Network propagation control method and device based on belief transfer

Abstract

The invention discloses a belief transfer-based network propagation control method and device, and the method comprises the steps: 1, combining nodes with small influence in a network to simplify a network structure, and obtaining a coarsened network Gc (Vc, Ec); 2, identifying key nodes from the coarsened network Gc (Vc, Ec) by using an improved belief transfer algorithm (BPD-v), and outputting a propagation control node sequence S; and step 3, partial node sequences are locally optimized in a fine tuning mode, the effect of the control method is further improved, and a better control sequence is further obtained. According to the method provided by the invention, on the premise of meeting the same propagation control effect, the resource consumption required by the control propagation is minimized; or on the premise of controlling the resource consumption in the same way, the propagation control effect is maximized. In addition, the method is low in time complexity and suitable for super-large-scale network propagation control.

Description

Technical Field

[0001] The present invention belongs to the technical field of network science and information processing, and particularly relates to a network propagation control method and device based on belief propagation. Background Art

[0002] In the field of network science, the research and understanding of complex systems occupy a core position, and network models provide a powerful framework for the characterization and analysis of numerous complex systems. For example, cascading events in power grids can be understood through network models, which helps us analyze their vulnerabilities, develop mitigation strategies, and design more robust power infrastructures. Network propagation control is a key mechanism for maintaining the stable operation of networks and ensuring the efficient transmission of information. It is directly related to the realization of many important functions such as reliable power supply in power grids and information diffusion in social networks, and plays a decisive role in improving the overall performance of networks and coping with challenges in complex dynamic environments.

[0003] To effectively solve the problem of network propagation control, researchers have focused on the key nodes of the network and abstracted the network decomposition problem. The core of this problem lies in finding a set of the most critical nodes, and through the precise control of these key nodes, the effective control of the entire network propagation process can be achieved. Specifically, when these key nodes are identified and controlled, the network can achieve maximum fragmentation with the minimum number of node removals, thereby achieving the purpose of controlling network propagation.

[0004] Traditional solutions usually measure the importance of each node based on the network topology and then remove those nodes with the highest importance. Although these methods are simple and easy to calculate, they are static and do not consider the impact of node removal on the network structure. To solve this problem, the adaptive centrality method on this basis will iteratively re-evaluate the importance of each node according to the structure of the remaining network after each removal of the most important node. However, these methods face difficulties due to high time complexity when dealing with large-scale networks. For example, the adaptive betweenness needs to recalculate the number of shortest paths in the remaining network. For a network with n nodes and m edges, its time complexity is O(n 2 m).

[0005] In the network decomposition method based on percolation theory, researchers use percolation theory to analyze the connectivity and robustness of networks. Specifically, it removes nodes or edges by selecting a predefined rule, and then calculates the size of the largest connected component in the remaining network after each removal operation, observing how the size of the largest connected component of the network changes. When the size of the largest connected component decreases significantly, it indicates that the network begins to disintegrate. By analyzing the disintegration process of the network under different attack strategies, the optimal node removal order is determined to fragment the network fastest. Existing methods have been proven to provide approximately optimal solutions in networks with only a small number of long-distance cycles, such as random networks. However, networks in practical scenarios usually contain a large number of cycles because cycles are crucial for maintaining the resilience of the network system. This poses a challenge to existing network decomposition methods because they often fail to effectively find the optimal solution when dealing with these complex network structures. Summary of the Invention

[0006] To overcome the deficiencies of the prior art, the present invention provides a network propagation control method and device based on belief propagation, including: Step 1: Merge nodes with less influence in the network to simplify the network structure, obtaining a coarsened network G c (V c , E c ); Step 2: Use the improved belief propagation algorithm (BPD-v) to identify key nodes from the coarsened network G c (V c , E c ), and output a propagation control node sequence S; Step 3: Locally optimize part of the node sequence by fine-tuning to further improve the effect of the control method, and then obtain a better control sequence. Through the method provided by the present invention, it is possible to minimize the resource consumption required for control propagation under the premise of meeting the same propagation control effect; or maximize the propagation control effect under the premise of the same control resource consumption. In addition, the method of the present invention has a low time complexity and is applicable to the propagation control of ultra-large-scale networks.

