A method for calculating the fault tolerance performance of interdependent circuit networks

Abstract

The invention discloses a method for calculating the fault tolerance performance of interdependent circuit networks, in the field of fault tolerance, robustness and reliability of interdependent circuit networks. Most dependent networks are not fragile in the real world. Therefore, in order to find a more accurate model to explain the mechanism of actual cascading failures, the relationship between fault tolerance allocation and the robustness of interdependent networks is deeply determined, thereby solving the problem of calculating the fault tolerance performance of interdependent circuit networks. Through the fault tolerance performance of the joint circuit calculated by the present invention, the circuit structure or the node capacity in the circuit is changed in real time, the fault tolerance of the joint circuit is maximized, and the robustness and stability of the joint circuit are increased.

Description

Technical Field

[0001] The present invention relates to the field of fault tolerance, robustness and reliability of interdependent circuit networks. Background Art

[0002] In recent years, interdependent networks (INs) or multiplexed networks composed of multiple layers of networks and their interconnections have attracted widespread attention. The typical topological structure diagram of an interdependent or interdependent network is shown in the figure below. Figure 5 As shown, A = {A1, A2, ..., A5} represents a network, which forms a dependent network with network B = {B1, B2, ..., B5}.

[0003] In August 2003, a power outage in the United States triggered a cascading failure of the power grid, leaving much of the population in Canada and the eastern United States in the dark. This devastating blackout sparked research on the cascading failure mechanisms and corresponding fault tolerance of interdependent networks. From the perspective of percolation theory, Parshani et al. proposed a top-level framework to study the robustness of interconnected networks (INs). (Parshani, R., Buldyrev, S. V. & Havlin, S., “Interdependent networks: Reducing the coupling strength leads to a change from a first to second order percolation transition”, Phys. Rev. Lett. 105, 048701 (2010)) and found that INs behave very differently from isolated networks and are fundamentally more sensitive to perturbations. Specifically, unlike the continuous phase transition of a single network, an abrupt phase transition, called a first-order phase transition, was observed in INs. This means that when INs operate near the phase transition point, even a small number of node failures can trigger a cascading failure, leading to the collapse of the entire system.

[0004] Generally speaking, when the interdependence between networks decreases below a critical value, a continuous phase transition occurs. Recently, Schneider et al. and Gao et al. conducted a series of studies, Gaogao Dong, Jianxi Gao, Lixin Tian, Ruijin Du, and Yinghuan He, "Percolation of partially interdependent networks under targeted attack", Phys. Rev. E 85, 016112 (2012), Valdez et al. showed that the robustness of interdependent systems can be enhanced by backing up highly interdependent nodes. They also observed that some special dependency schemes (i.e., classified dependency, which means that nodes with similarity in two networks are prone to dependency) have more robust characteristics than networks with random dependencies. In order to describe the fault tolerance of interdependent networks, the concept of network resilience was proposed. Previous research on resilience has focused primarily on the mechanisms of failure propagation and the collapse of interdependent systems. Many studies have focused on the design of mitigation and control strategies to mitigate and avoid catastrophic events and to remediate failures when they occur. Regardless, a thorough and complete understanding of cascading failures facilitates the design of strategies to enhance the robustness of interdependent systems. Percolation models have been used as a theoretical measure of robustness by monitoring how macroscopic correctness varies with the amount of microscopic damage to individual elements throughout the network. In particular, all models of the aforementioned tree-like networks can be easily and accurately solved using the generating function method. This approach also allows for in-depth theoretical analysis of percolation models to illustrate the difficulties of fault propagation in complex networks. Recently, scholars have proposed a number of extended percolation models to describe different ways in which nodes in interdependent networks are coupled or associated, such as the l-hop percolation model, the k-core percolation model, and the clique percolation model. Furthermore, in addition to direct connections, hidden dependencies between physically unconnected nodes are also considered as inherent fragments of complex interdependence in real complex systems. These percolation models exhibit discontinuous phase transitions, as opposed to the continuous phase transitions in classical percolation.

[0005] In fact, Buldyrev et al. first proposed using the generating function method to solve the phase transition problem in an interdependent network in their groundbreaking and original percolation model study. The network consists of two networks, named A and B, whose degree distributions are P and A (k) and P B(k). Generally speaking, network A contains N nodes, as does circuit B. Individual nodes in circuit A are connected to one and only one node in circuit B via dependent links, and vice versa. It is important to note that dependency links differ from connectivity edges within each network; if the other connected node at one end of a dependency link is first removed, the node in the network is also removed. This phenomenon reflects the reality that a power plant failure in a power grid can cause connected equipment to immediately cease operation due to power outages. Clearly, any node isolated from the largest interconnected branch in a single network is unable to function due to physical disconnection from the majority of other nodes. It is important to note that connectivity edges connect every node in a network giant component, and the properties of a mutually connected giant component (MCGC) ensure that every subordinate node in another network giant component is also connected. Therefore, an MCGC can be viewed as a stable structure for the remaining interdependent network, satisfying the condition that cascading failures are impossible. As mentioned earlier, numerous studies have shown that altering the dependencies between nodes within a network or between different networks significantly reduces the stability of interdependent systems.

