A robustness analysis method for the propagation of changes in the structure of a production line parameter association network.

Abstract

This invention relates to the field of production line parameter correlation network analysis technology, and particularly to a method for robustness analysis of the propagation of structural changes in production line parameter correlation networks. The method includes: constructing a production line parameter correlation network structure; forming a propagation model of structural changes in the production line parameter correlation network structure based on the change propagation process of an epidemic model under attack; constructing a propagation threshold to determine the outbreak or subsidence of infected change propagation in network nodes within the model; simulating the change propagation process of the model under two attack strategies and calculating the corresponding attack ratios; and using the normalized avalanche size as a robustness measure to analyze the impact of changes in the production line parameter correlation network structure on the robustness of multi-layer network structures. This invention addresses the problems of single modeling dimensions, insufficient dynamic analysis, and one-sided robustness assessment in existing technologies.

Description

Technical Field

[0001] This invention relates to the field of production line parameter correlation network analysis technology, and in particular to a method for robust analysis of the propagation of structural changes in production line parameter correlation networks. Background Technology

[0002] As a complex manufacturing network characterized by high investment and high integration, the production line exhibits multi-dimensional coupling characteristics, forming a non-linear interactive network across four design dimensions: configuration, dynamics, control, and optimization. This multi-dimensional coupling maps to a strongly correlated topology in the parameter space, where any change in design elements triggers cross-dimensional parameter fluctuations. Changes are ubiquitous in the production line design process; factors such as product upgrades and equipment replacements necessitate design modifications, leading to alterations in the production line's parameter-related network structure.

[0003] Due to the complex interrelationships between different design dimensions, even a minor change in a single design network node within the resulting network structures can propagate changes in the production line parameter-related network structure throughout the entire production line. Uncontrollable design changes lead to uncertainty in the changes to the production line parameter-related network structure. The resulting chain reaction may prevent the production line from completing the modification task on time, thereby affecting product production and processing, impacting order delivery dates, and causing significant losses to the company and related entities.

[0004] Numerous scholars have constructed various models to analyze the risk of changes in the production line parameter correlation network structure during the propagation process, based on different models. However, most of these models focus on the perspective of products or single network structures, lacking research on the risk propagation of changes in different production line parameter correlation network structures within the production line. Production line design parameters have higher dimensions, more complex parameter correlations, and more diverse propagation processes for parameter changes, posing a significant challenge to understanding the propagation of risks associated with changes in the production line parameter correlation network structure. Some studies have shown that parameter network structure changes are inherently dynamic, and while qualitative analyses have been conducted on the dynamic process of parameter network structure change propagation under specific circumstances, it is difficult to elucidate the general laws governing dynamic propagation.

[0005] In summary, traditional methods for modeling changes in production line parameter association networks have many shortcomings:

[0006] (1) Single modeling dimension: Traditional methods are mostly based on product or single network structure modeling, which cannot depict the coupling relationship of multi-dimensional parameters such as production line configuration, dynamic, control, and optimization.

[0007] (2) Insufficient dynamic propagation analysis: Existing models lack a quantitative description of the dynamic propagation path of parameter changes in multi-layer networks, making it difficult to predict the scope and intensity of cascade failures.

[0008] (3) One-sided robustness assessment: The robustness analysis is not systematically conducted by combining the network node state transition probability (such as infection rate and recovery rate) and topological characteristics (such as mesh and star topology), which cannot provide targeted optimization strategies for production line design changes. Summary of the Invention

[0009] The purpose of this invention is to propose a robustness analysis method for the propagation of changes in the network structure of production line parameters. By constructing a multi-layer network model with multi-dimensional parameter association and combining it with the theory of epidemic propagation to dynamically simulate the cascading effect of parameter changes, a quantitative assessment and optimization of the robustness of the production line can be achieved, solving the problems of single modeling dimension, insufficient dynamic analysis and one-sided robustness assessment in the existing technology.

[0010] To achieve this objective, the present invention adopts the following technical solution:

[0011] A robustness analysis method for the propagation of changes in the network structure of production line parameters includes the following steps:

[0012] S1. The design parameters of the production line in four dimensions—configuration, dynamics, control, and optimization—are abstracted into network nodes, and the interactions between parameters are mapped into network topology edges. This constructs a production line parameter association network structure that includes five multi-layer network structures: tree, mesh, hybrid, star, and ring.