[0007] The technical solution adopted by the present invention to solve its technical problems is as follows:

[0008] Step 1: Merge nodes with less influence in the network to simplify the network structure, obtaining a coarsened network G c (V c , E c );

[0009] Step 2: Use the improved belief propagation algorithm BPD-v to identify key nodes from the coarsened network G c (V c , E c ), and output a propagation control node sequence S;

[0010] Step 3: Locally optimize some node sequences through fine-tuning to further improve the effect of the control method and obtain a better control sequence.

[0011] Further, the specific content of the said Step 1 is as follows:

[0012] For a specific network G(V, E), V and E respectively represent the set of vertices and the set of edges in the network; given a node influence sequence S and the size t of the target node set, the PCS algorithm is adopted for the calculation of the coarsened network in the first stage PBF-I.

[0013] Furthermore, the specific PCS algorithm is as follows:

[0014] Step 1-1: Initialize the target node set V t as the first t nodes in S, and calculate the edge set E t among the nodes in V t , that is, the intersection of (V t ×V t ) and E;

[0015] Step 1-2: Based on the remaining network G t (V t ) or G r (V r , E r ) to obtain the set Ω t of connected components to be merged, where V r is the difference set between V and V t , and E r is the intersection of (V r ×V r ) and E;

[0016] Step 1-3: Construct the node set V′ c of the coarsened network, and merge each connected component ω t in Ω i into a new node V′ c contains all such new nodes;

[0017] Step 1-4: Construct the edge set E′ c of the coarsened network, and replace the edge e i connecting the node u in ω t with the node v in V uv by where u belongs to ω i , v belongs to V t , and ω i belongs to Ω t ;

[0018] Step 1-5: Obtain the coarsened network Gc (V c , E c ), where V c is the union of V′ c and V t , and E c is the union of E′ c and E t .

[0019] Furthermore, the specific steps of step 2 are as follows:

[0020] For a specific network G(V, E), given a control coarsening degree threshold θ c , the BPD-v algorithm is used for calculating the propagation control node sequence in the second stage PBF-II.

[0021] Furthermore, the specific BPD-v algorithm is as follows:

[0022] Step 2-1: Use the threshold θ for controlling the coarsening degree c to coarsen the network G(V, E) to obtain a coarsened network G c (V c , E c );

[0023] Step 2-2: Initialize the target node set V′ t as an empty set, set the counter t to 1, and the target node set size t′ to θ c multiplied by the total number of nodes n;

[0024] Step 2-3: When there is a cycle in the remaining network G′ t (V′ t ), perform the following steps: perform a decomposition operation D1 on the remaining network G′ t (V′ t ), remove all nodes with shell number 1 to obtain the largest connected component ^ω t ; on the largest connected component ^ω t , calculate the removal probability of each node v select the node u with the largest removal probability, add it to the target node set V′ t , and update the node influence sequence S, then increment the counter t by 1;

[0025] Step 2-4: End the loop, sort S(1:t - 1) and S(1:n) in descending order according to the node residual degree;

[0026] Step 2-5: Output the propagation control node sequence S.

[0027] Furthermore, the specific steps of step 3 are as follows:

[0028] Step 3-1: Obtain the propagation control node sequence S through the BPD-v algorithm;

[0029] Step 3-2: When the termination condition is not reached, execute the following loop: Calculate the critical threshold q c The corresponding number of nodes i = q c × n Randomly generate two integers t 1 and t 2 , satisfying 1 ≤ t 1 < t 2 ≤ n; Extract the subsequence S j = S(t 1 :t 2 ); If t 1 < i < t 2 , then use the PCS strategy to optimize the subsequence S j to make q c as small as possible; Otherwise, use the PCS strategy to optimize the subsequence S j to make R as small as possible, and then put the optimized subsequence S j back into the sequence S;

[0030] Step 3-3: End the loop to obtain the order parameter R and the critical threshold q c A smaller propagation control node sequence S.