[0006] Many previous literatures have assumed that the relationship between the two connected nodes in the two networks is completely correlated when studying cascading failures JianxiSV, BuldyrevG, Havlin Sand Stanley. “Robustness of a network formed by interdependent networks with a one-to-one correspondence of dependent nodes”. Phys. Rev. E 85 066134, which means that the failure of a node in one network will cause the immediate failure of its connected node in the other network. Summary of the Invention

[0007] Most dependency networks are not fragile in the real world. Therefore, in order to find a more accurate model to explain the mechanism of actual cascading failures and to deeply determine the relationship between fault tolerance allocation and the robustness of interdependent networks, we can solve the problem of calculating the fault tolerance performance of interdependent circuit networks.

[0008] A method for calculating the fault tolerance performance of interdependent circuit networks is proposed. The method is based on the following foundations:

[0009] First, node dependencies are hierarchical. That is, if the behavior of node a is partially affected by another node b, then node a is said to be partially dependent on node b. On the other hand, if the behavior of node a is completely controlled by node b, then node a is said to be completely dependent on node b.

[0010] Second: Every node in the network is fault-tolerant;

[0011] Third: Each node in the network has different fault tolerance capabilities;

[0012] Fourth: The fault tolerance of a node is closely related to its importance in the network;

[0013] Fifth: The degree of a node is related to its fault tolerance. This relationship is divided into three categories: positive correlation, negative correlation, and random correlation. Positive correlation means that nodes with high degree have high fault tolerance, while nodes with low degree have low fault tolerance. Negative correlation means that nodes with high degree have low fault tolerance, while nodes with low degree have high fault tolerance. Random correlation means that the relationship between degree and fault tolerance of a node is random.

[0014] The method includes:

[0015] Step 1: Calculate the degree distribution generating function of circuit A and circuit B;

[0016] G 0A (ξ)=∑ k P A (k)ξ k

[0017] G 0B (ξ)=∑ k P B (k)ξ k

[0018] Among them, P A (k) represents the node degree distribution of circuit A, P B (k) represents the node degree distribution of circuit B, ξ k represents the kth node variable;

[0019] Calculate the underlying branching process generation function of circuit A and circuit B,

[0020] G 1A (ξ)=G′ 0A (ξ) / G′ 0A (1)

[0021] G 1B (ξ)=G′ 0B (ξ) / G′ 0B (1)

[0022] Among them, G 1A (ξ) represents the underlying branching process generation function of circuit A, G 1B (ξ) represents the underlying branching process generation function of circuit B, G′ 0A (ξ), G′ 0B(ξ) represents G 0A (ξ), G 0B (ξ) The first-order derivative with respect to ξ, G′ 0A (1) G′ 0B (1) respectively represent G′ 0A (ξ) and G′ 0B (ξ) Function value when ξ = 1;

[0023] Step 2: Randomly select a node in circuit A and circuit B to start, and calculate the volume of the largest connected branch of circuit A and circuit B;

[0024] (1) When the degree of nodes in circuits A and B is positively correlated with their fault tolerance;

[0025] Step 2.1.1: Calculate the probability of a maximum connected branch appearing on a randomly selected link in circuits A and B using the following two equations:

[0026]

[0027] Among them, R A 、R B represents the probability of the largest connected branch appearing when a link is randomly selected in circuit A or circuit B, p[1-G 1A (1-R A )]、p[1-G 1B (1-R B )] represents the probability that the links randomly selected from the randomly selected nodes of circuit A and circuit B will lead to the maximum connected branch of circuit A. represents the probability that a randomly selected link starting from a randomly selected node in circuit a will lead to the maximum connected branch in circuit A, α represents the fault tolerance of the current circuit node, circuit a belongs to circuit A, the superscript * represents the random edge that can reach the maximum cluster, p[1-G 0A (1-R A )] represents the survival probability of the nodes that depend on a given node;

[0028]

[0029] Among them, k max represents the maximum order of the nodes in the corresponding circuit, p(k) represents the probability of occurrence of a node with degree k, represents the degree of a randomly selected node, <k>Indicates the average degree;