[0013] S2. Define the network node states as susceptible state S, infected state I, recovered state R, and reinfected state A, and define the network node state probabilities as infection rate β, recovery rate γ, reinfection rate μ, and reinfection rate λ.

[0014] Based on the change propagation process of the epidemic model under attack, the state transition of network nodes is simulated by infection rate β, recovery rate γ, reinfection rate μ and reinfection rate λ, so as to realize the change propagation process of the production line parameter associated network structure under cascading failure, and form a change propagation model of the production line parameter associated network structure.

[0015] S3. Construct a propagation threshold R0(t), and determine the outbreak or subsidence of network node infection change propagation in the production line parameter related network structure change propagation model based on the propagation threshold R0(t).

[0016] S4. Simulate the propagation process of changes in the production line parameter-related network structure under two attack strategies: random attack (RA) and intentional attack (IA). Calculate the random attack ratio τ. RA and the ratio of intentional attacks τ IA ;

[0017] S5. Using the normalized avalanche magnitude S(t) as a robustness measure, we analyze the impact of changes in the production line parameter-related network structure on the robustness of the multi-layer network structure.

[0018] Preferably, in S1, the five multilayer network structures include:

[0019] The tree-like network structure formed between the two dimensions of configuration and dynamics;

[0020] The mesh network structure formed between the dynamic and control dimensions;

[0021] The hybrid network structure formed between the two dimensions of control type and optimization type;

[0022] The star-shaped network structure formed between the internal components of the control dimension;

[0023] The ring network structure formed between the internal components of the optimal dimension.

[0024] Preferably, in S2, simulating network node state transitions using infection rate β, recovery rate γ, reinfection rate μ, and reinfection rate λ includes:

[0025] S21. Set the initial network node state of all network nodes in the production line parameter association network structure to the susceptible state S.

[0026] S22. A network node in a susceptible state S is attacked and then becomes an infected state I with an infection rate β.

[0027] S23. Network nodes in infected state I that transition to recovered state R with recovery rate γ.

[0028] S24. A network node in the recovery state R can be transformed back into a network node in the susceptible state S, and can be reinfected as a network node in the reinfection state A with a reinfection rate μ.

[0029] S25. A network node that is reinfected with network node A is transformed into a recovered state R with a recovery rate λ.

[0030] S26. Repeat S22-S25 until no network node in the production line parameter association network structure is infected due to a susceptible state S.

[0031] Preferably, in S2, the formation of the production line parameter association network structure change propagation model includes:

[0032] S27. Calculate the change in the number of network nodes in susceptible state S, infected state I, and recovered state R during the time interval Δt:

[0033] ΔS = -βSIΔt;

[0034] ΔR = γIΔt;

[0035] ΔI=|ΔS|-ΔR=βSIΔt-γIΔt;

[0036] Wherein, SI represents the number of all network nodes that may come into contact with each other in the production line parameter association network structure;

[0037] S28. Considering the possibility of network node reinfection, set the propagation probability μ = κβ. α And the probability of recovery λ=vγ θ Construct a set of epidemiological model equations for reinfection:

[0038]

[0039] Where N represents the total number of network nodes in the susceptible state S in the production line parameter association network structure, μ = κβ α The probability of reinfection is λ = vγ θ The probability of recovery for a network node in a reinfected state is given by κ, α, v, and θ, which are all variable parameters.

[0040] Preferably, in S3, the expression for the propagation threshold R0(t) is as follows:

[0041]

[0042] Where R0(t) represents the propagation threshold as it changes with time t, and αL(t) represents the inhibitory effect of network nodes on the recovery rate γ. <k>This indicates the average degree of correlation between production line parameters and the network structure.

[0043] When the propagation threshold R0(t) > 1, the infection and change propagation of network nodes in the production line parameter association network structure continues to erupt; when R0(t) < 1, the infection and change propagation of network nodes in the production line parameter association network structure subsides on its own.

[0044] Preferably, in S4, the random attack RA is a network node in the production line parameter associated network structure that is randomly selected, and the random attack ratio τ RA This is the ratio of the initial number of network nodes in a random attack to the total number of network nodes.

