A method for identifying key nodes of a time series network based on spectral normalization propagation accumulation

Abstract

This invention relates to the field of temporal network processing technology, specifically a method for identifying key nodes in temporal networks based on spectral normalization propagation accumulation. By extracting interactive node pairs within each consecutive time period and constructing a weighted adjacency matrix based on the union of nodes across the entire time domain, the method ensures that both the temporal characteristics of the interactions and the integrity of the nodes are considered. An adaptive propagation gain coefficient is calculated using the spectral radius of the cumulative adjacency matrix, adapting to the overall interaction intensity of the network without manual parameter tuning, avoiding over-amplification or underestimation of the propagation effect, and ensuring computational stability. By iteratively calculating the propagation state in different time periods and accumulating the results to obtain the total propagation state, the method accurately characterizes the transmission and cumulative reinforcement characteristics of node influence in temporal evolution, effectively improving the identification accuracy of key nodes.

Description

Technical Field

[0001] This invention relates to the field of temporal network processing technology, specifically a method for identifying key nodes in temporal networks based on spectral normalization propagation accumulation. Background Technology

[0002] Temporal networks, as a complex system modeling framework for depicting the dynamic evolution of interactive relationships over time, are widely used in practical scenarios such as crowd contact monitoring, communication network scheduling, and online social platform interaction. Unlike static networks, which only focus on the connections between nodes, the timing, sequence, and cumulative frequency of interactive events in temporal networks directly determine the final outcome of system dynamics such as information diffusion and propagation. That is, the interaction patterns of the same group of nodes at different times may trigger drastically different propagation effects. Therefore, accurately identifying the key nodes in temporal networks that can trigger significant propagation effects is a core issue in temporal network analysis and propagation dynamics research.

[0003] Currently, key node identification methods in temporal networks mainly fall into two categories. The first is the static aggregation strategy, which directly aggregates temporal interaction data across the entire time domain into a single static graph, or simply synthesizes the graph after independently calculating centrality on snapshots of each time segment. However, this approach loses the temporal cumulative characteristics of interaction events and the propagation correlation between time periods, only reflecting the instantaneous connection state of nodes and failing to characterize the cumulative effect of node interaction intensity across the entire time domain, leading to significant bias in the identification of continuously propagating key nodes. The second category is the temporal path-driven centrality method, which measures node influence by traversing propagation paths that satisfy time constraints. However, this method faces problems such as high computational complexity, sensitivity to parameters like path length, and a lack of adaptive adaptability to the overall network interaction intensity, requiring manual adjustment of propagation-related parameters. This results in poor stability in temporal networks with different densities and interaction frequencies, making large-scale application difficult.

[0004] Furthermore, existing methods for identifying key nodes in temporal networks have the following critical shortcomings: First, they do not fully utilize the cumulative interaction intensity across the entire time domain to calibrate the propagation gain, resulting in a lack of network adaptability in the quantification of propagation effects. Manually set parameters are prone to overestimating or underestimating the influence. Second, they do not reflect the transmission and cumulative reinforcement characteristics of node influence at different time periods through iterative calculation and accumulation of propagation states in different time periods, making it difficult to accurately characterize the continuous propagation capability of nodes in temporal evolution. Third, they lack adaptability to directed / undirected networks and have not formed a quantification calculation logic that matches the directionality of interaction, further affecting the accuracy of key node identification.

[0005] Therefore, there is an urgent need for a method for identifying key nodes in time-series networks to solve the problems of loss of time-series information, parameter sensitivity, and insufficient scalability of existing methods, and to provide efficient and reliable technical support for the identification of key nodes in various dynamic interactive systems. Summary of the Invention

[0006] To address the low recognition accuracy caused by the loss of interactive temporal accumulation characteristics in existing methods for identifying key nodes in temporal networks, this invention provides a method for identifying key nodes in temporal networks based on spectral normalized propagation accumulation. By integrating full-time-domain cumulative interaction strength, adaptively calibrating propagation gain, and iteratively accumulating propagation states in different time periods, the method effectively improves recognition accuracy. Based on this, this invention also provides a system for identifying key nodes in temporal networks based on the spectral radius of the cumulative adjacency matrix and an electronic device.

