A method for maximizing infrastructure network reliability based on nonlinear heat conduction model

Abstract

The present invention discloses a method for maximizing the reliability of an infrastructure network based on a nonlinear heat conduction model. First, the heat conduction model is mapped onto the infrastructure network model; then the original network topology is input into the mapped model; the original network is divided into several communities using a community partitioning algorithm, the nodes with the lowest betweenness centrality in each community are searched and sorted, and the nodes constituting the minimum connected graph among these nodes are connected to obtain the ideal optimal edge to be added for the first time; the network is totalized using a malicious attack model; and the algorithm iteration model is iterated to search and sort the sequence of edges in the largest connected component of the network after the attack, and the network resilience increment of the node with the lowest betweenness centrality in each community is calculated and sorted, and the optimal edge sequence to be added is obtained. After cyclic iteration, the optimal edge sequence is finally output. The present invention can maximize network resilience by modifying as little network topology as possible.

Description

Technical Field

[0001] The present invention belongs to the technical field of Internet of Things, and in particular relates to a method for maximizing the reliability of an infrastructure network. Background Art

[0002] As an important and challenging research issue in the field of complex networks, the reliability of large-scale infrastructure networks has attracted more and more attention from researchers in recent years. However, in real networks, especially large-scale infrastructure networks, network cascading failures play a vital role in improving network reliability. Once certain nodes or edges in the network are maliciously attacked, the entire network will face the risk of paralysis. For example, in power grid networks, airport networks, and road networks, network cascading failures are fatal to the entire network. Therefore, it is necessary to study how to restore the original function of the network as much as possible after a network cascading failure.

[0003] Through the summary of previous related research, it is found that the current research on improving network reliability is mainly based on dynamic or static network topology research, and is studied from three levels: evaluation, prediction, and recovery. Most of them only focus on the changes in network topology, or find the best data set of edge swap (ES) or edge addition (EA) to maximize network reliability. The ES strategy improves network reliability by changing the network topology to a specific onion-like structure. However, for large networks, the computational overhead is very large and has a great impact on the original network function. The EA strategy restores the network by adding appropriate edges, but it often brings the problem of computational overhead, such as the recently proposed backward addition algorithm PA (Posteriorly Addition). However, neither ES nor EA can globally optimize network reliability. In addition, they cannot maintain the original network topology characteristics.

[0004] There have been a lot of pioneering research works in this field, but most of the existing research focuses on providing protection measures for key nodes or using linear models to evaluate network reliability, and these methods have changed the original network structure to varying degrees. However, there are relatively few studies on nonlinear models to improve network reliability, mainly because nonlinear modeling methods will greatly increase the algorithm calculation complexity and cause greater damage to the original network topology.

[0005] Therefore, how to modify as few edges as possible through methods such as ES / EA to restore the original network functions to the maximum extent after a network collapse, and ensure that the maximum network resilience increment is considered during the network recovery process, is a difficult problem currently faced in improving the reliability of large-scale infrastructure networks. Summary of the invention

[0006] In order to overcome the shortcomings of the prior art, the present invention provides a method for maximizing the reliability of an infrastructure network based on a nonlinear heat conduction model. First, the heat conduction model is mapped onto the infrastructure network model; then the original network topology is input into the mapped model; the original network is divided into several communities using a community partitioning algorithm, the nodes with the lowest betweenness centrality in each community are searched and sorted, and the nodes that form the minimum connected graph among these nodes are connected to obtain the ideal optimal edge to be added for the first time; the network is totalized using a malicious attack model; the algorithm iteration model is then iterated to search and sort the sequence of edges in the largest connected component of the network after the attack, the network resilience increment of the node with the lowest betweenness centrality in each community is calculated and sorted, and the optimal edge sequence to be added is obtained. After cyclic iteration, the optimal edge sequence is finally output. The present invention can maximize network resilience by modifying as little network topology as possible.