[0045] Preferably, in S4, the intent attack IA is to attack network nodes with high out-degree or high in-degree after sorting them by their out-degree or in-degree, and the intent attack ratio τ IA This is the ratio of the initial number of network nodes intended for an attack to the total number of network nodes.

[0046] Preferably, in S5, the normalized avalanche magnitude S(t) is used to quantify the infection rate β and recovery rate γ of network nodes at time t, and the impact of changes in the network structure related to production line parameters on the robustness of the multilayer network is analyzed.

[0047] The expression for the magnitude of the avalanche, S(t), is:

[0048]

[0049] Where S(t) represents the normalized avalanche magnitude caused by network node i in the infected state I in the production line parameter association network structure after cascading propagation, and N represents the total number of network nodes in the production line parameter association network structure.

[0050] One of the above technical solutions has the following beneficial effects:

[0051] 1. Address the shortcomings of multi-layer network modeling and improve the accuracy of production line mapping.

[0052] Existing technologies only model from a single network or product perspective. This solution abstracts the four dimensions of the production line parameters into a multi-layered network, encompassing five topologies including tree and mesh structures. It is the first to achieve a systematic representation of the cross-dimensional coupling relationships between configuration, dynamics, control, and optimization. For example, the star-shaped structure of the control dimension (with key controller parameters as the central node) can accurately map the impact of control parameters on the overall situation in the actual production line, overcoming the problem that traditional methods are insufficient in characterizing the complexity of the production line.

[0053] 2. Breakthrough in dynamic propagation modeling reveals the laws governing cascading failures.

[0054] Traditional models lack a dynamic description of change propagation. This approach uses an epidemiological model combined with state transition probabilities (infection rate β, recovery rate γ, etc.) to quantify the temporal evolution of parameter changes. For example, by simulating the cycle of "parameter infection-recovery-reinfection," it reveals how changes can trigger persistent cascading failures when design margins are insufficient (e.g., low recovery rate γ), filling a gap in the study of the general laws of dynamic propagation.

[0055] 3. A robust indicator system is in place, providing a quantitative basis for decision-making.

[0056] A robustness assessment method based on normalized avalanche size S(t) and propagation threshold R0(t) is proposed, which can quantify system stability under different topologies (e.g., mesh networks have the greatest impact on robustness) and attack strategies. For example, by analyzing the impact of R0(t)β and recovery rate γ on vulnerable state S, enterprises can weigh protection costs (higher costs for larger β and smaller γ) against robustness requirements, providing data support for equipment redundancy design and parameter threshold optimization, and avoiding losses such as order delays caused by design changes.

[0057] 4. Enhanced adaptability to multiple scenarios, supporting full lifecycle optimization of the production line.

[0058] It is compatible with both random and intentional attack simulations, covering various risk scenarios such as equipment failure and human error. For example, in mobile phone production line applications, this method identifies the central node of the control-dimensional star structure as a critical vulnerability, guiding enterprises to prioritize strengthening the redundant design of such parameters, improving the production line's anti-interference capability during product upgrades or equipment replacements, and reducing the uncertainty risk of change propagation. Attached Figure Description

[0059] Figure 1 This is a schematic diagram illustrating the principle of the robustness analysis method for the propagation of changes in the network structure of production line parameter association according to the present invention.

[0060] Figure 2 This is a schematic diagram of five multi-layer network structures in the robustness analysis method for the propagation of changes in production line parameter association network structure of the present invention;

[0061] Figure 3 This is a schematic diagram illustrating the principle of the production line parameter association network structure change propagation model in the robustness analysis method for production line parameter association network structure change propagation of the present invention. Detailed Implementation

[0062] The technical solution of the present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0063] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "length," "width," "thickness," "upper," "lower," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "axial," "radial," and "circumferential," etc., indicating orientation or positional relationships, are based on the orientation or positional relationships shown in the accompanying drawings and are only for the convenience of describing this invention and simplifying the description, and are not intended to indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of this invention.

[0064] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this invention, unless otherwise stated, "a plurality of" means two or more.