[0007] To achieve the above objectives, the present invention provides the following technical solution:

[0008] A method for identifying key nodes in a temporal network based on spectral normalization propagation accumulation includes the following identification steps:

[0009] Extract the node pairs that interact within each consecutive time period from the original record of the target temporal network and store them in the node set of the corresponding time period;

[0010] Take the union of all node sets, and construct a weighted adjacency matrix for each node set based on the union. Each element in the matrix is ​​the sum of the number of interactions between corresponding node pairs within the corresponding time period.

[0011] The cumulative adjacency matrix is ​​obtained by summing all weighted adjacency matrices. The propagation gain coefficient α = 1 / (r + β) is calculated by taking its spectral radius r, where β is a constant to prevent division by zero.

[0012] Press x (n) =(I+αA (n) )x (n-1) Calculate and aggregate the propagation states of each node in each time period, and sum them to obtain the total propagation states of each node; sort them in descending order and select the nodes corresponding to the top K total propagation states as key nodes; x (n) x (n-1) Let I be the propagation state vector formed by combining the propagation states of each node in the union set during the nth and (n-1)th time periods; let A be the identity matrix; (n) It is the weighted adjacency matrix for the nth time period.

[0013] As a further improvement to the above scheme: if the target temporal network is a directed network, then the weighted adjacency matrix is ​​an asymmetric matrix, and the matrix elements correspond only to the total number of interactions between a node and another node; if the target temporal network is an undirected network, then the weighted adjacency matrix is ​​a symmetric matrix, and the matrix elements are the total number of interactions between two nodes.

[0014] As a further improvement to the above scheme: when the target temporal network contains the original interaction weights of the interaction events, the total number of interactions can be replaced with the total original interaction weights of the corresponding node pairs within the corresponding time period; if the interaction event has no original interaction weights, then the original interaction weight of the interaction event is set to 1.

[0015] As a further improvement to the above scheme: the initial propagation state vector x (0) It is a vector consisting entirely of 1s.

[0016] As a further improvement to the above scheme: β is set to 10. -12 .

[0017] As a further improvement to the above scheme: the time period is a time window divided by a fixed time step Δt, and the time interval of the nth time window is... ,in The observation start time of the original record; n=1,2,...,N, where N is the total number of time windows.

[0018] As a further improvement to the above scheme: K is set to 1%-10% of the total number of nodes.

[0019] As a further improvement to the above scheme, before extracting the interaction node pairs, it also includes preprocessing steps such as deduplication, error correction, timestamp unification, and node identifier standardization of the original records.

[0020] A key node identification system for time-series networks based on the spectral radius of the cumulative adjacency matrix includes:

[0021] The data preprocessing and extraction module is used to execute the above-mentioned key node identification method, perform deduplication, error correction, timestamp unification and node identifier standardization preprocessing on the original records of the target time series network, and extract the node pairs that interact within each consecutive time period and store them in the node set of the corresponding time period, wherein the time period is obtained through the above-mentioned key node identification method.

[0022] The weighted adjacency matrix construction module is used to execute the key node identification method described above, obtain the union of all node sets, and construct the weighted adjacency matrix for each time period based on the union. The matrix elements are the sum of the number of interactions between node pairs within the corresponding time period or the sum of the original interaction weights. The key node identification method described above is used to determine whether the target temporal network is a directed network or an undirected network. The weighted adjacency matrix is ​​asymmetric if the target temporal network is a directed network, and a symmetric matrix if the target network is an undirected network.

[0023] The propagation gain coefficient calculation module is used to execute the key node identification method described above, accumulate all weighted adjacency matrices to obtain the cumulative adjacency matrix, extract its spectral radius r, and calculate the propagation gain coefficient α.

[0024] The propagation state iteration and accumulation module is used to execute the key node identification method described above, using an all-one vector as the initial propagation state vector x. (0) Press x (n) =(I+αA (n) )x (n-1) Iteratively calculate the propagation state of each node in each time period, and sum them up to obtain the total propagation state of each node;

[0025] The key node screening module is used to execute the key node identification method described above. It sorts the total propagation state in descending order, determines K according to the accuracy requirements of the target application scenario, and selects the top K corresponding nodes as key nodes.