[0007] The technical solution adopted by the present invention to solve the technical problem includes the following steps:

[0008] Step 1: Map the heat conduction model onto the infrastructure network model;

[0009] Step 1-1: Thermal conduction model FPU-β model definition:

[0010]

[0011] in, is the kinetic energy of node i and all links to this node, is the potential energy of node i and all links to this node; H i represents the Hamiltonian of node i, p i represents the momentum of node i;

[0012] Step 1-2: Let the atoms of nodes and links have the same Hamiltonian:

[0013]

[0014] And the same potential energy:

[0015]

[0016] Among them, x i 、x j are the displacements of the ith node and the jth node from the equilibrium position, ki is the number of connections connected to the ith node, is the average weight on the link and ∈[0, 1], β is a nonlinear constant, and each link is regarded as a two-dimensional FPU-β chain. In this case, equation (3) is redefined as:

[0017]

[0018] Where q corresponds to the percentage of nodes removed after a malicious attack in an interdependent network, G(q) is the percentage of nodes that remain valid after the network is attacked, and the percentage of failed nodes is 1-G(q);

[0019] Step 1-3: For the two nodes i0 and j0 connected to the initial fault source, the fixed boundary method is used to simulate the fault source with the Langevin model, and the uncorrelated strength is determined by the Langevin model dynamic friction coefficient γ, and γ∈[1,100);

[0020] Then the motion of other nodes follows the canonical equation:

[0021]

[0022] Among them, q i represents the critical threshold of node i;

[0023] Let the Langevin source node act on nodes i0 and j0, and calculate their r ,q l and q t The number of edges added at the critical value K:

[0024]

[0025] Among them, p l 、p t and p r Represent the critical threshold q l ,q t and q r The nodal momentum at x l 、x t and x r Represent the critical threshold q l ,q t and q r The displacement of the node at the position away from the equilibrium position; Γ l , Γ t and Γ r Represent the critical threshold q l ,q t and q r Gaussian white noise at , and satisfies:

[0026]

[0027] Among them, k B is the Boltzmann constant, K qt is the critical value q tThe number of edges added to the network at; δ(.) is the error compensation function; Γ l，r，t (q t ) indicates q t Gaussian white noise at ;

[0028] Step 1-4: When the network reaches a balanced state, that is, when the network elasticity is the largest, the local edge increase of each node is defined as:

[0029]

[0030] The local energy density is defined as:

[0031]

[0032] Among them, x i-1 represents the displacement of node i backward from the equilibrium position;

[0033] The heat flow along the link is defined as:

[0034]

[0035] Among them, V represents the total potential energy of the current link;

[0036] Dimensional analysis:

[0037] Among them, u = u (t, x, y, z) is the characterization of thermal motion (in the rectangular coordinate system (x, y, z) in space); k is the diffusion rate, c is the specific heat capacity, ρ is the material density, t is the diffusion propagation time, is the rate of change of the diffusion propagation distance x, Indicates the heat production per unit time;

[0038]

[0039] Where J represents the Joule heat during diffusion; s, m, and K represent the diffusion area, material mass, and Fahrenheit temperature, respectively;

[0040] The relationship between diffusion propagation distance and diffusion time is:

[0041] x 2 ∝a 2 t (13)

[0042] Calculation of elastic potential energy increment:

[0043]

[0044] Among them, G I(q) represents the percentage of valid nodes in the I state after the network is attacked by malicious attacks and q nodes are removed, and G0(q) represents the percentage of valid nodes in the initial state after the network is attacked by malicious attacks and q nodes are removed;

[0045] In q t ,q r ,q l The changes in elastic potential energy after adding different numbers of edges are studied at three critical values:

[0046]

[0047] Among them, x i-Δ represents the displacement of the node with a position difference of Δ from the equilibrium position; x j-Δ represents the displacement of the node with a position difference of Δ from the equilibrium position of node j;

[0048] Substituting formula (15) into formula (1), the system kinetic energy and potential energy at different critical values ​​of heat in the network are obtained as shown in formula (16):

[0049]

[0050]

[0051] Step 2: Input the network topology of the infrastructure network model on the mapped model;

[0052] Step 3: Use the community partitioning algorithm to divide the network topology of the infrastructure network model into several communities, find and sort the nodes with the lowest betweenness centrality in each community, and connect the nodes that form the minimum connected graph among these nodes to obtain the ideal optimal edge sequence;

[0053] Step 4: Attack the infrastructure network model with the malicious attack model and use the breadth-first algorithm to store the failed node sequence;

[0054] Step 5: Iterate the model using the algorithm to restore the network;

[0055] Step 5-1: Find and sort the sequence of edges in the largest connected component of the network after the attack;

[0056] Step 5-2: Calculate and sort the network resilience increment ΔV of the nodes with the lowest betweenness centrality in each community i , that is, the elastic potential energy of the network, and obtain the actual optimal edge sequence;

[0057] Step 5-3: Determine whether the actual optimal edge sequence is less than or equal to the ideal optimal edge sequence. If so, execute step 6; if not, add a new edge sequence to change the network topology and maximize the network elasticity coefficient, and return to step 5-1;

[0058] Step 6: Output the optimal edge sequence.

