A network propagation control method and device based on belief propagation

Abstract

The application discloses a network propagation control method and device based on belief propagation, which comprises the following steps: step one: merging nodes with small influence in the network to simplify the network structure, and obtaining a coarsened network G c (V c ,E c ); step two: identifying key nodes from the coarsened network G c (V c ,E c ) by using an improved belief propagation algorithm (BPD-v), and outputting a propagation control node sequence S; and step three: locally optimizing part of the node sequence in a fine-tuning manner, further improving the effect of the control method, and then obtaining a better control sequence. By the method provided by the application, the resource consumption required for control propagation can be minimized under the premise of meeting the same propagation control effect, or the propagation control effect can be maximized under the premise of the same control resource consumption. In addition, the method has low time complexity and is suitable for super-large-scale network propagation control.

Description

TECHNICAL FIELD

[0001] The application 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

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

[0003] In order 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 is to find a set of key nodes, and through the precise control of these key nodes, the entire network propagation process can be effectively controlled. Specifically, after identifying these key nodes and controlling them, the network can be maximally fragmented with the smallest 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 the most important nodes. Although this method is simple and easy to calculate, it is static and does not consider the impact of node removal on the network structure. To solve this problem, adaptive centrality methods based on this 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 in dealing with large-scale networks due to high time complexity. For example, adaptive betweenness needs to recalculate the number of shortest paths in the remaining network, and for a network containing n nodes and m edges, its time complexity is O(n 2 m).

[0005] In network decomposition methods 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, then calculates the size of the largest connected component in the remaining network after each removal operation, and observes how the size of the largest connected component 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 make the network break down the fastest. Existing methods have been proven to provide near-optimal solutions in networks containing only a small number of long-distance cycles, such as random networks. However, networks in real-world scenarios often contain a large number of cycles, as cycles are essential to maintaining the resilience of network systems. This poses a challenge to existing network decomposition methods, as they often fail to find optimal solutions when dealing with these complex network structures. SUMMARY

[0006] In order to overcome the deficiencies of the prior art, the present application provides a network propagation control method and device based on belief propagation, which comprises the following steps: step 1: merging nodes with small influence in the network to simplify the network structure, obtaining a coarsened network G c (V c ,E c );Step 2: identifying key nodes from the coarsened network G c (V c ,E c ) using an improved belief propagation algorithm (BPD-v), and outputting a propagation control node sequence S;Step 3: locally optimizing part of the node sequence in a fine-tuning manner to further improve the effect of the control method, and thus obtaining a better control sequence. Through the method provided by the present application, the resource consumption required for control propagation can be minimized under the premise of meeting the same propagation control effect, or the propagation control effect can be maximized under the premise of the same control resource consumption. In addition, the method has low time complexity and is suitable for large-scale network propagation control.

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

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

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

[0010] Step 3: Partially optimize part of node sequence by fine-tuning to further improve the effect of the control method, and obtain a better control sequence.

[0011] Further, the step 1 is specifically:

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

[0013] Further, the PCS algorithm is specifically as follows:

[0014] Step 1-1: Initialize the target node set V t For the first t nodes in S, calculate the edge set E between the nodes in V t , i.e., the intersection of (V t ×V t ) and E; t

[0015] Step 1-2: Based on the remaining network G t (V t ) or G r (V r , E r ), obtain the set Ω of connected components that need to be merged t , where V r is the difference set of 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' of the coarsened network c , 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' of the coarsened network c , replace the edge e i connecting the node u in ω t and the node v in V uv in the original network with , 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 G​c (V c c ), wherein V c is the union of V′ c and V t , and E c is the union of E′ c and E t .