[0026] An electronic device includes a processor, a memory, and a computer program stored in the memory. When the processor executes the computer program, it implements the above-described method for identifying key nodes in a time-series network based on spectral normalization propagation accumulation.

[0027] Compared with the prior art, the beneficial effects of the present invention are:

[0028] 1. This invention ensures a balance between the temporal characteristics of interactions and the integrity of nodes by extracting interaction node pairs in each consecutive time period and constructing a weighted adjacency matrix based on the union of nodes across the entire time domain. It utilizes the spectral radius of the cumulative adjacency matrix to calculate the adaptive propagation gain coefficient, adapting to the overall interaction intensity of the network without manual parameter tuning, thus avoiding over-amplification or underestimation of propagation effects and ensuring computational stability. By iteratively calculating the propagation state in different time periods and accumulating the results to obtain the total propagation state, it accurately characterizes the transmission and cumulative reinforcement characteristics of node influence in temporal evolution. Compared to static aggregation strategies, this approach more closely reflects the real propagation mechanism, and compared to temporal path methods, it significantly reduces computational complexity. Therefore, this invention can stably improve the accuracy of key node identification while controlling computational overhead, and can be widely adapted to various temporal network scenarios.

[0029] 2. This invention employs an asymmetric matrix for directed networks, representing only the number of one-way interactions, and a symmetric matrix for undirected networks, merging the number of two-way interactions. This perfectly aligns with the real-world interaction logic of both types of networks, avoiding the problem of indiscriminately distinguishing network directionality and causing distortion in the quantification of interaction relationships. This targeted design allows the method to flexibly adapt to directed interaction scenarios such as social forwarding and private messaging, as well as undirected interaction scenarios such as offline contact and friend interaction. It can adapt to different network types without adjusting the core algorithm framework, expanding the application scope of the technical solution. Simultaneously, it ensures the accuracy of node interaction intensity quantification in different scenarios, further optimizing the reliability of key node identification results.

[0030] 3. This invention flexibly selects the sum of interaction counts or the sum of original interaction weights as matrix elements based on whether the original data carries interaction weights. This fully utilizes original interaction intensity information (such as chat duration, contact priority, etc.), making the quantification of node interaction intensity more closely match the actual scenario and improving the accuracy of propagation status and overall influence assessment. It is also compatible with unweighted original data; by setting a default weight of 1, it can operate normally without additional data supplementation or preprocessing, significantly reducing the requirements for input data. This flexible adaptability allows the invention to cover more diverse data scenarios while ensuring the consistency of recognition logic and the reliability of results under different data conditions, further expanding the applicability and practical value of the technical solution. Attached Figure Description

[0031] Figure 1 This is a flowchart of the key node identification method in this invention.

[0032] Figure 2 This is a diagram of the target temporal network structure in an embodiment of the present invention.

[0033] Figure 3 This is a diagram of the weighted time-series snapshot sequence structure in an embodiment of the present invention. Detailed Implementation

[0034] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0035] like Figure 1As shown, this invention first collects time-series interaction data, abstracting it into a quadruple containing initiating node, receiving node, timestamp, and weight. After cleaning and standardization, the network directionality is determined. Then, events are divided according to time windows, and the original interaction weights within the time window are statistically analyzed as edge weights to construct a weighted time-series snapshot sequence. Subsequently, the entire time-domain node set is unified, generating a corresponding weighted adjacency matrix sequence. The accumulated matrices are summed to obtain the cumulative adjacency matrix, and the adaptive propagation gain coefficient is calculated through spectral normalization. Next, the propagation state and cumulative vector are initialized, and the propagation state is iteratively updated and accumulated in time order to obtain the cumulative vector reflecting the total influence of the nodes. Finally, the TPC score is defined using this vector component, and the top K nodes are selected as the key node set according to the score.

[0036] I. Constructing an Interaction Event Set

[0037] The core objective of this section is to collect and organize temporal interaction data from the target system, ultimately constructing a structured set of interaction events. This serves as the "data foundation" for the entire temporal network's key node sorting method, and all subsequent snapshot construction and propagation calculations rely on the standardized data output by this section. Essentially, it transforms the chaotic raw interaction records from real-world scenarios into a unified format containing four core elements: "nodes, time, interaction relationships, and intensity," while preserving the key characteristics of temporal networks: temporal order, directionality, and interaction weights.