[0059] Preferably, β=2, γ=1, k B =1.

[0060] Preferably, the malicious attack model is a CI or HAD model.

[0061] Preferably, the algorithm iteration model is a JI or GS or SOR model.

[0062] Preferably, the community partitioning algorithm is Leiden.

[0063] Preferably, when adding a new edge sequence in step 5-3, the newly added edges must be located within the remaining largest connected patch of the network and added in a sequence from low to high betweenness centrality.

[0064] Preferably, the network elasticity coefficient R is:

[0065]

[0066]

[0067] in, represents the degree of unrelatedness between communities. is the number of overlapping nodes in the community, and N represents the total number of network nodes.

[0068] The beneficial effects of the present invention are as follows:

[0069] The present invention overcomes the shortcomings of the previous ES / EA algorithm that while improving network resilience, it also causes varying degrees of damage to the network topology. By adding multiple edges at one time, the network resilience is improved while ensuring the stability of the network topology and maximizing network elasticity. BRIEF DESCRIPTION OF THE DRAWINGS

[0070] Figure 1 It is a flow chart of the method of the present invention.

[0071] Figure 2 The following are result diagrams of the embodiments of the present invention under different attack models and different critical thresholds. (a) The results of the CI and HAD attack models under different critical thresholds under the JI iterative model; (b) The results of the CI and HAD attack models under different critical thresholds under the GS iterative model; (c) The results of the CI and HAD attack models under different critical thresholds under the SOR iterative model.

[0072] Figure 3The relationship between network reliability R and the added edge percentage q obtained by the embodiment of the present invention using the MA algorithm, PA algorithm and EA algorithm. (a) Experimental results under CI and HAD attacks in the ER network; (b) Experimental results under CI and HAD attacks in the US Powergrid network; (c) Experimental results under CI and HAD attacks in the OpenFlights network.

[0073] Figure 4 1 is a graph showing the relationship between network reliability R and network modularity Q under the action of the method of the present invention in the embodiment of the present invention. (a) Experimental results under different conditions in the ER network; (b) Experimental results under different conditions in the US Powergrid network; (c) Experimental results under different conditions in the OpenF1ights network. DETAILED DESCRIPTION

[0074] The present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0075] The purpose of the present invention is to provide a method for maximizing the reliability of large-scale infrastructure networks based on a nonlinear heat conduction model. Under the premise of ensuring network reliability, the original network topology is kept intact as much as possible, and the effectiveness of the proposed scheme is evaluated by introducing a new network reliability evaluation indicator (community irrelevance).

[0076] A method for maximizing infrastructure network reliability based on a nonlinear heat conduction model comprises the following steps:

[0077] Step 1: Map the heat conduction model onto the infrastructure network model;

[0078] Step 1-1: Thermal conduction model FPU-β model definition:

[0079]

[0080] in, is the kinetic energy of node i and all links to this node, is the potential energy of node i and all links to this node; p i represents the momentum of node i; n ij is the number of nodes on the link i→j. Each node on the link is connected only to its two nearest neighbors, and node i is connected to its k neighbors. i The nearest neighbor.

[0081] Step 1-2: Let the atoms of nodes and links have the same Hamiltonian:

[0082]

[0083] And the same potential energy:

[0084]

[0085] Among them, x i 、x j are the displacements of the ith node and the jth node from the equilibrium position, k i is the number of connections to the ith node, is the average weight on the link and β is a nonlinear constant, set to 2, and each link is regarded as a two-dimensional FPU-β chain. In this case, equation (3) is redefined as:

[0086]

[0087] Where q corresponds to the percentage of nodes removed after a malicious attack in an interdependent network, G(q) is the percentage of nodes that remain valid after the network is attacked, and the percentage of failed nodes is 1-G(q);

[0088] Step 1-3: For the two nodes i0 and j0 connected to the initial fault source, the fixed boundary method is used to simulate the fault source with the Langevin model. The uncorrelated strength is determined by the Langevin model dynamic friction coefficient γ, and γ∈[1,100), which is set to 1;

[0089] Then the motion of other nodes follows the canonical equation:

[0090]

[0091] Let the Langevin source node act on nodes i0 and j0, and calculate their r ,q l and q t The number of edges added at the critical value K:

[0092]

[0093] satisfy:

[0094]

[0095] Among them, k B is the Boltzmann constant, set to 1, is the critical value q t The number of network edges added at ;

[0096] Step 1-4: The experiment is run with a time step of 0.01 and a dimensionless time of 10 6 After that, when the network reaches a balanced state, that is, when the network elasticity is the largest, the local edge increase of each node is defined as:

[0097]

[0098] The local energy density is defined as:

[0099]

[0100] The heat flow along the link is defined as:

[0101]

[0102] Dimensional analysis:

[0103]

[0104] Where k is the diffusivity, c is the specific heat capacity, and ρ is the material density;

[0105]

[0106] The relationship between diffusion propagation distance and diffusion time is:

[0107] x 2 ∝a 2 t (13)

[0108] Calculation of elastic potential energy increment:

[0109]

[0110] In q t ,q r ,q l The changes in elastic potential energy after adding different numbers of edges are studied at three critical values:

[0111]

[0112] Substituting formula (15) into formula (1), the system kinetic energy and potential energy at different critical values ​​of heat in the network are obtained as shown in formula (16):

[0113]

[0114]

[0115] Step 2: Input the network topology of the infrastructure network model on the mapped model;

[0116] Step 3: Use the community partitioning algorithm to divide the network topology of the infrastructure network model into several communities, find and sort the nodes with the lowest betweenness centrality in each community, and connect the nodes that form the minimum connected graph among these nodes to obtain the ideal optimal edge sequence;

[0117] Step 4: Attack the infrastructure network model with the malicious attack model and use the breadth-first algorithm to store the failed node sequence;

[0118] Step 5: Iterate the model using the algorithm to restore the network;

[0119] Step 5-1: Find and sort the sequence of edges in the largest connected component of the network after the attack;

[0120] Step 5-2: Calculate and sort the network resilience increment ΔV of the nodes with the lowest betweenness centrality in each community i , that is, the elastic potential energy of the network, and obtain the actual optimal edge sequence;

[0121] Step 5-3: Determine whether the actual optimal edge sequence is less than or equal to the ideal optimal edge sequence. If so, execute step 6; if not, add a new edge sequence to change the network topology and maximize the network elasticity coefficient, and return to step 5-1;

[0122] Step 6: Output the optimal edge sequence. Specific embodiment:

[0124] For the convenience of description, the algorithm proposed in the present invention is abbreviated as MA (multiple addition) below.

[0125] The present invention designs a new edge adding algorithm MA, which adds multiple edges in each iteration to restore the target network according to the modular characteristics after community division. Figure 1 The whole process of the MA algorithm is shown. The present invention adopts two widely used node attack models CI (Collective Influence) and HDA (High Degree Adaptive) algorithm, and uses JI (Jacabi iterative model), GS (Gauss-Seidel method) and SOR (continuous overrelax) as iterative models. The present invention first divides the original network G (v, e) into α communities using a community partitioning algorithm, and divides the nodes with the lowest betweenness in each community into β communities. Then, the nodes in β are connected to form a minimum connected graph. Therefore, all edges in the minimum connected graph are regarded as ideal optimal edges. The actual execution process of the algorithm is shown in Algorithm 1.

[0126]

[0127] The breadth-first search (BFS) algorithm is used to store the network nodes that failed after the target network was attacked. Most importantly, after sorting the nodes of each LCC in descending order of inter-relationship, the lowest inter-relationship node of each LCC can be regarded as the real one. The actual network reliability increment ΔV of each node iIt can be calculated by formula (15). Use the existing iterative algorithm to execute step 4 and step 5 respectively until e real <e ideal , and at the same time real As the optimal output. Otherwise, return to step 2.

[0128] Experimental environment:

[0129] The experimental data sets used in the present invention include an artificially synthesized ER network and two real power grid networks, USpowergrid and OpenFlights.

[0130] Experimental results:

[0131] For a network, the function is usually determined by the characteristics of the network topology. It is crucial to keep the network topology unchanged while optimizing the network recovery capability. The present invention tests the topology changes of the ER network, the US Powergrid network and the OpenFlights network (as shown in Table 1). The results show that the topological characteristics of the network after being attacked do not change much by adopting the solution of the present invention.

