electronic device. The electronic device includes a processor, a memory, and a computer program stored in the memory. When the processor executes the computer program, it reads high-order network data, constructs a sparse high-order adjacency weight table, calculates high-order strength, high-order adjacency degree, high-order selection probability, and high-order adjacency information entropy score, and outputs a ranking sequence of key network nodes.

[0083] In practical implementation, adjacency lists or hash tables can be used to store higher-order adjacency weights. The main overhead of the algorithm comes from enumerating node pairs within the same higher-order interaction. In sparse higher-order networks, the scale of higher-order interactions is usually much smaller than the total number of nodes, so this implementation is suitable for fast candidate node selection in large-scale higher-order networks.

[0084] The present invention also provides a computer-readable storage medium having a computer program stored thereon. When the computer program is executed by a processor, it implements the network key node identification method based on high-order adjacency information entropy according to any embodiment of this specification.

[0085] Experimental example:

[0086] To validate the method, experiments can be conducted on high-order networks such as stochastic simplex, scale-free simplex, workplace contact simulation, hospital contact simulation, meeting contact simulation, and real-world social activity simulations. Evaluation metrics can include the propagation influence of high-order SIR models, inaccuracy functions, Kendall correlation coefficients, Top-K accuracy, structural failure curves, and runtime.

[0087] like Figure 2 As shown, the first-order edge and second-order simplex size of the validation network can be used to illustrate the high-order interaction density and the degree of group closure of different datasets, providing a network structure basis for subsequent comparisons.

[0088] like Figure 3 and Figure 4 As shown, the HOAIE score and the propagation influence of higher-order SIR can be compared using a normalized scatter plot, and the Kendall correlation coefficient can be used to quantify the consistency between different indicators and propagation influence. When HOAIE maintains a high correlation in multiple networks, it indicates that the present invention can more accurately identify nodes that have a significant impact on propagation or operational status.

[0089] like Figure 5 and Figure 6 As shown, the inaccuracy function and Top-K accuracy heatmap can further verify the recognition quality of the first few key nodes; such as Figure 7 and Figure 8 As shown, the excess destruction rate of higher-order closures and the proportion of remaining second-order simplexes can be used to explain the destructive effect of identified nodes on the group interaction structure, thus demonstrating the sensitivity of this invention to higher-order structures.

[0090] like Figures 9 to 11 As shown, comparisons were made under different first-order propagation rates, different second-order propagation rates, different components, and different network sizes, verifying that the present invention has good parameter stability, component effectiveness, and operational efficiency. The above results demonstrate that the present invention can screen candidate key nodes with low computational overhead in high-order networks such as transportation, communication, public health contact, and multi-agent collaboration, and provides a technical basis for resource scheduling, risk warning, and fault handling.

[0091] The validation results demonstrate that the higher-order adjacency information entropy score is positively correlated with the propagation influence of the higher-order SIR model and has explanatory power regarding the destruction of the second-order simplex and the group closure structure. This method is more suitable for networks with clear higher-order interaction semantics, where node influence depends on local group closures, and where low-cost and rapid screening of candidate key nodes is required, such as group contact networks, scientific research collaboration networks, label co-occurrence networks, and multi-agent collaborative relationship networks.

Claims

1. A method for identifying key network nodes based on high-order adjacency information entropy, characterized in that, Includes the following steps: Acquire the high-order network data to be identified, including the set of nodes and the set of high-order interactions; Obtain the interaction order weight, higher-order interaction weight, and scale normalization parameter. For each higher-order interaction in the higher-order interaction set, enumerate node pairs and construct the higher-order adjacency weights for each node pair. Based on the higher-order adjacency weights of each node pair, the higher-order strength of each node is calculated to reflect the overall connection strength formed by the target node through higher-order interactions. Based on the higher-order adjacency weights of each node and the higher-order strengths of its neighboring nodes, the higher-order adjacency degree of each node is calculated to characterize the quality of its higher-order neighbors. Based on the higher-order adjacency weights and higher-order strengths of each node, a higher-order selection probability distribution for each node is formed. Based on the higher-order selection probability distribution of each node, the higher-order adjacency information entropy of each node is calculated and used as the node importance score. The nodes are sorted in descending order based on their importance scores to obtain the ranking sequence of key nodes in the network.