[0038] 1. Collect interactive event data

[0039] Obtain raw interaction records from the time-series network of the target system (such as social platforms, communication networks, crowd contact monitoring systems, etc.), and abstract each raw interaction record into an event quadruple: Where u is the interaction initiating node; v is the interaction receiving node; is the timestamp of the interaction event; w is the original interaction weight of the interaction event. If there is no original interaction weight of the interaction event in the original interaction record, then let its original interaction weight w=1.

[0040] 2. Data cleaning and standardization.

[0041] The original interaction data is deduplicated, corrected for errors, and missing data is handled. Timestamps are unified (e.g., to the second or minute level), and entity identifiers are mapped to unified node IDs. The cleaned data is still represented by the event set ε.

[0042] 3. Determine the network direction

[0043] For a node pair (u, v), if there is an interaction direction (such as forwarding or sending a private message), it corresponds to a directed event (u→v) or a directed event (v→u); if there is no interaction direction (such as offline contact), it corresponds to an undirected event (u=v), and the quadruple is set... and Treated as the same "directed event", they will be counted together according to the undirected rule in subsequent statistics.

[0044] Therefore, all interactive events constitute an interactive event set: .

[0045] II. Constructing a weighted time-series snapshot sequence

[0046] The core objective of this section is to transform the obtained set of interaction events into a weighted temporal snapshot sequence arranged in chronological order. Essentially, this involves breaking down discrete interaction events on a continuous timeline into instantaneous network states within individual time windows. This preserves the temporal order of the interactions while quantifying the original interaction weights within each time window through edge weights, providing a structured network model for subsequent chronological calculation of propagation accumulation.

[0047] 1. Constructing a time window

[0048] Let the observation start time be If the time step of the time window is Δt>0, then the time interval T of the nth time window is... n Defined as:

[0049] ;

[0050] Where N is the total number of time windows.

[0051] 2. Construct a subset of interactive events within the current time window.

[0052] Define a subset ε of interactive events within the nth time window. (n) :

[0053] ;

[0054] 3. Calculate edge weights

[0055] For any pair of nodes (u,v), define its edge weight within the nth time window. The edge weight is the number of interactions between node pair (u,v) in the nth time window (if the event has original interaction weights, "number of interactions" can be replaced with the sum of the original interaction weights).

[0056] If there is an interaction direction between node pairs (u, v), then when u→v:

[0057] ;

[0058] When v→u:

[0059] ;

[0060] If there is no interaction direction between node pairs (u,v), then merge the undirected event (u,v) with the undirected event (v,u):

[0061] ;

[0062] If no interaction occurs between node pair (u,v) within the nth time window, then .

[0063] 4. Construct a weighted snapshot chart

[0064] Define the weighted snapshot graph G for the nth time window. (n) for:

[0065] ;

[0066] ;

[0067] ;

[0068] In the formula, V (n) E represents the set of nodes within the nth time window. (n) W represents the set of valid edges within the nth time window; edges with interaction are valid, and edges without interaction are invalid. (n) Let E represent the set of valid edge weights within the nth time window. (n) One-to-one correspondence.

[0069] Statistics of each weighted snapshot G (n) To obtain the weighted time-series snapshot sequence {G} (n) (n=1...N).

[0070] III. Constructing a weighted adjacency matrix sequence

[0071] The core objective of this section is to address the inconsistency among snapshot nodes and transform the graphical weighted temporal snapshot sequence into a standardized weighted adjacency matrix sequence. Essentially, it converts the discrete instantaneous network state into a numerically computable matrix form, providing a unified input for subsequent mathematical operations such as spectral normalization and linear propagation updates.

[0072] 1. Construct a unified set of nodes across all time domains

[0073] Find the union of all weighted snapshot graphs to obtain a unified set of nodes U:

[0074] ;

[0075] Here, ∪ represents the union symbol.

[0076] And the nodes in the node set U are numbered sequentially in a fixed order as follows:

[0077] ;

[0078] In the formula, v i v j Let i and j represent the i-th and j-th nodes in the node set U, respectively, where i and j ∈ [1, M], and M is the total number of nodes in the node set U.