[0065] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "joining" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal communication between two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0066] A robustness analysis method for the propagation of changes in the network structure of production line parameters includes the following steps:

[0067] S1. The design parameters of the production line in four dimensions—configuration, dynamics, control, and optimization—are abstracted into network nodes, and the interactions between parameters are mapped into network topology edges. This constructs a production line parameter association network structure that includes five multi-layer network structures: tree, mesh, hybrid, star, and ring.

[0068] S2. Define the network node states as susceptible state S, infected state I, recovered state R, and reinfected state A, and define the network node state probabilities as infection rate β, recovery rate γ, reinfection rate μ, and reinfection rate λ.

[0069] Based on the change propagation process of the epidemic model under attack, the state transition of network nodes is simulated by infection rate β, recovery rate γ, reinfection rate μ and reinfection rate λ, so as to realize the change propagation process of the production line parameter associated network structure under cascading failure, and form a change propagation model of the production line parameter associated network structure.

[0070] S3. Construct a propagation threshold R0(t), and determine the outbreak or subsidence of network node infection change propagation in the production line parameter related network structure change propagation model based on the propagation threshold R0(t).

[0071] S4. Simulate the propagation process of changes in the production line parameter-related network structure under two attack strategies: random attack (RA) and intentional attack (IA). Calculate the random attack ratio τ. RA and the ratio of intentional attacks τ IA ;

[0072] S5. Using the normalized avalanche magnitude S(t) as a robustness measure, we analyze the impact of changes in the production line parameter-related network structure on the robustness of the multi-layer network structure.

[0073] The working principle of this technical solution is as follows:

[0074] Step S1: Through network node attributes and edge relationships, the system characterizes the dynamic coupling mechanism and hierarchical evolution law between cross-dimensional parameters.

[0075] Step S2: The propagation of changes in the production line parameter-related network structure is essentially a process where changes in one or more parameters lead to changes in the network structure, triggering adaptive changes in downstream parameters through the relationships between parameters. If downstream parameters lack the ability to absorb changes, they may fail to complete the design changes, leading to malfunctions. Whether design parameters can absorb changes depends on their maximum design margin while ensuring performance or meeting original functional requirements.

[0076] Step S3: When the production line design is changed, each parameter in the production line parameter association network structure has a certain degree of resilience. To illustrate the cascading failure phenomenon caused by network node propagation in the production line parameter association network structure, a propagation threshold is constructed.

[0077] Step S4: In the actual production line, workers prioritize adjusting equipment with greater margin of error to ensure that the equipment can maintain normal operation without redesign. Therefore, the initial fault in the production line parameter association network structure starts from the attacked nodes, considering both random attack (RA) and intentional attack (IA) strategies.

[0078] Step S5: During design changes, excessive changes to parameter nodes can increase change costs. Considering the dynamic process of cascading propagation, the normalized avalanche magnitude S is used to quantify the robustness of the interdependent production line parameter network structure to cascading failures.

[0079] In summary, the beneficial effects of this technical solution include:

[0080] 1. Address the shortcomings of multi-layer network modeling and improve the accuracy of production line mapping.

[0081] Existing technologies only model from a single network or product perspective. This solution abstracts the four dimensions of the production line parameters into a multi-layered network, encompassing five topologies including tree and mesh structures. It is the first to achieve a systematic representation of the cross-dimensional coupling relationships between configuration, dynamics, control, and optimization. For example, the star-shaped structure of the control dimension (with key controller parameters as the central node) can accurately map the impact of control parameters on the overall situation in the actual production line, overcoming the problem that traditional methods are insufficient in characterizing the complexity of the production line.

[0082] 2. Breakthrough in dynamic propagation modeling reveals the laws governing cascading failures.

[0083] Traditional models lack a dynamic description of change propagation. This approach uses an epidemiological model combined with state transition probabilities (infection rate β, recovery rate γ, etc.) to quantify the temporal evolution of parameter changes. For example, by simulating the cycle of "parameter infection-recovery-reinfection," it reveals how changes can trigger persistent cascading failures when design margins are insufficient (e.g., low recovery rate γ), filling a gap in the study of the general laws of dynamic propagation.

[0084] 3. A robust indicator system is in place, providing a quantitative basis for decision-making.