[0019] Further, the step 2 is specifically as follows:

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

[0021] Further, the BPD-v algorithm is specifically as follows:

[0022] Step 2-1: using the control roughness degree threshold θ c , roughen the network G(V,E) to obtain the rough 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 as 1, and the target node set size t′ as θ c times the total number of nodes n;

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

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

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

[0027] Further, the step 3 is specifically as follows: ​

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

[0029] Step 3-2: When the termination condition is not reached, perform the following loop: calculate the critical threshold q c The corresponding number of nodes i = q c Two integers t1 and t2 satisfying 1 ≤ t1 < t2 ≤ n are randomly generated; a subsequence S j = S(t1:t2) is extracted from the sequence S; if t1 < i < t2, the subsequence S j is optimized using the PCS strategy so that q c is as small as possible; otherwise, the subsequence S j is optimized using the PCS strategy so that R is as small as possible, and then the optimized subsequence S j is put back into the sequence S.

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

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

[0032] A coarsening unit adopts a PCS strategy to find and merge nodes with influence less than a set threshold according to the topological structure information of a basic network, efficiently coarsens the network structure, and provides a coarsened network G c (V c , E c ) for subsequent processes.

[0033] A disassembling unit identifies key nodes in the coarsened network using a belief propagation algorithm, determines a propagation control node sequence S according to the influence degree of the nodes on the maximum connected component, and

[0034] An optimization unit performs local optimization adjustment on the node sequence obtained by the disassembling unit, further improves the control effect, and obtains a better control sequence.

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

[0036] The method of the present application can 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, by using a phased strategy combined with network percolation theory and belief propagation algorithm. In addition, the method of the present application has low time complexity and is suitable for large-scale network propagation control. BRIEF DESCRIPTION OF DRAWINGS

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

[0038] Figure 2 This is a schematic diagram of the PCS algorithm. (a) is the network G under study, and (b) is the remaining network G after excluding the dashed edges. t (V t ), where V r V is the set of nodes under the shadow. t For the remainder, and Ω t ={ω1,ω2}, the dashed side represents G t (V t ) and V t The remaining connections, (c) the coarsened network G after merging ω1 and ω2 into new nodes u1 and u2 respectively. c ;

[0039] Figure 3 This is a schematic diagram of the FFS algorithm. (a) Two consecutive slices S1 and S2 are randomly selected from the sequence S to be optimized. The dashed line represents the critical threshold q. c (a) the position of the sequence after optimizing S1 and S2, and (b) the new sequence within the range including the critical threshold q. c The slice S3,(d) at the location is optimized to make the critical threshold q c By moving the position forward, a better solution was obtained.

[0040] Figure 4 This is a diagram showing the calculation time (in seconds) of PBF-1 and PBF-3 compared with MSRG, FINDER, and BPD.

[0041] Figure 5 Under different q values, the remaining network G t (V t Connected component ω in ) i ∈Ωt 的 Average infection frequency and its size |ω i Box plots of Pearson correlation coefficient (PCC) for |, (a)(e) p2p-Gnutella08 network, (b)(f) Email-Enron network, (c)(g) loc-Gowalla network, (d)(h) twitter-L network;

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

[0043] Figure 7 It concerns the coarsening threshold θ c Critical threshold q c The box plot, where θ c ∈[0.000,0.009] and with equal intervals of 0.001. Detailed Implementation

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

[0045] This invention provides a network propagation control method and apparatus based on belief transmission.

[0046] Step 1: Simplify the network structure by merging nodes with less influence in the network to obtain the coarsened network G. c (V c E c );

[0047] For a specific network G(V,E), V and E represent the set of vertices and the set of edges in the network, respectively; given the node influence sequence S and the size t of the target node set, the PCS algorithm is used to calculate the coarsening network in the first stage PBF-I;

[0048] Step 1-1: Initialize the target node set V t Given the first t nodes in S, calculate V. t The edge set E between nodes t That is (V) t ×V t The intersection of E and E;

[0049] Steps 1-2: Based on the remaining network G t (V t ) or G r (V r E r ) Obtain the set Ω of the connected components that need to be merged. t V r For V and V t The difference set, E r For (V) r ×V r The intersection of E and E;

