Network reliability analysis method based on mixed K / Q-cascade model

Abstract

The embodiment of the invention provides a network reliability analysis method based on a mixed K / Q-cascade model. The network reliability analysis method is based on a mixed K / Q-cascade model, and comprises the following steps: 1, initializing model parameters; step 2, generating an initial random network and extracting a node initial degree; step 3, first-round failure detection is carried out, and the first-round failure detection is carried out only for k-cascade nodes; 4, recording the initial change of the network scale; step 5, cascade failure iteration judgment and network updating: carrying out loop iteration until no node fails; and step 6, visualizing network scale evolution. According to the hybrid K / Q cascade model provided by the invention, the local dependence and heterogeneity of network nodes are considered, and the node difference is also considered. Analysis is carried out according to a mixed K / Q cascade model, and besides information rapid propagation and information rapid stop, information propagation also has a third state, namely a metastable state. When the complex network is in the state, information spreading almost stops for a long period of time in the earlier stage until sudden collapse occurs at a certain moment in the middle. The phase change phenomenon is more suitable for some actual scenes.

Description

Technical Field

[0001] This application relates to the field of robustness analysis technology for complex networks, and in particular to a network reliability analysis method based on a hybrid K / Q-cascade model. Background Technology

[0002] Q-cascade and K-cascade models are the most commonly used and classic models for studying complex networks. In the real world, when individuals make decisions, they often refer to the advice of those around them due to incomplete information and limited personal abilities. Furthermore, this decision-making mechanism is also widely applicable in group behavior problems or decision-making dilemmas.

[0003] (1) Q-cascade model

[0004] When there are only two strategies, the above mechanism is also called "binary decisions with externalities." Externalities refer to the fact that individuals can refer to the decisions of their neighbors, and binary means that each individual can only choose one of the two decisions and must choose one. Based on this decision-making mechanism, Watts proposed a simple global cascade model. This model exhibits a score threshold characteristic, usually denoted by q, hence it is also called the Q-cascade model. The Q-cascade model has the following three characteristics:

[0005] 1. Local dependency: The state change of a node is only related to the state of its neighboring nodes and is independent of other nodes in the network.

[0006] 2. The score threshold characteristic, that is, whether a node's state has changed is related to the proportion of its neighbors whose states have changed out of the total number of neighbors, is the most significant difference from K-cascade.

[0007] 3. Heterogeneity: Unlike previous research models, this model can be applied to heterogeneous networks. The global cascade is defined as follows: initially, some nodes are disturbed, eventually leading to a large-scale impact on the network.

[0008] In network science, complex networks are commonly used to characterize social and economic systems. Nodes represent individuals within the system, and edges between nodes represent the relationships between individuals. For example, in a complex network, nodes represent individuals, and the relationship between two people is represented by edges; if they know each other, an edge exists, otherwise it doesn't. In an economic network, nodes represent institutions, and edges represent the cooperation between institutions. Using network-based modeling methods can reduce analytical complexity and de-emphasize individual details, allowing focus on the global properties of the system.

[0009] In the Q-cascade model, 0 and 1 are used to represent two states of a node: state 0 is the original state, and state 1 is the changed state. For example, in an information propagation network, 0 represents not knowing the information, and 1 represents knowing the information; in a decision network, 0 represents not supporting the decision, and 1 represents supporting it; in a disease transmission network, 0 represents not infected, and 1 represents infected. Initially, all nodes are in state 0, and the network remains stable. When the network is disturbed, some nodes change their state to 1. At different times, a node can observe the states of its neighbors. If the number of neighbors in state 1 exceeds a threshold q, the node's state changes to 1; otherwise, it remains unchanged. k represents the degree of the node, and p... k Let f(q) represent the network degree distribution, q represent the threshold for node state changes, and f(q) represent the probability distribution of the score threshold. Furthermore, each node's state change occurs only once, and nodes with a state of 1 no longer change their state.