[0085] A robustness assessment method based on normalized avalanche size S(t) and propagation threshold R0(t) is proposed, which can quantify system stability under different topologies (e.g., mesh networks have the greatest impact on robustness) and attack strategies. For example, by analyzing the impact of R0(t)β and recovery rate γ on vulnerable state S, enterprises can weigh protection costs (higher costs for larger β and smaller γ) against robustness requirements, providing data support for equipment redundancy design and parameter threshold optimization, and avoiding losses such as order delays caused by design changes.

[0086] 4. Enhanced adaptability to multiple scenarios, supporting full lifecycle optimization of the production line.

[0087] It is compatible with both random and intentional attack simulations, covering various risk scenarios such as equipment failure and human error. For example, in mobile phone production line applications, this method identifies the central node of the control-dimensional star structure as a critical vulnerability, guiding enterprises to prioritize strengthening the redundant design of such parameters, improving the production line's anti-interference capability during product upgrades or equipment replacements, and reducing the uncertainty risk of change propagation.

[0088] To further explain, in S1, the five multilayer network structures include:

[0089] The tree-like network structure formed between the two dimensions of configuration and dynamics;

[0090] The mesh network structure formed between the dynamic and control dimensions;

[0091] The hybrid network structure formed between the two dimensions of control type and optimization type;

[0092] The star-shaped network structure formed between the internal components of the control dimension;

[0093] The ring network structure formed between the internal components of the optimal dimension.

[0094] It is necessary to explain the definitions related to network nodes, network topology edges, dimensions, and production line parameters:

[0095] Network nodes: The four core dimensional parameters of the production line are abstracted into network nodes, including:

[0096] Configuration parameters: spatial structural parameters such as equipment layout and production line topology;

[0097] Dynamic parameters: dynamic performance parameters such as equipment motion trajectory, speed, and acceleration;

[0098] Control parameters: control logic, sensor thresholds, PID parameters, and other control parameters;

[0099] Optimization parameters: Optimization target parameters such as production efficiency, energy consumption, and pass rate.

[0100] Network topology edges: Causal relationships (such as temperature of device A → processing accuracy of device B) or cooperative relationships (such as process sequence constraints) between parameters are defined as directed or undirected edges, forming a multi-layered network with cross-dimensional coupling.

[0101] To further explain, in S2, the simulation of network node state transitions using infection rate β, recovery rate γ, reinfection rate μ, and reinfection rate λ includes:

[0102] S21. Set the initial network node state of all network nodes in the production line parameter association network structure to the susceptible state S.

[0103] S22. A network node in a susceptible state S is attacked and then becomes an infected state I with an infection rate β.

[0104] S23. Network nodes in infected state I that transition to recovered state R with recovery rate γ.

[0105] S24. A network node in the recovery state R can be transformed back into a network node in the susceptible state S, and can be reinfected as a network node in the reinfection state A with a reinfection rate μ.

[0106] S25. A network node that is reinfected with network node A is transformed into a recovered state R with a recovery rate λ.

[0107] S26. Repeat S22-S25 until no network node in the production line parameter association network structure is infected due to a susceptible state S.

[0108] Specifically, the working principle of state transition is as follows:

[0109] Step S21: Simulate the normal operating state of the production line parameters when they are not subject to change attacks. At this time, the parameters are not affected by any design changes or faults, and have the potential to respond to potential changes, but have not yet been infected (i.e., no abnormal parameter values ​​or failure of correlations have occurred).

[0110] Step S22: When a network node is subjected to a design change attack (such as parameter mutation or equipment failure), it transitions from a susceptible state S to an infected state I with an infection rate β. The infection rate β represents the probability that a parameter change propagates through associated edges per unit time, reflecting the coupling strength between parameters (such as the probability of an increase in the temperature of device A affecting the pressure parameter of device B).

[0111] Step S23: If infected state node I has sufficient design margin (e.g., a large parameter threshold range), it will automatically recover to the normal state R with recovery rate γ. Recovery rate γ reflects the self-repair capability of parameters under the impact of changes, such as the probability that the robot arm speed parameter will return to the normal range through feedback control after overload.