[0079] 2. Construct a weighted adjacency matrix

[0080] For each weighted snapshot graph, an M×M weighted adjacency matrix is ​​constructed:

[0081] ;

[0082] ;

[0083] In the formula, A (n) This represents the weighted adjacency matrix within the nth time window; A represents (n) The element in the i-th row and j-th column; This represents node v within the nth time window. i and v j The boundary weights between them.

[0084] If the current weighted snapshot is an undirected snapshot, A (n) Then it is a symmetric matrix; otherwise, A (n) It is then an asymmetric matrix.

[0085] Statistical weighted adjacency matrix A (n) To obtain the weighted adjacency matrix sequence {A} (n) (n=1...N).

[0086] IV. Calculating the propagation gain coefficient

[0087] The core objective of this section is to calculate an adaptive propagation gain coefficient based on the cumulative original interaction weights across the entire time domain using "spectral normalization." Essentially, this provides a scale-controllable adjustment factor for subsequent linear propagation updates, avoiding excessive amplification of propagation effects or numerical instability caused by manual parameter tuning, while ensuring that the propagation gain coefficient adapts to the characteristics of the original interaction weights of different networks, thus guaranteeing the rationality and stability of the propagation calculation.

[0088] 1. Calculate the cumulative adjacency matrix

[0089] The weighted adjacency matrix sequence {A (n) The corresponding elements in all weighted adjacency matrices of} (n=1...N) are summed one by one to obtain the cumulative adjacency matrix A. sum :

[0090] ;

[0091] 2. Calculate the spectral radius r

[0092] Define A sum The spectral radius r is:

[0093] ;

[0094] In the formula, ρ(A) sum ) represents the cumulative adjacency matrix A sum The spectral radius is the cumulative adjacency matrix A. sum The maximum absolute value of all eigenvalues ​​in λ. k Represents the cumulative adjacency matrix A sum The k-th eigenvalue; |λ k | represents the eigenvalue λ k The absolute value of.

[0095] 3. Calculate the propagation gain coefficient

[0096] Let β > 0 to avoid division by zero and enhance numerical stability; it is usually taken as 10. -12 The propagation gain coefficient is defined as:

[0097] ;

[0098] The propagation gain coefficient α is adaptively determined by the original interaction weights of the entire network, avoiding excessive amplification or numerical instability caused by manual parameter tuning.

[0099] Where β takes the value 10 -12 This is mainly because the magnitude of the principal eigenvalue of the matrix in this invention is 10. 0 Up to 10 1 , and 10 -12 The value is much smaller than this order of magnitude, applying only a small perturbation to the matrix without changing its topological connectivity or the distribution of core eigenvalues, thus specifically eliminating the computational risks caused by zero eigenvalues. At the same time, this value is much lower than the computational accuracy requirements of the main parameters of this application (propagation gain coefficient α, propagation state vector x, and propagation cumulative vector c), and will not have a perceptible impact on key calculation results such as propagation gain. It can also adapt to extreme working conditions such as dynamic topological changes and node disconnection, ensuring the stable execution of processes such as matrix inversion and eigenvalue decomposition.

[0100] V. Calculate the propagation cumulative vector

[0101] The core objective of this section is to iteratively calculate the propagation state of nodes in chronological order and accumulate the state values ​​across all time windows, ultimately obtaining a propagation accumulation vector that reflects the total influence of the nodes. Essentially, it simulates the real process of influence evolving over time in a temporal network, spreading through interactions, and accumulating and strengthening through repeated interactions, transforming the results of the previous processing into a quantitative result of node influence.

[0102] 1. Construct the propagation state vector and the cumulative vector

[0103] Define the propagation state vector x∈R M With propagation cumulative vector c∈R M Among them, R M Let U be an M-dimensional vector space over the real number field R, where M is the total number of nodes in the node set U. The propagation state vector x∈R M In the middle, the i-th component x i For the i-th node v i The propagation state values; the propagation accumulation vector c∈R M In the i-th component c i For the i-th node v i The cumulative influence value, both of which are M-dimensional real vectors, therefore belong to R. M Space. This invention uses R M The purpose of defining a state vector in space is to uniformly map the propagation states of discrete nodes into a linear space, providing a rigorous mathematical foundation for subsequent spectral decomposition of the cumulative adjacency matrix and calculation of propagation gain.