[0031] A network propagation control device based on belief propagation, comprising:

[0032] A coarsening unit, according to the topological structure information of the basic network, uses the PCS strategy to find and merge nodes with influence less than the set threshold, efficiently coarsen the network structure, and provide the coarsened network G c (V c , E c );

[0033] A disassembling unit, using the belief propagation algorithm to identify key nodes in the coarsened network, and determining the propagation control node sequence S according to the influence degree of the nodes on the largest connected component;

[0034] An optimization unit, performing local optimization adjustment on the node sequence obtained by the disassembling unit, further improving the control effect, and obtaining a better control sequence.

[0035] The beneficial effects of the present invention are as follows:

[0036] The method of the present invention can minimize the resource consumption required for controlling the spread or maximize the spread control effect under the premise of satisfying the same spread control effect through a phased strategy, combining the network percolation theory and the belief propagation algorithm. In addition, the method of the present invention has a low time complexity and is applicable to the spread control of ultra-large-scale networks. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 is a flowchart of the method of the present invention;

[0038] Figure 2 is a schematic diagram of the PCS algorithm, (a) the studied network G, (b) the remaining network G after excluding the dotted edges t (V t ), where V r is the set of nodes under the shadow, V t is the rest, and Ω t ={ω 1 , ω 2}, the dotted edges represent the remaining connections between G t (V t ) and V t , (c) the coarsened network G 1 and ω 2 are respectively merged into new nodes u 1 and u 2 ; c ;

[0039] Figure 3 is a schematic diagram of the FFS algorithm (a) two consecutive slices S 1 and S 2 randomly selected from the sequence S to be optimized, the dotted line is the position of the critical threshold q c , (b) the new sequence after optimizing S 1 and S 2 , (c) the slice S c within the range including the position of the critical threshold q 3 , (d) obtaining a better solution by optimizing S 3 to advance the position of the critical threshold q c ;

[0040] Figure 4 is a schematic diagram comparing the computation times (in seconds) of PBF-1 and PBF-3 with those of MSRG, FINDER, and BPD;

[0041] Figure 5 is the number of connected components ω t (V t ) in the remaining network G i ∈Ωt 的Average infection frequency and its magnitude |ω i Box plots of the Pearson correlation coefficient (PCC) between (a)(e) the p2p-Gnutella08 network, (b)(f) the Email-Enron network, (c)(g) the loc-Gowalla network, and (d)(h) the twitter-L network;

[0042] Figure 6 is at an average degree of <k>Critical threshold q of BPD and BPD-v on an ER random network with = 3.50 c And the variation of the calculation time (in seconds) with the network size n, (a) critical threshold q c , (b) calculation time;

[0043] Figure 7 Is the box plot of the critical threshold q c Regarding the coarsening threshold θ c Of the critical threshold q c Where θ Specific implementation mode

[0044] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0045] The present invention provides a network propagation control method and device based on belief propagation.

[0046] Step 1: Merge the nodes with relatively small influence in the network to simplify the network structure, and obtain the coarsened network G c (V c , E c );

[0047] For a specific network G(V, E), V and E respectively represent the vertex set and edge set in the network; given the node influence sequence S and the target node set size t, the PCS algorithm is used for the calculation of the coarsened network in the first stage PBF-I;

[0048] Step 1-1: Initialize the target node set V t As the first t nodes in S, calculate the edge set E t Among the nodes in V t , that is, the intersection of (V t ×V t ) and E;

[0049] Step 1-2: Based on the remaining network G t (V t ) or G r (V r , E r ) to obtain the set Ω t Of the connected components to be merged, where V r Is the difference set of V and V t , E r Is the intersection of (V r ×V r ) and E;

[0050] Step 1-3: Construct the node set V' c Of the coarsened network, and set Ω t Each connected component ω in i is merged into a new node V′ c which contains all such new nodes;

[0051] Steps 1 - 4: Construct the edge set E′ of the coarsened network c , and replace the edge e connecting node u in ω i with node v in V t in the original network with uv where u belongs to ω , v belongs to V i , ω t belongs to Ω i ; t ;

[0052] Steps 1 - 5: Obtain the coarsened network G c (V c , E c ), where V c is the union of V′ c and V t , and E c is the union of E′ c and E t .

[0053] Step 2: Use the improved belief propagation algorithm (BPD - v) to identify key nodes from the coarsened network G c (V c , E c ) and output the propagation control node sequence S;

[0054] For a specific network G(V, E), given the control coarsening degree threshold θ c , the BPD - v algorithm is used for calculating the propagation control node sequence in the second - stage PBF - II.