[0050] Steps 1-3: Construct the node set V′ of the coarsened network c , will Ω t Each connected component ω in i Merge into a new node V′ c Includes all such new nodes;

[0051] Steps 1-4: Construct the edge set E′ of the coarsened network c Connect ω in the original network i Middle nodes 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 Belongs to Ω t ;

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

[0053] Step 2: Utilize the improved belief transfer algorithm (BPD-v) from the coarsened network G c (V c E c Identify key nodes in the process and output the propagation control node sequence S;

[0054] For a specific network G(V,E), given a control coarsening 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: Use a threshold θ to control the degree of coarsening. c Coarsening the network G(V,E) yields the coarsened network G. c (V c E c );

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

[0057] Steps 2-3: When the remaining network G′ t (V′ t If a loop exists in the remaining network G′, perform the following steps: t (V t ) performs a decomposition operation D1 to remove all nodes with shell number 1, obtaining the maximum connected component ^ω t . On the maximum connected component ^ω t , the removal probability of each node v is calculated The node u with the largest removal probability is selected and added to the target node set V' t , and the node influence sequence S is updated, and then the counter t is incremented by 1;

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

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

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

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

[0062] Step 3-2: When the termination condition is not met, perform the following loop: calculate the critical threshold q c , and 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 j from the sequence S. If t1<i<t2, optimize the subsequence S j using the PCS strategy so that q c is as small as possible; otherwise, optimize the subsequence S j using the PCS strategy so that R is as small as possible. Then put the optimized subsequence S j back into the sequence S.

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

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

[0065] A coarsening unit that uses the PCS strategy to find and merge nodes with small influence based on the topology information of the base network, efficiently coarsens the network structure, and provides a coarsened network G c (V c , E c );

[0066] The disassembling unit identifies key nodes in the coarsened network by using the belief propagation algorithm, and determines the propagation control node sequence S according to the influence degree of the nodes on the maximum connected component;

[0067] The optimization unit locally optimizes and adjusts the node sequence obtained by the disassembling unit, further improves the control effect, and obtains a better control sequence.

[0068] Embodiments:

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

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

[0071] The present application can minimize the resource consumption required for control under the premise of meeting the same propagation control effect by using the phased strategy combined with the network percolation theory and the belief propagation algorithm, or maximize the propagation control effect under the premise of the same control resource consumption, and is suitable for various large-scale network propagation control problems, such as information propagation control in social networks, vulnerability analysis of power networks, and intervention strategy formulation of disease transmission networks, etc. The present application provides an effective tool and means for optimization and management of network structure.

[0072] As Figure 1 The present application provides a network propagation control device based on belief propagation, which mainly uses the network coarsening strategy (PCS) proposed in the PBF-I phase to coarsen the network structure, uses the belief propagation algorithm (BPD-v) based on the coarsened network proposed in the PBF-II phase to obtain an initial propagation control node sequence of the network, and uses the fragmentation optimization fine-tuning strategy (FFS) proposed in the PBF-III phase to obtain a better propagation control node sequence, thereby solving the problem of large-scale network propagation control, and specifically comprising 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 edge set, respectively. Then define the residual network Gr(Vr,Er), considering a set of nodes, denoted as V. t And t = |V t | is targeted for removal or immunity, i.e., V r =V\V t And E r =(V r ×V r )∩E. Also use G. t (V t ) or G r (V r ) to represent G r (V r E r Let Ω t For G t (V t A set of components, including G t (V t Let q represent the proportion of nodes in Vt, where q = t / n. Then, according to percolation theory, the order parameter R(q) of q can be expressed as R(q) = |^ω t | / n, where G represents t (V t The largest connected component (LCC) of . Using a critical threshold q c The earliest moment when R(q)∝O(0) is when q>q c Then G t GCC does not exist in q; if q ≤ q c If so, then it exists. Define a sufficiently small given tolerance θ (e.g., θ = 0.01) to approximate the critical threshold, i.e. Furthermore, it is assumed that if R(q) < θ, then GCC does not exist;