[0104] The propagation state vector x is used to record the instantaneous influence state of each node in the current time window, and is initialized to x. (0) =1, that is, x (0) i =1 for all nodes v i All of these are true, and each element in x is 1.

[0105] Before the propagation calculation begins, no interactions have occurred, and the initial influence of all nodes should be considered "equal". An all-one vector is the simplest and unbiased initial assumption, ensuring that subsequent differences in influence are determined solely by the "temporal order and intensity of interactions", rather than by initial bias.

[0106] The propagation accumulation vector c is used to record the sum of the instantaneous influence of nodes within all time windows, and is initialized to c=0, i.e., c i =0 for all nodes v i Both are true, and each element in c is 0.

[0107] The core of the cumulative vector is "superimposing the instantaneous states of each window". At the beginning of the calculation, no window has been passed, so the cumulative value is 0. It is gradually accumulated with subsequent iterations.

[0108] 2. Dissemination and updates

[0109] Perform propagation state updates for each time window in the order of n=1,2,…,N:

[0110] ;

[0111] In the formula, x (n-1) x (n) Let represent the propagation state vectors of the (n-1)th and nth time windows, respectively; I is an M×M identity matrix.

[0112] The propagation state update consists of two parts:

[0113] First, Ix (n-1) This indicates the self-sustaining / persistent nature of the node's influence.

[0114] Second, αA (n) x (n-1) Indicates along the weighted snapshot G (n) The effective edges in the middle are used for diffusion and propagation.

[0115] 3. Accumulate the propagation status at each moment.

[0116] After each update is completed, the propagation state is accumulated in c:

[0117] ;

[0118] After traversing all the weighted snapshot images, we obtain:

[0119] ;

[0120] By accumulation, we can see that the i-th element c in the propagation accumulation vector c is... i The i-th node v in the unified node set U i The total influence across all time domains, i.e., v within all time windows i The sum of instantaneous influence.

[0121] VI. Output TPC scores and sort and select key nodes.

[0122] The core objective of this section is to transform the obtained propagation accumulation vector into a directly applicable ranking result of node criticality. Essentially, through the logic of "defining quantitative indicators → ranking → filtering," the modeling and propagation calculation of temporal networks described earlier are translated into concrete conclusions of "identifying which nodes can trigger significant propagation effects," directly serving real-world scenarios.

[0123] 1. Define the TPC score for each node.

[0124] For any node v in the node set Ui Its temporal propagation centrality TPC(v i ) is defined as:

[0125] ;

[0126] Among them, c i To propagate the i-th element in the cumulative vector c, and connect it with node v i One-to-one correspondence.

[0127] 2. Obtain the sequence of key nodes

[0128] Press TPC(v) on all nodes i Sort the nodes from largest to smallest to get the sorted result.

[0129] 3. Output the set of key nodes

[0130] Based on application requirements, select the first K nodes (K is between 1% and 10% of the total number of nodes) to form the key node set S. K :

[0131] ;

[0132] Among them, v (1) v (2) ..., v (K) These represent the 1st, 2nd, ..., Kth key nodes, respectively.

[0133] VII. Examples

[0134] like Figure 2 As shown, this embodiment takes a target temporal network with 5 nodes (numbered 1-5) as the research object. The target temporal network contains 3 consecutive time windows (e.g., Figure 3 As shown, the observation start time is divided according to a fixed time step Δt = 2s. The network is an undirected network (with a symmetric weighted adjacency matrix) without original interaction weights (the default weight of a single interaction is 1). The method of this invention identifies the Top-2 key nodes with the strongest propagation influence in the target temporal network, verifying the feasibility and effectiveness of the method.

[0135] Figure 2 The value in the square brackets on the edge between two nodes indicates the time when the two nodes interact. For example, the value [1,4] for node pair (1,2) indicates that node 1 and node 2 interacted at the 1st and 4th seconds.

[0136] Figure 3 The value on the middle edge represents the number of interactions between the two nodes within the corresponding time window, and Time represents the time.