[0010] Due to the perturbation, assume that initially ρ0% of the nodes change from state 0 to state 1. After several state updates, the network reaches equilibrium, at which point the node states no longer change. Let ρ be the proportion of nodes whose states change at this point, and let u be the average probability that a node initially in state 0 can be connected to a node in state 1 via any edge. Assuming a node has k edges, the probability that this node changes from state 0 to state 1 is:

[0011]

[0012] Let G0 represent the probability that a node's state changes to 1 when m of its k neighbors are in state 1. For the entire network, the proportion of nodes whose state changes from 0 to 1 is defined as follows: ∞ ):

[0013]

[0014] Solving for the node state change ratio ρ is equal to solving the following system of equations:

[0015]

[0016] This set of equations is the general equations for solving the Q-cascade model. Given the degree distribution of the network, the probability distribution of the score threshold, and the proportion of nodes with initial perturbations, it is possible to solve for the proportion of nodes whose state has changed in the final steady state. Similarly, changing the F function yields different cascade models; when the score threshold is changed to an integer threshold, the model becomes a K-cascade model.

[0017] (2) K-cascade model

[0018] Similar to Q-cascade, K-cascade is also a commonly used model for analyzing the reliability of complex networks, and it also satisfies local dependency and heterogeneity. The biggest difference between the two is that W-cascade is a fractional threshold model, while K-cascade is an integer threshold model, typically using k... s This represents the threshold. When updating the node policy, assuming the node degree is k, when the node has at least k neighbors... s A node's state changes to 1 only if all its neighbors are in state 1; otherwise, it remains in state 0. This is achieved using F(m,k,k). s This represents a node of degree k with m neighbors in state 1, and a threshold value of k. s In the model, this represents the probability of the node's state changing from 0 to 1. Generally, k... s If it is a constant, then we have

[0019] F(m,k,k s )=P(m≥kk s )=P(k s ≥km)

[0020] Note that when k≤k s At that time, k s ≥km must hold true, in which case P(k) s ≥km)=1. Therefore, in the K-cascade model, the degree is less than or equal to k. s The state of each node will inevitably change, and eventually, when a stable state is reached, the degree of all nodes in the network will be greater than k. s Similarly, the solution system of equations for K-cascade can be written out:

[0021]

[0022] For classic complex networks (such as ER networks and BA networks), the propagation of opinions or decisions within the network is usually analyzed based on Q-cascade and K-cascade models, thereby analyzing the network's robustness. However, these analytical models provide a relatively simplistic description of changes in node states and do not take into account the differences between nodes. Summary of the Invention

[0023] To address the aforementioned issues, this invention proposes a "hybrid K / Q-cascade model," which, based on the differences in the rules followed by nodes, allows some node state changes to follow K-cascade rules while others follow Q-cascade processes.

[0024] This invention provides a network reliability analysis method based on a hybrid K / Q-cascade model, the method comprising:

[0025] The first step is to initialize the model parameters;

[0026] The second step is to generate an initial random network and extract the initial degree of the nodes;

[0027] The third step, the first round of failure detection, only targets k-cascade nodes;

[0028] The fourth step is to record the initial changes in network size;

[0029] Step 5, Cascade Failure Iteration and Network Update: Iterate repeatedly until no node fails;

[0030] Step 6: Visualize the network's scale evolution.

[0031] In some embodiments, the first step, model parameter initialization, includes:

[0032] Define core parameters:

[0033] q: The failure threshold ratio of the q-cascade rule, which causes a node to fail when its current degree is less than q times its initial degree;

[0034] ks: The absolute threshold for failure of the k-cascade rule; a node fails when its current degree is below ks.

[0035] N: Total number of network nodes;

[0036] z: Average network degree;

[0037] rou: The proportion of nodes using the k-cascade rule;

[0038] Derivation of derived parameters:

[0039] The total number of edges in the network, M = int(N*z / 2), is calculated based on the average degree of the random graph.

[0040] The number of nodes using the k-cascade rule is Nk = int(N*rou);

[0041] The number of nodes using the q-cascade rule is Nq = N – Nk.

[0042] In some embodiments, the second step, generating an initial random network and extracting the initial degree of nodes, includes:

[0043] Constructing the basic network: Based on the number of nodes N and the total number of edges M, generate an undirected random graph G;

[0044] Record initial degree features: Calculate the initial degree of each node in the network and form an initial degree list Degree, where Degree[i] represents the initial degree of node i.