[0030] Step 2.1.2: Calculate the volume of the largest connecting branch of circuit A and circuit B;

[0031]

[0032] Among them, p[1-G 0A (1-R A )] represents the survival probability of the nodes that depend on a given node, represents the survival probability of the node that depends on the randomly selected node;

[0033] (2) When the degree of nodes in circuit A and circuit B is negatively correlated with their fault tolerance;

[0034] Step 2.2.1: Calculate the probability of a maximum connected branch appearing on a randomly selected link in circuits A and B using the following two equations:

[0035]

[0036] Step 2.2.2: Calculate the volume of the largest connecting branch of circuit A and circuit B;

[0037]

[0038] (3) When the degrees of nodes in circuits A and B are randomly correlated with their fault tolerance;

[0039] In step 2.3.1, calculate the probability of a maximum connected branch appearing on a randomly selected link in circuits A and B using the following two equations:

[0040] R A =p[G1(1-R A )]·[1-G0(1-R B )]+p·α[1-G1(1-αR A )]·G0(1-R B )·(1-p)

[0041] R B =p[1-G1(1-R B )]·[1-G0(1-R A )]+α[1-G1(1-αR B )]·{1-p[1-G0(1-R A )]}·(1

[0042] -p)

[0043] Step 2.3.2: Calculate the volume of the largest connecting branch of circuit A and circuit B;

[0044] S A =p[G 0A (1-R A )]·[1-G 0B (1-R B )]+p·[1-G 0A (1-αR A )]·G 0B (1-R B )·(1-p)

[0045] S B =p[1-G 0B (1-R B )]·[1-G 0A (1-R A )]+[1-G 0B (1-αR B )]·{1-p[1-G 0A (1-R A )]}

[0046] (1-p)

[0047] Step 3: Add the volumes of the largest connected branches of circuits A and B calculated in step 2 and compare them with the threshold. If the volume is greater than the threshold, it is considered that the joint fault tolerance performance of circuits A and B is poor; otherwise, the fault tolerance performance is good.

[0048] By calculating the fault tolerance performance of the combined circuit according to the present invention, the circuit structure or the node capacity in the circuit is changed in real time, the fault tolerance of the combined circuit is maximized, and the robustness and stability of the combined circuit are increased. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 For <k A >= <k B >=4 interdependent coupled random networks, with the largest connected branch (S A and S B ) and the fraction of the original retained vertex p. The solid line represents the theoretical prediction, and the discrete symbols (squares and triangles) represent the simulation results of 20 implementations on a network of 105 nodes.

[0050] Figure 2 Schematic diagram of the percolation characteristics of interdependent scale-free networks under different fault tolerance dependencies; for the interdependent scale-free network with kmin = 4, kmax = 316 and λ = 2.7, the maximum connected branch (S A and S B ) corresponds to the portion of the original retained vertex p, the solid line represents the theoretical prediction, and the discrete symbols (squares and triangles) represent the schematic diagram of the simulation results implemented 20 times on a network of 105 nodes;

[0051] Figure 3 For <k A >= <k B Different values of α and ρ for interdependent coupled random networks with >=4, the set and The percolation model transition region, including the phase transition boundary; Region I and Region II represent the simulation results of the first and second phase transition regions, respectively, after 20 realizations on a random network of 105 vertices. The boundary between Regions I and II represents a schematic diagram of the theoretically predicted results.

[0052] Figure 4 For the interdependent scale-free network with kmin = 4, kmax = 316 and λ = 2.7, for different values of α and ρ, the set and The percolation transition zone, including the phase transition boundary, is shown in Figure 1. Regions I and II represent the simulation results for the first and second phase transition regions, respectively, after 20 implementations on a scale-free network with 105 vertices. The boundary between Regions I and II represents a schematic diagram of the theoretically predicted results.

[0053] Figure 5 Schematic diagram of the typical topological structure of interdependent or interdependent networks. DETAILED DESCRIPTION

[0054] Through simulation results and theoretical predictions, Figure 1 The maximum connection branch size (given by S) of the interdependent coupled random network is shown. A and S B The relationship between the original retained vertex p is shown in Figure 2. The theoretical and simulation results are completely consistent. In addition, it can be found that for a fixed ρ value, S A or S B There is a mutation from zero to non-zero value. At the same time, for the case of high degree nodes with high fault tolerance, that is, positive dependency, the change of the maximum connection branch size S A and S B In fact, similar results can be found for negative and random dependencies as for positive dependencies, but their percolation model transition points are different.