[0137] 1. Data preprocessing and node pair extraction

[0138] After preprocessing the original interaction records of the target temporal network by deduplication, error correction, timestamp unification, and node identifier standardization, as follows: Figure 3 As shown, node pairs that interact within each of the three time windows are extracted and stored in the node set corresponding to that time period:

[0139] Weighted snapshot G of the first time window ([0,2s)) (1) The node pairs that interact include (1,2), (1,5), (2,3), (3,5), (3,4), and (5,5).

[0140] Weighted snapshot G of the second time window ([2s, 4s)) (2) The interactive node pairs in the middle include (1,2), (1,5), (2,3), (2,4), and (4,5).

[0141] Weighted snapshot G of the third time window ([4s, 6s)) (3) The interactive node pairs in the middle include (2,5), (2,4), (3,5), and (4,5).

[0142] 2. Construct the union and weighted adjacency matrix of nodes across the entire time domain.

[0143] Take the union of all node sets to obtain the unified node set U={1,2,3,4,5} (M=5) across the entire time domain.

[0144] Based on the node set U, construct a 5×5 weighted adjacency matrix for each time window (the matrix elements are the sum of the number of interactions between node pairs; the matrix is ​​symmetric in the undirected network):

[0145] The weighted adjacency matrix A for the first time window (1) :

[0146] ;

[0147] The second time window weighted adjacency matrix A (2) :

[0148] ;

[0149] The third time window weighted adjacency matrix A (3) :

[0150] ;

[0151] 3. Calculate the cumulative adjacency matrix and propagation gain coefficient.

[0152] By summing the three weighted adjacency matrices, we obtain the cumulative adjacency matrix A.sum :

[0153] ;

[0154] Calculate the cumulative adjacency matrix A sum The spectral radius r = 7.555715954993894.

[0155] Take the zero constant β=10 -12 Substituting into the formula α=1 / (r+β), the propagation gain coefficient α≈0.13235013 is calculated.

[0156] 4. Iteratively calculate and accumulate the propagation states.

[0157] Initialize the initial propagation state vector x (0) =(1,1,1,1,1) T The propagation accumulation vector is c=(0,0,0,0,0). T ;

[0158] The propagation state vectors for each time period are calculated iteratively according to the time window sequence and accumulated into the cumulative vector:

[0159] First time window:

[0160] x (1) =(I+αA (1) )x (0) =(1.26470026,1.26470026,1.39705039,1.26470026,1.39705039) T .

[0161] Second time window:

[0162] x (2) =(I+αA (2) )x (1) =(1.61698331,1.78436656,1.89920013,1.98678292,1.73181689) T .

[0163] Third time window:

[0164] x (3) =(I+αA (3) )x (2) =(1.61698331,2.50572992,2.12840633,2.45215026,2.71844955) T .

[0165] Overall propagation status:

[0166] c=x (1) +x (2) +x (3) =(4.49866688,5.55479674,5.42465685,5.70363344,5.84731683) T .

[0167] 5. Select key nodes

[0168] Sort the elements of the cumulative vector c (i.e. the total propagation state of each node) in descending order: 5 (5.84731683) > 4 (5.70363344) > 2 (5.55479674) > 3 (5.42465685) > 1 (4.49866688).

[0169] Select the first two nodes with K=2 (since the current total number of nodes is small, the first two are selected; when the number is large, select according to a ratio of 1%-10%) to obtain the key node set {5,4}.

[0170] 6. Implementation Results

[0171] In this embodiment, the method of the present invention successfully identified the Top-2 key nodes with the strongest propagation influence in the target temporal network as nodes 5 and 4. This result verifies that the method of the present invention can effectively characterize the propagation and accumulation characteristics of nodes in temporal evolution, accurately select key nodes, and the calculation process requires no manual parameter tuning, demonstrating strong stability and operability.