[0112] Step S24: The recovered node R may revert to a susceptible state S due to continuous change pressure or secondary disturbances, and then transition to a reinfected state A with a reinfection rate μ. The reinfection rate μ characterizes the increased sensitivity of a node to new changes given its history of infection (e.g., equipment parameters after multiple adjustments are more susceptible to disturbances).

[0113] Step S25: The reinfected node A recovers to the normal state R with a recovery rate λ, where λ has a non-linear relationship with the initial recovery rate γ, reflecting that the node's recovery ability may decrease after multiple infections (e.g., equipment aging leads to a reduced probability of repair).

[0114] Step S26: By repeating S22-S25, simulate the cascading propagation of changes in the production line parameter association network structure until stabilization. This iterative process follows the "contact-infection-recovery" dynamic equilibrium of the epidemic model, and finally terminates the simulation by determining whether new susceptible nodes have been infected, ensuring coverage of the entire cycle of change propagation.

[0115] To further explain, in S2, the formation of the production line parameter association network structure change propagation model includes:

[0116] S27. Calculate the change in the number of network nodes in susceptible state S, infected state I, and recovered state R during the time interval Δt:

[0117] ΔS = -βSIΔt;

[0118] ΔR = γIΔt;

[0119] ΔI=|ΔS|-ΔR=βSIΔt-γIΔt;

[0120] Wherein, SI represents the number of all network nodes that may come into contact with each other in the production line parameter association network structure;

[0121] At any given time point, each network node in a susceptible state S has an equal probability of coming into contact with a downstream network node and triggering infection transmission. Let the total number of all possible network nodes be SI. Within an interval of Δt, the expected number of actually infected network nodes is SI × βΔt. The decrease in the number of network nodes in a susceptible state is expressed as: ΔS = -βSIΔt.

[0122] In the production line parameter association network structure, some network nodes in the infected state I will recover to the recovered state R at regular intervals. Within the interval Δt, the recovery probability of network nodes in the infected state I is γΔt, the expected value of the number of network nodes in the recovered state is I×γΔt, and the increase in the total number of network nodes in the recovered state is expressed as: ΔR=γIΔt.

[0123] Therefore, the change in the total number of network nodes in infected state I is: ΔI=|ΔS|-ΔR=βSIΔt-γIΔt, where β is the probability that a network node in susceptible state S actually comes into contact and becomes infected per unit time, and γ is the probability that each network node in infected state I recovers per unit time.

[0124] S28. Considering the possibility of network node reinfection, set the propagation probability μ = κβ. α And the probability of recovery λ=vγ θ Construct a set of epidemiological model equations for reinfection:

[0125]

[0126] Where N represents the total number of network nodes in the susceptible state S in the production line parameter association network structure, μ = κβ α λ represents the probability of reinfection, νγθ represents the probability of recovery for network nodes in the reinfected state, and κ, α, v, and θ are all variable parameters.

[0127] In order to obtain the mathematical model formula for the propagation of changes in the network structure related to production line parameters, the basic infection number is a very important indicator.

[0128] Suppose a network node in a susceptible state S is attacked at t = 0 and becomes an infected network node I. Let l(t) be the probability that the network node is still infected at time t. In the following interval Δt, the probability that the network node in infected state I recovers is γΔt. Then, the probability that it remains in an unrecovered state is 1 - γΔt. From this, we can deduce:

[0129] l(t)(t+Δt)=l(t)(1-γΔt)=l(t)+Δl

[0130] During the time interval from t to t+Δt, for a network node in a susceptible state S to be infected, two conditions must be met. First, the upstream network node in susceptible state S must be in infected state I at this time. Second, the adjacent network node in susceptible state S must have a parameter association with it and be infected. Therefore, the probability of each network node in susceptible state S being infected during this time interval is l(t)×βΔt. Note that in the definition of the basic reproduction number, the state of all network nodes is considered to be in susceptible state S, so the total number of network nodes in susceptible state S is N. During the time interval from t to t+Δt, the expected value of newly infected nodes is N×l(t)×βΔt. The expected value of the number of infected nodes ultimately affected by the network node initially in infected state I, i.e., the basic reproduction number, is:

[0131] N0=∫0 ∞ Nβl(t)dt

[0132] Next, we need to discuss the fixed points (Si) of the epidemiological model equations. * I * R * ):

[0133]