[0045] In some embodiments, the third step, initial failure detection, is only applicable to k-cascade nodes and includes:

[0046] Partitioning the rule nodes: Randomly select NK nodes from all N nodes and mark them as the kl list. The nodes in kl follow the k-cascade rule, and the remaining nodes follow the q-cascade rule;

[0047] Initial failure determination: Traverse each node in kl. If the initial degree Degree[ch] of the node is less than the absolute threshold ks, then determine that the node has failed and add it to the list dd of nodes to be deleted;

[0048] Recording and deletion operations:

[0049] - Initialize the list L to record the number of nodes deleted in each round. The first element of L is the number of nodes that failed in the first round, len(dd);

[0050] - Delete all failed nodes in dd from the network G;

[0051] - Extract the surviving nodes of the network G after deletion to form a new node list nodelist.

[0052] In some embodiments, the fourth step, recording the initial change in network scale, includes:

[0053] Initialize the network scale record list l:

[0054] - The first element of l is the initial number of nodes in the network, N;

[0055] - The second element is the number of surviving nodes after the first round of deletion, len(nodelist).

[0056] In some embodiments, the fifth step, cascading failure iterative determination and network update: Iterate in a loop until no nodes fail, including:

[0057] Reset the list dd of nodes to be deleted to be empty;

[0058] Traverse all current surviving nodes, each node ch in nodelist:

[0059] - If the node ch belongs to kl, a k-cascade rule node: Calculate the current real-time degree d = G.degree[ch] of the node. If d < ks, determine that it has failed and add it to dd;

[0060] - If the node ch does not belong to kl, a q-cascade rule node: Calculate the current real-time degree d = G.degree[ch] of the node. If d < Degree[ch] × q, q times the initial degree, determine that it has failed and add it to dd;

[0061] Determine the termination condition for the iteration:

[0062] - If dd is empty, no node fails in this round, and the loop terminates;

[0063] - If dd is not empty and a node fails in this round: remove all failed nodes from network G, update the list of surviving nodes nodelist to the current list of nodes in G, and add the updated number of surviving nodes len(nodelist) to the network size list l.

[0064] In some embodiments, the sixth step, visualizing network scaling evolution, includes:

[0065] A line graph with scatter plots is drawn with the iteration rounds as the horizontal axis and the number of surviving nodes as the vertical axis to visually demonstrate the dynamic changes in network size during cascading failures.

[0066] In some embodiments, the hybrid K / Q-cascade model includes: some node state changes following K-cascade rules, and other node state changes following Q-cascade rules.

[0067] The beneficial effects of the above embodiments include:

[0068] Compared to traditional Q-cascade and K-cascade models, the hybrid K / Q cascade model proposed in this invention considers not only the local dependencies and heterogeneity of network nodes but also their differences. Traditional models analyze information propagation in complex networks in two states: rapid information propagation and rapid cessation of information flow. However, according to the hybrid K / Q cascade model, information propagation also involves a third state: metastable state. In this state, information propagation in the complex network almost stagnates for a long period in the early stages, until it suddenly collapses at some point in the middle. This phase transition phenomenon is more consistent with some real-world scenarios. Attached Figure Description

[0069] The accompanying drawings illustrate, by way of example and not limitation, the various embodiments discussed herein.

[0070] Figure 1 It is a stochastic mixture K / Q-cascade model;

[0071] Figure 2 This is a cascaded phase transition thermogram with k_s = 15 fixed in a stochastic K / Q-cascade model. Detailed Implementation

[0072] In order to gain a more detailed understanding of the features and technical content of the embodiments of this application, the implementation of the embodiments of this application will be described in detail below with reference to the accompanying drawings. The accompanying drawings are for reference and illustration only and are not intended to limit the embodiments of this application.

[0073] In the embodiments described in this application, it should be noted that, unless otherwise stated and limited, the term "connection" should be interpreted broadly. For example, it can be an electrical connection, or a connection between two internal components. It can be a direct connection or an indirect connection through an intermediate medium. Those skilled in the art can understand the specific meaning of the above term according to the specific circumstances.

[0074] It should be noted that the terms "first," "second," and "third" used in the embodiments of this application are merely used to distinguish similar objects and do not represent a specific ordering of objects. It is understood that "first," "second," and "third" can be interchanged in a specific order or sequence where permitted. It should be understood that the objects distinguished by "first," "second," and "third" can be interchanged where appropriate so that the embodiments of this application described herein can be implemented in an order other than those illustrated or described herein.