[0172] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A method for identifying key nodes in a temporal network based on spectral normalization propagation accumulation, characterized in that, The identification steps include the following: Extract the node pairs that interact within each consecutive time period from the original record of the target temporal network and store them in the node set of the corresponding time period; Take the union of all node sets, and construct a weighted adjacency matrix for each node set based on the union. Each element in the weighted adjacency matrix is ​​the sum of the number of interactions between the corresponding node pairs within the corresponding time period. The cumulative adjacency matrix is ​​obtained by summing all weighted adjacency matrices. The propagation gain coefficient α = 1 / (r+β) is calculated by taking its spectral radius r, where β is a constant to prevent division by zero. Press x (n) =(I+αA (n) )x (n-1) Calculate and aggregate the propagation states of each node in each time period, and sum them to obtain the total propagation states of each node; sort them in descending order and select the nodes corresponding to the top K total propagation states as key nodes; x (n) x (n-1) Let I be the propagation state vector formed by combining the propagation states of each node in the union set during the nth and (n-1)th time periods; let A be the identity matrix; (n) It is the weighted adjacency matrix for the nth time period.

2. The method for identifying key nodes in a temporal network based on spectral normalization propagation accumulation as described in claim 1, characterized in that, If the target temporal network is a directed network, the weighted adjacency matrix is ​​an asymmetric matrix, and the matrix elements correspond only to the total number of interactions between a node and another node; if the target temporal network is an undirected network, the weighted adjacency matrix is ​​a symmetric matrix, and the matrix elements are the total number of interactions between two nodes.

3. The method for identifying key nodes in a temporal network based on spectral normalization propagation accumulation as described in claim 1, characterized in that, When the target temporal network contains the original interaction weights of the interaction events, the sum of the number of interactions is replaced with the sum of the original interaction weights of the corresponding node pairs within the corresponding time period; if the interaction event has no original interaction weights, then the original interaction weight of the interaction event is set to 1.

4. A method for identifying key nodes in a temporal network based on spectral normalization propagation accumulation according to any one of claims 1-3, characterized in that, Initial propagation state vector x (0) It is a vector consisting entirely of 1s.

5. A method for identifying key nodes in a temporal network based on spectral normalization propagation accumulation according to any one of claims 1-3, characterized in that, β takes the value of 10 -12 .

6. A method for identifying key nodes in a temporal network based on spectral normalization propagation accumulation according to any one of claims 1-3, characterized in that, The time period is a time window divided by a fixed time step Δt, and the time interval of the nth time window is... ,in The observation start time of the original record; n=1,2,...,N, where N is the total number of time windows.

7. A method for identifying key nodes in a temporal network based on spectral normalization propagation accumulation according to any one of claims 1-3, characterized in that, K takes values ​​ranging from 1% to 10% of the total number of nodes.

8. A method for identifying key nodes in a temporal network based on spectral normalization propagation accumulation according to any one of claims 1-3, characterized in that, Before extracting the interactive node pairs, the process also includes preprocessing steps such as deduplication, error correction, timestamp unification, and node identifier standardization of the original records.

9. A system for identifying key nodes in a temporal network based on the spectral radius of a cumulative adjacency matrix, characterized in that, A method for identifying key nodes in a temporal network based on spectral normalization propagation accumulation, as described in any one of claims 1-8, comprises: The data preprocessing and extraction module is used to perform deduplication, error correction, timestamp unification and node identifier standardization on the raw records of the target time series network, and extract the node pairs that interact within each consecutive time period and store them in the node set of the corresponding time period. The weighted adjacency matrix construction module is used to obtain the union of all node sets and construct the weighted adjacency matrix for each time period based on the union. The matrix elements are the sum of the number of interactions between node pairs in the corresponding time period or the sum of the original interaction weights. If the target temporal network is a directed network, the weighted adjacency matrix is ​​an asymmetric matrix; if it is an undirected network, it is a symmetric matrix. The propagation gain coefficient calculation module is used to accumulate all weighted adjacency matrices to obtain the cumulative adjacency matrix, extract its spectral radius r, and calculate the propagation gain coefficient α. The propagation state iteration and accumulation module is used to take an all-ones vector as the initial propagation state vector x. (0) Press x (n) =(I+αA (n) )x (n-1) Iteratively calculate the propagation state of each node in each time period, and sum them up to obtain the total propagation state of each node; The key node filtering module is used to sort the total propagation status in descending order, determine K according to the accuracy requirements of the target application scenario, and select the top K corresponding nodes as key nodes.

10. An electronic device, characterized in that, The method includes a processor, a memory, and a computer program stored in the memory. When the processor executes the computer program, it implements the method for identifying key nodes in a time-series network based on spectral normalization propagation accumulation as described in any one of claims 1-8.

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