[0074] Step 2: For network G(V,E), given the node influence sequence S and the target node set size t, the PCS algorithm is used for the calculation of the coarsening network in the first stage PBF-I. Figure 2 This is a simple example of the PCS algorithm flow. The specific steps of the PCS algorithm are as follows: First, initialize the target node set V. t Given the first t nodes in S, calculate V. t The edge set E between nodes t E t =(V t ×V t )∩E t Next, based on the remaining network G t (V t ) Obtain the set Ω of the connected components that need to be merged. t Then, construct the node set V of the coarsened network. c , will Ω t Each connected component in the network is merged into a new node, containing all such new nodes. Then, the edge set E of the coarsened network is constructed. c Connecting V in the original network t Middle nodes u and V r Edge e of node v uv Replace 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 greedily suppress the residual of each coarsening component. To simplify the network structure, this algorithm utilizes the residual value of nodes to select nodes for priority removal, thereby gradually reducing the network size while preserving its main structural features. The specific steps are as follows: Given G(V,E), S, and δ1, first initialize V... t =V, θ=1, S′=S, t=t′=t″=j=n, i=0. When t>0, perform the following operations in a loop: When t″>nj, perform the following operations in a loop: Select node u=S′(t″) and decrement t″ by 1. If in G t-1 (V t \{u}) in Then u from V t Remove the element from S', update S(t) = u, and 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 × δ1, reset t' and t″ to n, update j to i, and reset i to 0. The loop continues until t = 0, and finally outputs 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 each selected node can effectively reduce the connectivity of the network through the greedy strategy. The specific steps are as follows: Given G(V, E), S, and π1, initialize the target node set V t = V and t = n. Then, randomly select π1 nodes from V t 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 Update S(t) = u, and decrement t by 1. 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 a candidate node set and greedily removing key nodes. This algorithm takes into account both the residual degree of nodes and the size of connected components, and selects the optimal nodes to remove through the adaptive strategy. The specific steps are as follows: Given G(V, E), S, ^π2, and T, initialize the target node set V t = V, S′ = S, t = n, j = 1, π2 = 2. Then, enter the outer loop. When j < T, perform the following operations: Enter the inner loop. When t > 0, randomly select π2 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 Update S′(t), and decrement t by 1. After the inner loop ends, increment j by 1, update π2 to min{j, ^π2}, 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 identify key nodes from the coarsened network G c (V c , E c ) and output the propagation control node sequence S. The specific operation of BPD-v is as follows: First, use the threshold θ c that controls the coarsening degree to coarsen the network G(V, E) to obtain the coarsened network G c (V c , E​ c ), where V c is the set of nodes after coarsening, E c is the set of edges after coarsening. Then, the set of target nodes V′ t is initialized as an empty set, and a counter t is set as 1, and the size of the set of target nodes t′ is When the remaining network G′ t (V′ t ) is not a loop-free graph, the following operations are performed: performing a decomposition operation D1(G′ t (V′ t )) on the remaining network G′ t (V′ t ), removing all nodes with shell number 1, to obtain a maximum connected component ^ω t . On ^ω t , the removal probability of each node v is calculated by the formula The node u with the largest removal probability is selected and added to the set of target nodes V′ t , and the node sequence S(t) is updated, and then the counter t is incremented 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 removed correctly after removing node v in G t (V t ), is the probability that node i will become the root node of a tree-structured component 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, the PCS-1 strategy is used to sort the sequences S(1:t-1) and S(t:n) respectively, and the two sequences are merged to output the propagation control node sequence S;