[0132] In this embodiment, the MA, PA and EA algorithms are iterated 1000 times under the most widely used iteration models JI, GS and SOR. Under the JI iteration model, the number of added edges accounts for 3.4% of all edges (e.g. Figure 2 (a)), under the GS iterative model, the number of added edges is 1.4% (e.g. Figure 2 (b)), under the SOR iterative model, the number of added edges is 2.3% (e.g. Figure 2 (c)). It should be noted that the proposed MA algorithm only adds 29 edges to obtain the maximum reliability under different attack modes, which accounts for only 0.5% of all edges in the original network. In addition, the efficiency of the MA algorithm is 2.2 times that of the PA algorithm and 6.8 times that of the EA algorithm. Finally, under different iteration models, the absolute error of the edge added by the algorithm is less than 12.

[0133] exist Figure 3 In this paper, the present invention uses curves of different colors to simulate the impact of MA, PA and EA algorithms attacked by CI attack model on network reliability R. The corresponding three light curves represent the effects under HDA attack. Figure 3 Specifically, the present invention obtains the best network reliability performance in the US Powergrid network, Q=0.83, and the number of edges added after the CI attack is 3.4% of all the edges in the original network.

[0134] The present invention analyzes the optimal, suboptimal and worst results of the proposed MA algorithm respectively, where The values ​​are set to 0.2, 0.4, 0.6 and 0.8 respectively (such as Figure 4 Note that the ideal result is when both the attack method and the number of attacked edges are known. The intermediate result is achieved when the attack method is known but the number of attacked edges is unknown. When both the attack method and the number of attacked edges are unknown, a fragile result is obtained.

[0135] Depend on Figure 4 It can be seen that the network reliability R of different networks increases monotonically with the increase of community irrelevance. When the value is 0.8, the network reliability R reaches the optimal value. In particular, the proposed MA algorithm performs best in the highly modular US Powergrid network, followed by the OpenFlights network, and the ER network is the worst. It can be seen that the degree of network modularity has an important impact on network reliability. In addition, the higher the degree of network modularity, the greater the difference in network performance.

[0136] In order to further prove that the scheme of the present invention has non-negligible superiority in all benchmark tests, the present invention analyzes the algorithm time complexity of different recovery algorithms (see Table 2). It can be concluded that the most widely used EA algorithm has the lowest time complexity, but at the expense of the worst network reliability. In addition, the performance of the PA algorithm is much higher than that of the ES algorithm, and the time complexity of the two is basically the same. Obviously, the MA algorithm proposed in the present invention takes time complexity into account while maintaining a high recovery rate.

[0137] Table 1 Changes in network topology characteristics after using the proposed model

[0138]

[0139] is the network threshold, K is the average clustering coefficient of the network, L is the average path length of the network, Q is the degree of network modularity, and S is the number of strongly connected components in the network.

[0140] Table 2: Performance comparison of different algorithms.

[0141]

[0142] α is the number of iterations of the algorithm, K is the number of newly added edges, and w is the number of edges that need to be exchanged.