[0055] Step 2 - 1: Coarsen the network G(V, E) using the threshold θ c for controlling the coarsening degree to obtain the coarsened network G c (V c , E c );

[0056] Step 2 - 2: Initialize the target node set V′ t as an empty set, set the counter t to 1, and the size t′ of the target node set to θ c multiplied by the total number of nodes n;

[0057] Step 2 - 3: When there are loops in the remaining network G′ t (V′ t ), perform the following steps: For the remaining network G′ t (V' t ) Perform the decomposition operation D1 to remove all nodes with shell number 1, and obtain the largest connected component ^ω t . On the largest connected component ^ω t , calculate the removal probability of each node v Select the node u with the largest removal probability, add it to the target node set V', t and update the node influence sequence S, then increment the counter t by 1;

[0058] Step 2-4: End the loop, and sort S(1:t-1) and S(1:n) in descending order according to the node residual degree;

[0059] Step 2-5: Output the propagation control node sequence S.

[0060] Step 3: Locally optimize part of the node sequence by fine-tuning to further improve the effect of the control method and obtain a better control sequence.

[0061] Step 3-1: Obtain the propagation control node sequence S through the BPD-v algorithm;

[0062] Step 3-2: When the termination condition is not reached, execute the following loop: Calculate the critical threshold q c The corresponding number of nodes i = q c × n Randomly generate two integers t 1 and t 2 , satisfying 1 ≤ t 1 < t 2 ≤ n. Extract the subsequence S j = S(t 1 :t 2 ) from the sequence S. If t 1 < i < t 2 , then use the PCS strategy to optimize the subsequence S j to make q c as small as possible; otherwise, use the PCS strategy to optimize the subsequence S j to make R as small as possible. Then put the optimized subsequence S j back into the sequence S;

[0063] Step 3-3: End the loop to obtain a better propagation control node sequence S.

[0064] A network propagation control device based on belief propagation, comprising:

[0065] A coarsening unit, according to the topological structure information of the basic network, uses the PCS strategy to find and merge nodes with smaller influence, efficiently coarsen the network structure, and provide the coarsened network G c (V c ,E c );

[0066] A disassembling unit that uses the belief propagation algorithm to identify key nodes in the coarsened network and determines a propagation control node sequence S based on the influence degree of the nodes on the largest connected component;

[0067] An optimization unit that performs local optimization and adjustment on the node sequence obtained by the disassembling unit to further improve the control effect and obtain a better control sequence.

[0068] Embodiment:

[0069] The present invention is used to solve the problems of large resource consumption or poor propagation control effect in the process of ultra-large-scale network propagation control in the prior art. To overcome these technical problems, the present invention provides a network propagation control method based on belief propagation, and the specific implementation scheme is as follows:

[0070] In the network coarsening stage (PBF-I), the present invention simplifies the network structure by merging nodes with less influence in the network to obtain a coarsened network, thereby improving the efficiency of subsequent calculations. In the key node identification stage (PBF-II), the present invention uses an improved belief propagation algorithm (BPD-v) to identify key nodes from the coarsened network, outputs a propagation control node sequence S, pays special attention to the largest connected component and obtains key nodes from the filtered candidate set. In the local optimization and adjustment stage (PBF-III), the present invention locally optimizes part of the node sequence by fine-tuning to further improve the effect of the control method and obtain a better propagation control node sequence.

[0071] The present invention can combine the network percolation theory and the belief propagation algorithm through a phased strategy, and can minimize the resource consumption required for control on the premise of meeting the same propagation control effect; or maximize the propagation control effect on the premise of the same control resource consumption, and is applicable to various large-scale network propagation control problems, such as information transfer control in social networks, vulnerability analysis of power grids, and formulation of intervention strategies for disease transmission networks, etc., providing effective tools and means for the optimization and management of network structures.