[0086] Step 4: In the fragmentation optimization fine-tuning (PBF-III) stage, the FFS algorithm is used to locally optimize part of the node sequence in a fine-tuned manner, further improving the effectiveness of the control method and obtaining a better control sequence. Figure 3 is a simple example of the FFS algorithm. The specific operation is as follows: first, obtain the propagation control node sequence S by the BPD-v algorithm. Then, when the termination condition is not reached, execute the following loop: calculate the critical threshold q c The corresponding node number i = q c ×n randomly generate two integers t1 and t2, satisfying 1 ≤ t1 < t2 ≤ n. Extract the subsequence S j = S(t1:t2) from the sequence S. If t1 < i < t2, optimize the subsequence S j using the PCS strategy so that q c is as small as possible; otherwise, optimize the subsequence S j using the PCS strategy so that R is 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 The comparison of different network fragmentation methods in computing time is given, mainly showing the comparison of PBF-1 and PBF-3 methods with MSRG, FINDER and BPD methods in computing time, especially in processing large-scale networks. Among them, the cross (x) means that MSRG and FINDER either take too long to get the result or run out of memory on the LiveJournal network.

[0088] It can be seen from Figure 4 that the PBF-1 method has a significant advantage in computing time, especially when dealing with large-scale networks. PBF-1 method not only outperforms other methods in the quality of the solution, but also performs well in 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 PBF-1.

[0089] Figure 5 The immune effect evaluation of different network fragmentation methods under different networks and transmission models is given. Through the box plot, the Pearson correlation coefficient (PCC) between the average infection frequency and the size of the connected component in the remaining network |ω i | is shown, where and the value range of q is [0.01, q c The study was conducted at equal intervals of 0.02, i.e., box plots of PCCs under different q values. Figure (ad)ζ follows Covid-19 dynamics, and (eh)ζ follows SIR dynamics.

[0090] Depend on Figure 5 It can be seen that under all network and transmission models, the PCC value of the PBF method is close to 1, indicating that it performs well in controlling the spread of the epidemic and can effectively reduce the size of connected components, thereby reducing the infection frequency. Other methods: HD, CI, BPD, MSRG, and FINDER methods perform well under some network and transmission models, but perform poorly under other conditions, with large fluctuations in PCC values.

[0091] Figure 6 The performance comparison between the improved belief transfer algorithm (BPD-v) and the original belief transfer algorithm (BPD) on synthetic networks is presented, demonstrating the critical threshold q of the two algorithms under different network sizes. c Comparison with calculation time.

[0092] Depend on Figure 6 It is evident that BPD-v has a significant advantage in computation time, especially when processing large-scale networks. Although the critical threshold q of BPD-v... c Slightly inferior to BPD, but with higher computational efficiency, it can complete calculations in a shorter time. Specifically, the critical threshold q of BPD-v... c Slightly higher than BPD, but the difference is not large, indicating that BPD-v is comparable to BPD in network decomposition performance. However, BPD-v's computation time is significantly less than that of BPD, especially on large-scale networks, where BPD-v's computation time is about half that of BPD, indicating that BPD-v has a significant advantage in computational efficiency.

[0093] Figure 7 Different coarsening thresholds θ are given. c The impact on network decomposition performance is illustrated using box plots at different coarsening thresholds θ. c Below, the critical threshold q c The changes.

[0094] Depend on Figure 7 It can be seen that at different θ c Below, 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 θ c Changes in q c The impact is not significant, but choosing a larger θ c Values ​​can reduce computation time and improve computation efficiency to some extent.

[0095] The application is based on seepage theory and belief propagation algorithm, and develops a network propagation control device based on belief propagation. The device can minimize the resource consumption required for control 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, and can be widely applied to various real scenes, such as information propagation control in social networks, vulnerability analysis of power networks, and intervention strategy formulation of disease transmission networks.

[0096] In summary, the device realizes network propagation control based on belief propagation, and the three-stage method can effectively reduce the calculation cost on a large-scale network, and realize efficient and high-quality network propagation control.

[0097] The application also provides a network propagation control device based on belief propagation, which comprises:

[0098] The coarsening unit adopts a PCS strategy to find and merge nodes with small influence according to the topological structure information of the basic network, efficiently coarsens the network structure, and provides a simplified model for the subsequent disassembly process.