[0075] This invention proposes a "hybrid K / Q-cascade model," which, based on the differences in node adherence rules, allows some node state changes to follow K-cascade rules while others follow Q-cascade processes. An example is... Figure 1 As shown. The left figure shows the state of the node at time step n, and the right figure shows the state of the node at time step (n+1), using a synchronous update method. Taking the state updates of node A and node B as examples, assuming that the state update method of node A is Q-cascade model, q = 0.5, and the state update method of node B is K-cascade model, k s =3. Yellow nodes represent neighbors of node A with state 1, and blue nodes represent neighbors of node B with state 1. In the nth round, 35 of node A's neighbors have state 1 (excluding node B), exceeding the threshold q; node B has 1 neighbor with state 1 and the remaining three neighbors have state 0 (including node A), which is less than or equal to k. s Therefore, in round n+1, node A's state becomes 1, represented by yellow, and node B's state becomes 1, represented by blue.

[0076] For ease of description, nodes whose state updates follow the K-cascade model are called K nodes, and the other group of nodes are called Q nodes.

[0077] Let α be the proportion of nodes in the network whose state updates follow the K-cascade model, and 1-α be the proportion of nodes whose state updates follow the Q-cascade model. Randomness means that the selection of nodes according to the α proportion is random. When α = 0.5, the model becomes a completely random model, where each node has an equal probability of updating its state according to either the Q-cascade or K-cascade model. The randomized mixed K / Q-cascade model has the following properties:

[0078] (1) Basic properties such as local dependence and heterogeneity;

[0079] (2) Node differences, namely the differences in the rules followed by node state changes. Some nodes have integer threshold characteristics, while other nodes satisfy fractional threshold characteristics. This is the biggest difference between this model and the above models.

[0080] (3) Randomness: Each node has a probability of α for node K and a probability of 1-α for node Q.

[0081] The model is theoretically analyzed using seepage theory. The following variables are defined: ρ0, the proportion of nodes whose state changes from 0 to 1 during the initial perturbation; ρ, the proportion of nodes in state 1 when the network reaches stability; u ∞ When the network is stable, the average probability that a node in state 0 can connect to a neighbor in state 1 through any edge. In the Q-cascade model, the probability of a node with an initial state of 0 becoming a node with a state of 1; The probability that a neighbor of a node in state 0 will have its state changed to 1 through an additional connection. The same applies to K-cascade. Here, the variables ρ and u... ∞ The calculation of ρ is the key to solving the problem. The following explanation will be based on the variable ρ.

[0082] ρ represents the proportion of nodes in state 1 when the network is stable. It consists of two parts: one part is the initially disturbed nodes, with a proportion of ρ0; the other part is the nodes that evolve from state 0 to state 1 over time. The second term can be further subdivided into two terms: Q nodes and K nodes. Because the proportion of Q nodes is 1-α, and the probability of a node in state 0 becoming a node in state 1 is... Therefore, this part of the contribution is Similarly, the contribution of node K is Therefore, the formula for calculating ρ is as follows:

[0083]

[0084] Combining the background techniques, the following equations are obtained for solving the stochastic mixed K / Q-cascade model:

[0085]

[0086] in, and F(m,k,k) s These correspond to the threshold functions in Q-cascade and K-cascade, respectively.

[0087] Compared to traditional Q-cascade and K-cascade models, the hybrid K / Q cascade model proposed in this invention considers not only the local dependencies and heterogeneity of network nodes but also their differences. Traditional models analyze information propagation in complex networks in two states: rapid information propagation and rapid cessation of information flow. However, according to the hybrid K / Q cascade model, information propagation also involves a third state: metastable state. In this state, information propagation in the complex network almost stagnates for a long period in the early stages, until it suddenly collapses at some point in the middle. This phase transition phenomenon is more consistent with some real-world scenarios.

[0088] Fixed k s =15, select ER random network, number of nodes N=10000, average degree z=20; in the random mixed K / Q-cascade model, the proportion of nodes whose state changes from 0 to 1 given the initial perturbation is ρ0=0.2.

[0089] The following is a detailed implementation of the algorithm:

[0090]

[0091]

[0092]

[0093] Figure 2 These are simulation results from a stochastic mixed K / Q-cascade model. The horizontal axis represents the threshold q, and the vertical axis represents the proportion of K nodes α. Figure 2 (a) is a heatmap showing the proportion of nodes whose state has not changed in the network (i.e., cascading failure). The blue area indicates that the proportion of nodes whose state has not changed tends to 0, and the yellow area indicates that the proportion of nodes whose state has not changed tends to 1. Figure 2 (b) shows a heatmap of the total time steps of the cascaded network. The blue area indicates that the network reaches the final state after a relatively short number of time steps, while the yellow area indicates that the network reaches the final state after a relatively long number of time steps.