[0072] Such as Figure 1 , the present invention provides a network propagation control device based on belief propagation, mainly through the network coarsening strategy (PCS) proposed in the PBF-I stage to coarsen the network structure, the belief propagation algorithm (BPD-v) based on the coarsened network proposed in the PBF-II stage to obtain an initial propagation control node sequence of the network, and the fragmentation optimization and fine-tuning strategy (FFS) proposed in the PBF-III stage to obtain a better propagation control node sequence, and solves the large-scale network propagation control problem, specifically including the following steps:

[0073] Step 1: Define an unweighted undirected network \(G(V, E)\) consisting of \(n = |V|\) nodes and \(m = |E|\) edges, where \(V\) and \(E\) represent the node set and the edge set respectively. Then define the remaining network \(G_r(V_r, E_r)\) considering a set of nodes, denoted as \(V\) t and \(t = |V\) t |, which are removed or immunized purposefully, i.e., \(V\) r = V\V t and \(E\) r = (V r × V r ) ∩ E. Also denote \(G\) t (V t ) or \(G\) r (V r ) to represent \(G\) r (V r , E r ). Let \(\Omega\) t be the set of components of \(G\) t (V t ), containing all the independent connected components in \(G\) t (V t ), and \(q = t / n\) represents the proportion of nodes in \(V_t\). Then, according to percolation theory, the order parameter \(R(q)\) of \(q\) can be expressed as \(R(q)=|\hat{\omega}\) t | / n, where represents the largest connected component (LCC) of \(G\) t (V t ). Use the critical threshold \(q\) c to measure the earliest moment when \(R(q)\propto O(0)\), i.e., if \(q > q\) c , then there is no GCC in \(G\) t ; if \(q\leq q\) c , then there exists. Define a given tolerance \(\theta\) small enough (e.g., \(\theta = 0.01\)) to approximate the critical threshold, i.e., and assume that if \(R(q)<\theta\), then there is no GCC;

[0074] Step 2: For the network \(G(V, E)\), given the node influence sequence \(S\) and the target node set size \(t\), adopt the PCS algorithm for the calculation of the coarsened network in the first stage PBF-I. Figure 2 is a simple example of the PCS algorithm process. The specific steps of the PCS algorithm are as follows: First, initialize the target node set \(V\) t as the first \(t\) nodes in \(S\), and calculate the edge set \(E\) t between the nodes in \(V\) t , i.e., \(E\) t =(V t × V t ) ∩ E t . Then, based on the remaining network \(G\) t (V t ) to obtain the set Ω of connected components to be merged t . Then, construct the node set V of the coarsened network c , merge each connected component in Ω t into a new node, and include all such new nodes. After that, construct the edge set E of the coarsened network c , and replace the edge e t connecting node u in V r and node v in V uv with where u ∈ ω i , v ∈ V t , ω i ∈ Ω t . Finally, obtain the coarsened network G c (V c , E c ), where V c = V′ c ∪ V t , E c = E′ c ∪ E t .

[0075] More specifically, the PCS algorithm has three coarsening strategies, PCS-1, PCS-2, and PCS-3:

[0076] The core idea of PCS-1 is to simplify the network structure by greedily suppressing the residual degree of each coarsened component . This algorithm uses the residual degree of nodes to select the nodes to be removed preferentially, thereby gradually reducing the scale of the network while maintaining the main structural features of the network. The specific steps are as follows: Given G(V, E), S, and δ 1 , first initialize V t = V, θ = 1, S′ = S, t = t′ = t″ = j = n, i = 0. When t > 0, loop and execute the following operations: When t″ > n - j, loop and execute the following operations: Select node u = S′(t″), and decrement t″ by 1. If in G t-1 (V t \{u}) then remove u from V t , update S(t) = u, decrement t by 1; otherwise, keep u in S′, update S′(t′) = u, decrement t′ by 1, and increment i by 1. After the inner loop ends, update θ = θ × δ 1 , reset t′ and t″ to n, update j to i, and reset i to 0. The loop continues until t = 0, and finally output S;

[0077] PCS-2 is based on the explosive immune method and simplifies the network structure by randomly selecting a candidate node set and greedily removing the node with the smallest residual degree. By randomly selecting the candidate node set, this algorithm avoids the high computational complexity of global search, and at the same time ensures that the nodes selected each time can effectively reduce the connectivity of the network through the greedy strategy. The specific steps are as follows: Given G(V,E), S, and π 1 , initialize the target node set V t = V and t = n. Then, randomly select π t nodes from V 1 to form a candidate set Next, calculate the residual degree of each node and select the node u with the smallest residual degree. Remove the node u from V and update S(t) = u, and subtract 1 from t. Repeat the above steps until t = 0, and finally output S; t