[0099] The disassembly unit identifies key nodes in the coarsened network by using a belief propagation algorithm, and determines a propagation control node sequence S according to the influence degree of the nodes on the maximum connected component.

[0100] The optimization unit locally optimizes and adjusts the node sequence obtained by the disassembly unit, further improves the control effect, and obtains a better control sequence.< / k>

Claims

1. A network propagation control method based on belief propagation, characterized by, The method comprises the following steps: Step 1: Combine the nodes with less influence in the network to simplify the network structure, and obtain a coarsened network ; Step 2: Identify key nodes from the coarsened network using the improved belief propagation algorithm BPD-v, output the propagation control node sequence ;​ The BPD-v is specifically as follows: Step 2-1: using a threshold to control the roughening degree performing roughening processing on the network to obtain a roughened network ; Step 2-2: Initialize target node set Set counter to 0 Set target node set size to 1 Set Multiply by total number of nodes ; Step 2-3: When there are loops in the remaining network , the following steps are performed: for the remaining network , perform a decomposition operation D1 to remove all nodes with shell number 1 to obtain the maximal connected component ; on the maximal connected component , calculate the removal probability of each node ; select the node with the largest removal probability and add it to the target node set and update the node influence sequence , then add 1 to the counter . Step 2-4: End the loop, sort the nodes by the residual degree from large to small respectively and ​ Step 2-5: Output the propagation control node sequence ; Step 3: locally optimizing the partial node sequence.

2. The belief propagation based network control method of claim 1, wherein, The step 1 is specifically as follows: For a particular network , , respectively denote the set of vertices and edges in the network; given a node influence sequence and a target node set size , the PCS algorithm is used for the calculation of the coarsened network in the first phase PBF-I.

3. The belief propagation based network control method of claim 2, wherein, The PCS algorithm is specifically as follows: Step 1-1: Initialize the target node set For each node, compute the edge set between it and the target node set , i.e. the intersection of the node's edge set and the target node set Steps 1-2: Remaining network based or get the set of connected components that need to be merged where is the difference set of and is the intersection of and Steps 1-3: Construct the node set of the coarsened network ,Will Each connected component in Merge into a new node , Includes all such new nodes; Step 1-4: Constructing the edge set of the roughed network , the edges between the nodes in the original network are replaced by , where belongs to , the edges between the nodes in the original network are replaced by , where belongs to , belongs to , belongs to ; Step 1-5: Obtain roughed network wherein is and the union of is and the union of 4. The method of claim 3, wherein, The step 2 is specifically as follows: For a particular network , given a control roughness threshold , employing the BPD-v algorithm for the computation of the sequence of propagating control nodes in the second stage PBF-II.

5. The belief propagation based network propagation control method of claim 4, wherein, The step 3 is specifically as follows: Step 3-1: Obtain the propagation control node sequence by BPD-v algorithm ; Step 3-2: When the termination condition is not reached, perform the following loop: calculate the critical threshold The corresponding number of nodes Randomly generate two integers And , satisfying ; Extract the subsequence from the sequence ; if , optimize the subsequence so that is as small as possible; otherwise, optimize the subsequence so that is as small as possible, and then put the optimized subsequence back into the sequence ; Step 3-3: End the cycle, resulting in the sequence parameter and critical threshold smaller sequence of propagation control nodes .

6. An apparatus for using the network traffic control method of claim 1, wherein The method comprises the following steps: The roughening unit adopts a PCS strategy to find and merge nodes with influence less than a set threshold according to topology structure information of a basic network, efficiently roughens the network structure, and provides a roughened network for a subsequent process ; The disassembly unit identifies key nodes in the coarsened network by using a belief propagation algorithm, and determines a sequence of propagation control nodes according to the influence degree of the nodes on the maximum connected component ; An optimization unit locally optimizes and adjusts the node sequence obtained by the disassembling unit, further improves the control effect, and obtains a better control sequence.

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