Claims

1. A method for maximizing infrastructure network reliability based on a nonlinear heat conduction model, characterized in that: The steps include: Step 1: Map the heat conduction model onto the infrastructure network model; Step 1-1: Thermal conduction model FPU-β model definition: in, is the kinetic energy of node i and all links to this node, is the potential energy of node i and all links to this node; Hi represents the Hamiltonian of node i, p i represents the momentum of node i; Step 1-2: Let the atoms of nodes and links have the same Hamiltonian: And the same potential energy: Among them, x i 、x j are the displacements of the ith node and the jth node from the equilibrium position, k i is the number of connections to the ith node, is the average weight on the link and β is a nonlinear constant, and each link is regarded as a two-dimensional FPU-β chain. In this case, equation (3) is redefined as: Where q corresponds to the percentage of nodes removed after a malicious attack in an interdependent network, G(q) is the percentage of nodes that remain valid after the network is attacked, and the percentage of failed nodes is 1-G(q); Step 1-3: For the two nodes i0 and j0 connected to the initial fault source, the fixed boundary method is used to simulate the fault source with the Langevin model, and the uncorrelated strength is determined by the Langevin model dynamic friction coefficient γ, and γ∈[1,100); Then the motion of other nodes follows the canonical equation: Among them, q i represents the critical threshold of node i; Let the Langevin source node act on nodes i0 and j0, and calculate their r ,q l and q t The number of edges added at the critical value K: Among them, p l 、p t and p r Represent the critical threshold q l ,q t and q r The nodal momentum at x l 、x t and x r Represent the critical threshold q l ,q t and q r The displacement of the node at the position away from the equilibrium position; Γ l , Γ t and Γ r Represent the critical threshold q l ,q t and q r Gaussian white noise at , and satisfies: Among them, k B is the Boltzmann constant, is the critical value q t The number of edges added to the network at; δ(.) is the error compensation function; Γ l，r，t (q t ) indicates q t Gaussian white noise at ; Step 1-4: When the network reaches a balanced state, that is, when the network elasticity is the largest, the local edge increase of each node is defined as: The local energy density is defined as: Among them, x i-1 represents the displacement of node i backward from the equilibrium position; The heat flow along the link is defined as: Where V represents the total potential energy of the current link; Dimensional analysis: Among them, u = u (t, x, y, z) is the characterization of thermal motion, and the rectangular coordinate system (x, y, z) is determined in space; κ is the diffusion rate, c is the specific heat capacity, ρ is the material density, and t is the diffusion propagation time. is the rate of change of the diffusion propagation distance x, Indicates the heat production per unit time; Where J represents the Joule heat during diffusion; s, m, and K represent the diffusion area, material mass, and Fahrenheit temperature, respectively; The relationship between diffusion propagation distance and diffusion time is: x 2 ∝a 2 t (13) Calculation of elastic potential energy increment: Among them, G I (q) represents the percentage of valid nodes in the I state after the network is attacked by malicious attacks and q nodes are removed, and G0(q) represents the percentage of valid nodes in the initial state after the network is attacked by malicious attacks and q nodes are removed; In q t ,q r ,q l The changes in elastic potential energy after adding different numbers of edges are studied at three critical values: Among them, x i-Δ represents the displacement of the node with a position difference of Δ from the equilibrium position; x j-Δ represents the displacement of the node with a position difference of Δ from the equilibrium position of node j; Substituting formula (15) into formula (1), the system kinetic energy and potential energy at different critical values ​​of heat in the network are obtained as shown in formula (16): Step 2: Input the network topology of the infrastructure network model on the mapped model; Step 3: Use the community partitioning algorithm to divide the network topology of the infrastructure network model into several communities, find and sort the nodes with the lowest betweenness centrality in each community, and connect the nodes that form the minimum connected graph among these nodes to obtain the ideal optimal edge sequence; Step 4: Attack the infrastructure network model with the malicious attack model and use the breadth-first algorithm to store the failed node sequence; Step 5: Iterate the model using the algorithm to restore the network; Step 5-1: Find and sort the sequence of edges in the largest connected component of the network after the attack; Step 5-2: Calculate and sort the network resilience increment ΔV of the nodes with the lowest betweenness centrality in each community i , that is, the elastic potential energy of the network, and obtain the actual optimal edge sequence; Step 5-3: Determine whether the actual optimal edge sequence is less than or equal to the ideal optimal edge sequence. If so, execute step 6; if not, add a new edge sequence to change the network topology and maximize the network elasticity coefficient, and return to step 5-1; Step 6: Output the optimal edge sequence.

2. A method for maximizing infrastructure network reliability based on a nonlinear heat conduction model according to claim 1, characterized in that: β=2,γ=1,k B =1.

3. The method for maximizing infrastructure network reliability based on a nonlinear heat conduction model according to claim 1, characterized in that: The malicious attack model is a CI or HAD model.

4. The method for maximizing infrastructure network reliability based on a nonlinear heat conduction model according to claim 1, characterized in that: The algorithm iteration model is a JI or GS or SOR model.

5. The method for maximizing infrastructure network reliability based on a nonlinear heat conduction model according to claim 1, characterized in that: The community partitioning algorithm is Leiden.

6. The method for maximizing infrastructure network reliability based on a nonlinear heat conduction model according to claim 1, characterized in that: When adding a new edge sequence in step 5-3, the newly added edges must be located in the largest remaining connected piece of the network and added in a sequence from low to high betweenness centrality.

7. The method for maximizing infrastructure network reliability based on a nonlinear heat conduction model according to claim 1, characterized in that: The network elasticity coefficient R is: in, represents the degree of unrelatedness between communities. is the number of overlapping nodes between communities, and N represents the total number of network nodes.

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