2. The network key node identification method based on high-order adjacency information entropy according to claim 1, characterized in that, The higher-order adjacency weights for each node pair are constructed as follows: For any pair of nodes, iterate through all higher-order interactions that contain that pair of nodes; For each higher-order interaction that includes the node pair, multiply the interaction weight of the higher-order interaction by the order weight of the corresponding order, and then divide by the scale normalization term corresponding to the scale of the higher-order interaction to obtain the adjacency weight contribution value of the higher-order interaction to the node pair; sum up the adjacency weight contribution values ​​of each higher-order interaction that includes the node pair to obtain the higher-order adjacency weight of the node pair.

3. The network key node identification method based on high-order adjacency information entropy according to claim 1, characterized in that, Calculating higher-order adjacency includes: For the target node, sum the higher-order adjacency weights of all its neighboring nodes in its higher-order neighborhood to obtain the higher-order strength of the target node. For the target node, traverse all neighboring nodes in its higher-order neighborhood, multiply the higher-order adjacency weight between the target node and each neighboring node by the higher-order strength of the corresponding neighboring node to obtain the weighted strength value of each neighboring node, and sum the weighted strength values ​​of each neighboring node to obtain the higher-order adjacency degree of the target node.

4. The network key node identification method based on high-order adjacency information entropy according to claim 1, characterized in that, Based on the higher-order adjacency weights and neighbor higher-order strengths of each node, the higher-order selection probability distribution of each node is formed as follows: Multiply the higher-order adjacency weights between the target node and each higher-order neighbor node by the higher-order strength of the corresponding higher-order neighbor node, and then divide by the higher-order adjacency degree of the target node.

5. The network key node identification method based on high-order adjacency information entropy according to claim 1, characterized in that, Calculating the higher-order adjacency information entropy of each node includes: For a target node with a non-empty higher-order neighborhood and a higher-order adjacency greater than zero, take the negative value of the base-2 logarithm of each higher-order selection probability in its higher-order selection probability distribution, multiply it by the corresponding higher-order selection probability, and sum the results of each product to obtain the higher-order adjacency information entropy of the target node; for a node with an empty higher-order neighborhood or a higher-order adjacency equal to zero, set its higher-order adjacency information entropy to zero.

6. The network key node identification method based on high-order adjacency information entropy according to claim 1, characterized in that, Calculating the higher-order adjacency information entropy of each node also includes a second-order higher-order adjacency information entropy extension step: For the target node, its first-order higher-order adjacency and the higher-order adjacency of each of its higher-order neighbor nodes are weighted and fused by adjusting the weight parameters of direct and indirect higher-order adjacency to obtain the target node's second-order higher-order adjacency. Based on the higher-order adjacency weights, neighbor higher-order strengths, and neighbor higher-order adjacency of each node, a second-order higher-order selection probability distribution for each node is formed.

7. The network key node identification method based on high-order adjacency information entropy according to claim 6, characterized in that, The second-order higher-order adjacency information entropy expansion step also includes: multiplying the first-order higher-order adjacency degree of the target node by the weight adjustment parameter to obtain the direct adjacency term; multiplying the higher-order adjacency weights between the target node and each higher-order neighbor node by the higher-order adjacency degree of the corresponding neighbor node, summing the results, and then multiplying by the complement value of the weight adjustment parameter to obtain the indirect adjacency term; and adding the direct adjacency term and the indirect adjacency term to obtain the second-order higher-order adjacency degree.

8. A network key node identification system based on high-order adjacency information entropy, used to implement the network key node identification method and steps based on high-order adjacency information entropy as described in claims 1-7, characterized in that, include: The data acquisition module is used to acquire the high-order network data to be identified, including the node set and the high-order interaction set; The weight construction module is used to obtain the interaction order weight, higher-order interaction weight and scale normalization parameter. For each higher-order interaction in the higher-order interaction set, it enumerates node pairs and constructs the higher-order adjacency weight of each node pair. Based on the higher-order adjacency weights of each node pair, the higher-order strength of each node is calculated to reflect the overall connection strength formed by the target node through higher-order interactions. The entropy calculation module is used to calculate the higher-order adjacency degree of each node based on the higher-order adjacency weights of each node and the higher-order strengths of its neighboring nodes, which is used to characterize the quality of higher-order neighbors; it forms the higher-order selection probability distribution of each node based on the higher-order adjacency weights of each node and the higher-order strengths of its neighbors; and it calculates the higher-order adjacency information entropy of each node based on the higher-order selection probability distribution of each node, which serves as the node importance score. The node sorting module is used to sort the nodes in descending order according to their importance scores, thus obtaining the sorting sequence of key network nodes.

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