[0094] The technical solutions described in the embodiments of this application can be combined arbitrarily without conflict.

[0095] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A network reliability analysis method, characterized in that, Based on the hybrid K / Q-cascade model, the method includes: The first step is to initialize the model parameters; The second step is to generate an initial random network and extract the initial degrees of nodes; The third step is the first-round failure detection, only for k-cascade nodes; The fourth step is to record the initial change in network size; The fifth step is the iterative determination of cascade failure and network update: loop until no nodes fail; The sixth step is to visualize the evolution of network size.

2. The network reliability analysis method according to claim 1, characterized in that, In the first step, the initialization of model parameters includes: Define the core parameters: q: The failure threshold ratio of the q-cascade rule. A node fails when its current degree is lower than q times the initial degree; ks: The absolute failure threshold of the k-cascade rule. A node fails when its current degree is lower than ks; N: The total number of network nodes; z: The average degree of the network; rou: The proportion of nodes adopting the k-cascade rule; Derive the derived parameters: The total number of network edges M = int(N * z / 2), calculating the total number of edges of the random graph based on the average degree; The number of nodes adopting the k-cascade rule Nk = int(N * rou); The number of nodes adopting the q-cascade rule Nq = N - Nk.

3. The network reliability analysis method according to claim 2, characterized in that, In the second step, generating an initial random network and extracting the initial degrees of nodes includes: Construct the basic network: Based on the number of nodes N and the total number of edges M, generate an undirected random graph G; Record the initial degree characteristics: Calculate the initial degree of each node in the network to form an initial degree list Degree, where Degree[i] represents the initial degree of node i.

4. The network reliability analysis method according to claim 3, characterized in that, In the third step, the first-round failure detection, only for k-cascade nodes, includes: Divide the rule nodes: Randomly select Nk nodes from all N nodes and mark them as the kl list. Nodes in kl follow the k-cascade rule, and the remaining nodes follow the q-cascade rule; Initial failure determination: Traverse each node in kl. If the initial degree Degree[ch] of the node < the absolute threshold ks, then determine that the node fails and add it to the list of nodes to be deleted dd; Record and deletion operations: - Initialize the list L to record the number of nodes deleted in each round. The first element of L is the number of nodes that failed in the first round len(dd); - Delete all failed nodes in dd from the network G; - Extract the surviving nodes of the network G after deletion to form a new node list nodelist.

5. The network reliability analysis method according to claim 4, characterized in that, In the fourth step, recording the initial change in network size includes: Initialize the network size record list l: - The first element of l is the initial number of network nodes N; - The second element is the number of surviving nodes after the first-round deletion len(nodelist).

6. The network reliability analysis method according to claim 5, characterized in that, In the fifth step, the iterative determination of cascade failure and network update: loop until no nodes fail, includes: Reset the list of nodes to be deleted dd to be empty; Traverse all current surviving nodes, each node ch in nodelist: - If node ch belongs to kl, a k-cascade rule node: Calculate the current real-time degree d = G.degree[ch] of the node. If d < ks, determine that it fails and add it to dd; - If the node ch does not belong to kl, q-cascade rule node: Calculate the current real-time degree d of the node = G.degree[ch]. If d < d times the initial degree, which is Degree[ch] × q, it is determined to be失效 and added to d. Determine the iteration termination condition: - If dd is empty, no nodes fail in this round, and the loop is terminated. - If dd is non-empty and nodes fail in this round: Delete all failed nodes in dd from the network G, update the list of surviving nodes nodelist to the list of nodes in the current G, and add the updated number of surviving nodes len(nodelist) to the network scale list l.

7. The network reliability analysis method according to claim 6, characterized in that, The sixth step, visualizing the evolution of the network scale, includes: Taking the iteration round as the horizontal axis and the number of surviving nodes in the network as the vertical axis, draw a line chart with scatter markers to visually display the dynamic change of the network scale during the cascade failure process.

8. The network reliability analysis method according to claim 1, characterized in that, The hybrid K / Q-cascade model includes: The state changes of some nodes follow the k-cascade rule, and the state changes of another part of the nodes follow the Q-cascade rule.

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