[0055] Figure 2 The percolation characteristics of interdependent scale-free networks under different tolerances are shown. It can be found that for the positive dependency case, S A or S B There is a continuous percolation transition from zero to non-zero, which means that under the positive dependence assumption, there is a crossover point to ensure a smooth percolation transition from first order to second order, while for the negative dependence case and the random dependence case, S A or S B The transition becomes abrupt, as will be explained in the next paragraph.

[0056] Figure 3 The conclusion is that enhancing fault tolerance and increasing the proportion of nodes with zero fault tolerance in the entire interdependent network can lead to a shift from a first-order phase transition to a second-order phase transition. A first-order phase transition is a sudden change in the state of a system under environmental attack, while a second-order phase transition is a smooth, gradual change in the state of an interdependent system. If a system cannot avoid failure, in practical applications, it is more effective if the system undergoes a second-order phase transition because the slow-moving nature of the fault can control its propagation.

[0057] Figure 3 and Figure 4 All of these studies show that if the fault tolerance of nodes falls below a certain value, the fraction of nodes with complete dependence, or ρ, dominates the percolation phase transition of the interdependent network. Consequently, regardless of the value of ρ, the entire system undergoes a first-order phase transition, which, due to its abrupt nature, is detrimental to system maintenance. Furthermore, if the fault tolerance of nodes exceeds a threshold, the system with a well-chosen ρ value will undergo a second-order phase transition. All of this suggests that by appropriately adjusting ρ and α, phase transitions can be controlled, helping the system transition from a detrimental first-order phase transition to a more acceptable second-order phase transition.

[0058] It is well known that many real networks are small-world networks. Figure 3 and Figure 4 Another observation in is that small-world networks are always prone to undergo second-order phase transitions due to the weak dependencies between nodes in real small-world networks, so our results can reasonably explain why real networks are usually not so weak under attacks.

[0059] The problem of cascading failures in interdependent networks composed of fault-tolerant nodes in scale-free and coupled random networks is studied. In known interdependent network models, it is assumed that each pair of interconnected nodes has full interdependence strength, that is, if one node fails, its partner node will also fail. Compared with previous models, the present invention incorporates more realistic considerations: first, each node is fault-tolerant, and each node in one network corresponds to one and only one partner vertex in the other network. Second, the failure of a node in one network causes its partner node to fail with a certain probability. When its interdependent partner node fails, the node coupling strength can be determined by the link retention probability of the vertex.

[0060] The present invention considers the fault tolerance α and degree k of randomly selected node i in three cases: i First, if a high-degree node has low fault tolerance, it can be said that the fault tolerance of the node is negatively correlated with its degree, which is usually referred to as negative dependence. Second, if a node has high fault tolerance, it can be said that the fault tolerance of the node is positively dependent on its degree. Third, if the fault tolerance of a node is neither negatively nor positively dependent on its degree, it is named random dependence. Finally, if a node has no fault tolerance, it means that if one of its connected nodes fails, the node will fail immediately. Nodes with no fault tolerance are named completely dependent nodes.

[0061] Assume that the number of nodes with fully dependent properties is ρ. It is found that when the parameter ρ changes, a large number of phase transition phenomena will occur. The interdependent system has a strong robustness. If ρ < ρ c , it can be characterized by a second-order transition, otherwise, if ρ>ρ c , then the interdependent system is destructible, and the interdependent network will undergo a first-order percolation transition triggered by a cascading failure. The results show that the interdependent system composed of nodes with positive dependencies always has good robustness, while the interdependent system composed of nodes with negative dependencies is always fragile, and the robustness of the interdependent network based on random dependent nodes is moderate. At the same time, for scale-free networks, it is found that if the high-degree nodes in the interdependent system have high fault tolerance, the robustness of the entire interdependent system can be greatly enhanced, which means that strengthening the fault tolerance of high-degree nodes can significantly enhance the robustness of the entire interdependent system. In addition, it is found that the point value of the second-order percolation transition is small at any time and is not affected by the change of the model parameter ρ, which means that if ρ is lower than the critical value ρ c , then interdependent networks exhibit strong robustness. The results of this study of interconnected systems composed of nodes with weak fault tolerance are consistent with the observation that interdependent networks in the real world are generally more robust. Therefore, the study provides a possible approach to explain the stability of interdependent networks in the real world.< / k>