[0078] PCS-3 combines the relational method and the adaptive strategy, and simplifies the network structure by dynamically selecting the candidate node set and greedily removing the key nodes. This algorithm takes into account both the residual degree of the nodes and the size of the connected components, and selects the optimal nodes to be removed through the adaptive strategy. The specific steps are as follows: Given G(V,E), S, ^π 2 and T, initialize the target node set V t = V, S′ = S, t = n, j = 1, π 2 = 2, then, enter the outer loop. When j < T, perform the following operations: Enter the inner loop. When t > 0, randomly select π 2 nodes from {S′(i), i ∈ [max{1, t - n / j}, t]} to form a candidate set Select the optimal node u, whose or |ω i (u)| is the smallest in the candidate set. Remove u from V t and update S′(t), and subtract 1 from t. After the inner loop ends, add 1 to j, π 2 is updated to min{j, ^π 2}, and reset t and V t . If S is better than S, update S; otherwise, restore S to S. The outer loop continues until j reaches the iteration limit T, and finally output S;

[0079] Step 3: In the disassembly (PBF-II) stage, use the improved belief propagation algorithm (BPD-v) to obtain from the coarsened network G c (V c , E c ​)Identify key nodes and output the propagation control node sequence S. The specific operations of BPD-v are as follows: First, use the threshold θ for controlling the coarsening degree c Coarsen the network G(V, E) to obtain the coarsened network G c (V c , E c ), where V c is the set of coarsened nodes and E c is the set of coarsened edges. Then, initialize the target node set V′ t as an empty set, set the counter t to 1, and the size of the target node set t′ to When the remaining network G′ t (V′ t ) is not an acyclic graph, perform the following operations: Decompose the remaining network G′ t (V′ t ) using the decomposition operation D1(G′ t (V′ t )) and remove all nodes with shell number 1 to obtain the largest connected component ^ω t . On ^ω t , calculate the removal probability of each node v through the formula Select the node u with the largest removal probability, add it to the target node set V′ t , update the node sequence S(t), and then increment the counter t by 1. The formula for calculating the removal probability of node v is as follows:

[0080]

[0081] where Γ′(v) is the set of neighbors of node v in G t (V t ), x is a given hyperparameter, is the probability that node i can be correctly removed after removing node v in G t (V t ), is the probability that node i will become the root node of a component with a tree structure after removing node v in G t (V t ). and are obtained by solving the following belief propagation equations:

[0082]

[0083] where z i→v is defined as:

[0084]

[0085] After the loop ends, use the PCS-1 strategy to sort the sequences S(1:t-1) and S(t:n) respectively, and merge the two to output the propagation control node sequence S;

[0086] Step 4: In the fragmented optimization fine-tuning (PBF-III) stage, use the FFS algorithm to locally optimize part of the node sequence by fine-tuning, further improve the effect of the control method, and obtain a better quality control sequence. Figure 3 This is a simple example of the FFS algorithm. The specific operations are as follows: First, obtain the propagation control node sequence S through the BPD-v algorithm. Then, when the termination condition is not reached, execute the following loop: Calculate the critical threshold q c The corresponding number of nodes i = q c ×n Randomly generate two integers t 1 and t 2 , satisfying 1 ≤ t 1 < t 2 ≤ n. Extract the subsequence S j = S(t 1 :t 2 ). If t 1 < i < t 2 , then use the PCS strategy to optimize the subsequence S j to make q c as small as possible; otherwise, use the PCS strategy to optimize the subsequence S j to make R as small as possible. Then put the optimized subsequence S j back into the sequence S. After the loop ends, output the final propagation control node sequence S.

[0087] Figure 4 shows the comparison of different network decomposition methods in terms of computing time, mainly demonstrating the comparison of the PBF-1 and PBF-3 methods with the MSRG, FINDER, and BPD methods in terms of computing time, especially the efficiency in dealing with large-scale networks. The cross (x) indicates that either MSRG and FINDER take too long to obtain results or run out of memory on the LiveJournal network.