Claims

1. A method for calculating the fault tolerance performance of an interdependent circuit network, the method being based on the following foundations: First, node dependencies are hierarchical. That is, if the behavior of node a is partially affected by another node b, then node a is said to be partially dependent on node b. On the other hand, if the behavior of node a is completely controlled by node b, then node a is said to be completely dependent on node b. Second: Every node in the network is fault-tolerant; Third: Each node in the network has different fault tolerance capabilities; Fourth: The fault tolerance of a node is closely related to its importance in the network; Fifth: The degree of a node is related to its fault tolerance. This relationship is divided into three categories: positive correlation, negative correlation, and random correlation. Positive correlation means that nodes with high degree have high fault tolerance, while nodes with low degree have low fault tolerance. Negative correlation means that nodes with high degree have low fault tolerance, while nodes with low degree have high fault tolerance. Random correlation means that the relationship between degree and fault tolerance of a node is random. The method includes: Step 1: Calculate the degree distribution generating function of circuit A and circuit B; G 0A (ξ)=∑ k P A (k)ξ k G 0B (ξ)=∑ k P B (k)ξ k Among them, P A (k) represents the node degree distribution of circuit A, P B (k) represents the node degree distribution of circuit B, ξ k represents the kth node variable; Calculate the underlying branching process generation function of circuit A and circuit B, G 1A (ξ)=G′ 0A (ξ) / G′ 0A (1) G 1B (ξ)=G′ 0B (ξ) / G′ 0B (1) Among them, G 1A (ξ) represents the underlying branching process generation function of circuit A, G 1B (ξ) represents the underlying branching process generation function of circuit B, G′ 0A (ξ), G′ 0B (ξ) represents G 0A (ξ), G 0B (ξ) The first-order derivative with respect to ξ, G′ 0A (1) G′ 0B (1) respectively represent G′ 0A (ξ) and G′ 0B (ξ) Function value when ξ = 1; Step 2: Randomly select a node in circuit A and circuit B to start, and calculate the volume of the largest connected branch of circuit A and circuit B; (1) When the degree of nodes in circuits A and B is positively correlated with their fault tolerance; Step 2.1.1: Calculate the probability of a maximum connected branch appearing on a randomly selected link in circuits A and B using the following two equations: Among them, R A 、R B represents the probability of the largest connected branch appearing when a link is randomly selected in circuit A or circuit B, p[1-G 1A (1-R A )]、p[1-G 1B (1-R B )] represents the probability that the links randomly selected from the randomly selected nodes of circuit A and circuit B will lead to the maximum connected branch of circuit A. represents the probability that a randomly selected link starting from a randomly selected node in circuit a will lead to the maximum connected branch in circuit A, α represents the fault tolerance of the current circuit node, circuit a belongs to circuit A, the superscript * represents the random edge that can reach the maximum cluster, p[1-G 0A (1-R A )] represents the survival probability of the nodes that depend on a given node; Among them, k max represents the maximum order of the nodes in the corresponding circuit, p(k) represents the probability of occurrence of a node with degree k, represents the degree of a randomly selected node, <k> Indicates the average degree;< / k> Step 2.1.2: Calculate the volume of the largest connecting branch of circuit A and circuit B; Among them, p[1-G 0A (1-R A )] represents the survival probability of the nodes that depend on a given node, represents the survival probability of the node that depends on the randomly selected node; (2) When the degree of nodes in circuit A and circuit B is negatively correlated with their fault tolerance; Step 2.2.1: Calculate the probability of a maximum connected branch appearing on a randomly selected link in circuits A and B using the following two equations: Step 2.2.2: Calculate the volume of the largest connecting branch of circuit A and circuit B; (3) When the degrees of nodes in circuits A and B are randomly correlated with their fault tolerance; In step 2.3.1, calculate the probability of a maximum connected branch appearing on a randomly selected link in circuits A and B using the following two equations: R A =p[G1(1-R A )]·[1-G0(1-R B )]+p·α[1-G1(1-αR A )]·G0(1-R B )·(1-p) R B =p[1-G1(1-R B )]·[1-G0(1-R A )]+α[1-G1(1-αR B )]·{1-p[1-G0(1-R A )]}·(1-p) Step 2.3.2: Calculate the volume of the largest connecting branch of circuit A and circuit B; S A =p[G 0A (1-R A )]·[1-G 0B (1-R B )]+p·[1-G 0A (1-αR A )]·G 0B (1-R B )·(1-p) S B =p[1-G 0B (1-R B )]·[1-G 0A (1-R A )]+[1-G 0B (1-αR B )]·{1-p[1-G 0A (1-R A )]}·(1-p) Step 3: Add the volumes of the largest connected branches of circuits A and B calculated in step 2 and compare them with the threshold. If the volume is greater than the threshold, it is considered that the joint fault tolerance performance of circuits A and B is poor; otherwise, the fault tolerance performance is good.

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