[0088] As can be seen from Figure 4 , the PBF-1 method has a significant advantage in terms of computing time, especially when dealing with large-scale networks. The PBF-1 method not only outperforms other methods in terms of the quality of the solution but also shows excellent performance in terms of computing efficiency. Specifically, the average computing time of PBF-1 is about 1,223 times faster than MSRG, about 39 times faster than FINDER, and about 126 times faster than BPD. Although PBF-3 performs better on some networks, the overall computing time is slightly longer than that of PBF-1.

[0089] Figure 5 Evaluates the immune effects of different network dismantling methods under different networks and transmission models, and shows the average infection frequency through box plots at different q values. and the size of the connected component |ω i | in the remaining network, where and the value range of q is [0.01, q c , and the research is carried out at equal intervals of 0.02, that is, the box plot of PCCs at different q values. Figures (a - d) ζ follows the Covid - 19 dynamics, and (e - h) ζ follows the SIR dynamics.

[0090] It can be seen from Figure 5 that under all networks and transmission models, the PCC values of the PBF method are close to 1, indicating that it performs excellently in controlling the spread of the epidemic, can effectively reduce the size of the connected component, and thus reduce the infection frequency. Other methods: HD, CI, BPD, MSRG, and FINDER methods perform better in some networks and transmission models, but perform poorly in other cases, and the PCC values fluctuate greatly.

[0091] Figure 6 Compares the performance of the improved belief propagation algorithm (BPD - v) and the original belief propagation algorithm (BPD) on synthetic networks, and shows the comparison of the critical threshold q c and the calculation time of these two algorithms under different network scales.

[0092] It can be seen from Figure 6 that BPD - v has a significant advantage in calculation time, especially when dealing with large - scale networks. Although the critical threshold q c of BPD - v is slightly inferior to that of BPD, its calculation efficiency is higher and it can complete the calculation in a shorter time. Specifically, the critical threshold q c of BPD - v is slightly higher than that of BPD, but the gap is not large, indicating that BPD - v is equivalent to BPD in terms of network dismantling effect, while the calculation time of BPD - v is significantly less than that of BPD, especially on large - scale networks. The calculation time of BPD - v is about half of that of BPD, indicating that BPD - v has a significant advantage in calculation efficiency.

[0093] Figure 7 Shows the influence of different coarsening thresholds θ c on the network dismantling performance, and shows the change of the critical threshold q c under different coarsening thresholds θ c through box plots.

[0094] It can be seen from Figure 7 that under different θ c the critical threshold q c The change is not significant, indicating that the PBF method has a certain robustness to the selection of the coarsening threshold. From another perspective, although the change in θ c has little impact on q c , choosing a larger value of θ c can, to a certain extent, reduce the calculation time and improve the calculation efficiency.

[0095] Based on percolation theory and belief propagation algorithm, the present invention develops a network propagation control device based on belief propagation. Through coarsening the network and improving the belief propagation algorithm, the present invention can minimize the resource consumption required for control on the premise of meeting the same propagation control effect; or maximize the propagation control effect on the premise of the same control resource consumption, and can be widely applied to various real-world scenarios, such as information transmission control in social networks, vulnerability analysis of power grids, and formulation of intervention strategies for disease transmission networks.

[0096] In summary, the device of the present invention realizes network propagation control based on belief propagation, and the proposed three-stage method can effectively reduce the calculation cost on large-scale networks and achieve efficient and high-quality network propagation control.

[0097] The present invention also provides a network propagation control device based on belief propagation, including:

[0098] A coarsening unit, according to the topological structure information of the basic network, adopts the PCS strategy to find and merge nodes with less influence, coarsen the network structure efficiently, and provide a simplified model for the subsequent disassembly process;

[0099] A disassembly unit, uses the belief propagation algorithm to identify key nodes in the coarsened network, and determines the propagation control node sequence S according to the influence degree of the nodes on the largest connected component;

[0100] An optimization unit, performs local optimization adjustment on the node sequence obtained by the disassembly unit to further improve the control effect and obtain a better control sequence.< / k>

Claims

1. A network communication control method based on belief transfer, characterized in that: The steps include: Step 1: Merge nodes with less influence in the network to simplify the network structure and obtain a coarsened network G c (V c ,E c ); Step 2: Use the improved belief transfer algorithm BPD-v to extract the coarsened network G c (V c ,E c ) to identify key nodes and output a propagation control node sequence S; Step 3: Locally optimize some node sequences through fine-tuning to further improve the effect of the control method and obtain a better control sequence.

2. A network communication control method based on belief transfer according to claim 1, characterized in that: The step 1 is specifically as follows: For a specific network G(V,E), V and E represent the vertex set and edge set in the network respectively; given the node influence sequence S and the target node set size t, the PCS algorithm is used to calculate the coarsened network in the first stage PBF-I.

3. A network communication control method based on belief transfer according to claim 2, characterized in that: The PCS algorithm is specifically as follows: Step 1-1: Initialize the target node set V t For the first t nodes in S, calculate V t The edge set E between the midpoints t , that is (V t ×V t ) and E; Step 1-2: Based on the residual network G t (V t ) or G r (V r ,E r ) Get the set of connected components that need to be merged Ω t , where V r V and V t The difference of r =V r ×V r ) and E; Step 1-3: Construct the node set V′ of the coarsened network c , Ω t Each connected component ω in i Merge into a new node V′ c Contains all such new nodes; Step 1-4: Construct the edge set E′ of the coarsened network c , connect ω in the original network i Midpoint u and V t Edge e of node v uv Replace with where u belongs to ω i , v belongs to V t ,ω i Belong to Ω t ; Step 1-5: Get the coarsened network G c (V c ,E c ), where V c V′ c With V t The union of c E′ c With E t The union of .

4. A network communication control method based on belief transfer according to claim 3, characterized in that: The step 2 is specifically as follows: For a specific network G(V,E), given a control coarsening threshold θ c ,The BPD-v algorithm is adopted for the calculation of the propagation control node sequence in the second stage PBF-II.

5. A network communication control method based on belief transfer according to claim 4, characterized in that: The BPD-v algorithm is as follows: Step 2-1: Use a threshold value θ to control the degree of coarsening c The network G(V,E) is coarsened to obtain the coarsened network G c (V c ,E c ); Step 2-2: Initialize the target node set V′ t is an empty set, set the counter t to 1, and the target node set size t′ to θ c Multiply by the total number of nodes n; Step 2-3: When the residual network G′ t (V′ t ) when there is a cycle, perform the following steps: t (V′ t ) Perform decomposition operation D1, remove all nodes with shell number 1, and obtain the maximum connected component ^ω t ; In the largest connected component ^ω t Calculate the removal probability of each node v Select the node u with the highest probability of removal and add it to the target node set V′ t , and update the node influence sequence S, and then add 1 to the counter t; Step 2-4: End the loop and sort S(1:t-1) and S(1:n) by node residual degree from large to small; Step 2-5: Output the propagation control node sequence S.

6. A network communication control method based on belief transfer according to claim 5, characterized in that: The step 3 is as follows: Step 3-1: Obtain the propagation control node sequence S through the BPD-v algorithm; Step 3-2: When the termination condition is not met, execute the following loop: Calculate the critical threshold q c The corresponding number of nodes i = q c ×n Randomly generate two integers t1 and t2, satisfying 1 ≤ t1 < t2 ≤ n; Extract the subsequence S from the sequence S j = S(t1:t2); If t1 < i < t2, optimize the subsequence S using the PCS strategy j Make q c As small as possible; Otherwise, optimize the subsequence S using the PCS strategy j Make R as small as possible, and then put the optimized subsequence S j Back into the sequence S; Step 3-3: End the loop and get the order parameter R and critical threshold q c A smaller sequence S of propagation control nodes.

7. A device using the network transmission control method as claimed in claim 1, characterized in that: include: The coarsening unit uses the PCS strategy to find and merge nodes whose influence is less than the set threshold according to the topological structure information of the basic network, and efficiently coarsens the network structure to provide a coarsened network G for subsequent processes. c (V c ,E c ); Disassemble the unit, use the belief transfer algorithm to identify key nodes in the coarsened network, and determine the propagation control node sequence S according to the influence of the node on the maximum connected component; The optimization unit performs local optimization and adjustment on the node sequence obtained by the disassembly unit to further improve the control effect and obtain a better control sequence.

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