[0134] Combining the basic reproduction number and the fixed points of the epidemiological model equations, it can be concluded that when I * A value of 0 is a necessary condition for a fixed point. At this point, no network node is in the infected state I, and therefore, network nodes in the susceptible state S will not face the possibility of being exposed to infection transmission. As for network nodes in the recovered state R, they naturally remain healthy. In this way, the total number of network nodes will necessarily be in a stable state, that is:

[0135] S * +I * +R * =1

[0136] Based on the stable state and the necessary conditions for a fixed point, the fixed point can be represented by only one parameter:

[0137] (S * I * R * )=(S * ,0,1-S * )

[0138] Considering the stability of the fixed point, suppose that a small subset of network nodes in susceptible state S are infected and become infected state I. Note the relationship between the fixed point and I. * The equation is:

[0139]

[0140] If N0S * -1>0, the proportion of infected network nodes I * It will immediately begin to rise exponentially, at which point I... * =0 is an unstable fixed point, and cascading failures in parametric association networks will spread. Therefore, the necessary condition for the epidemic spread of cascading failures in parametric association networks is:

[0141] N0S * >1

[0142] If a network node has been infected before, recovered, and then become infected again, the network node that was originally in a susceptible state S will become in a reinfected state A. Since each network node in a reinfected susceptible state S has an equal probability of coming into contact with downstream nodes, and there is a certain probability of transmission during contact, but these network nodes have already developed corresponding antibodies, the transmission probability is now μ = κβ. α Where κ and α are infection rate variables, the probability that each network node in susceptible state S will be infected during this period is l(t)×μΔt. Since all network nodes are considered to be in susceptible state S, the total number of network nodes in susceptible state S is N. The expected number of newly infected network nodes during the period from t to t+Δt is N×l(t)×μΔt. The expected number of infected nodes ultimately affected by a network node in re-infected state A, i.e., the basic reproduction number, is:

[0143] N0=∫0 ∞ Nμl(t)dt

[0144] If a network node that was originally in a susceptible state S changes to a re-infected state A, there is a certain probability that it will recover and change from the re-infected state A to the recovered state R. At this time, the recovery probability is λ = vγ. θ Where v and θ are recovery rate variables. Within the same time interval Δt, a network node in the infected state I has a definite recovery probability of λΔt, and the expected number of network nodes in the recovered state R during this time is A×λΔt.

[0145] Therefore, the fixed point (S) of the epidemiological model equations for reinfection * A * R * )for:

[0146]

[0147] Combining the epidemiological model equations of basic reproduction number and reinfection, it can be concluded that when A * A value of 0 is a necessary condition for a fixed point. At this point, no network node is in a reinfected state A, and therefore nodes in a susceptible state S will not face the possibility of being exposed to infection transmission. As for network nodes in a recovered state R, they naturally remain healthy. In this way, the total number of network nodes will necessarily remain stable.

[0148] If a network node in a susceptible state S can infect and spread to more than one downstream network node in a susceptible state S, then in the production line parameter association network structure, the propagation chain will inevitably expand like a snowball, eventually evolving into a large-scale cascading failure.

[0149] To further explain, in S3, the expression for the propagation threshold R0(t) is as follows:

[0150]

[0151] Where R0(t) represents the propagation threshold as it changes with time t, and αL(t) represents the inhibitory effect of network nodes on the recovery rate γ. <t>This indicates the average degree of correlation between production line parameters and the network structure.

[0152] When the propagation threshold R0(t) > 1, the infection and change propagation of network nodes in the production line parameter association network structure continues to erupt; when R0(t) < 1, the infection and change propagation of network nodes in the production line parameter association network structure subsides on its own.

[0153] When production line design changes, each parameter in the production line parameter association network structure possesses a certain degree of resilience. Take a robotic arm, for example; its extension distance and speed can be adjusted to meet the requirements of modified designs. In the production line parameter association network structure, the propagation threshold becomes increasingly important for parameter nodes throughout the entire production line. To minimize the risk of design changes at critical nodes, a lower propagation threshold is preferable. To illustrate the cascading failure phenomenon caused by node propagation in the production line parameter association network structure, a propagation threshold is constructed.

[0154] To further explain, in S4, the random attack RA refers to randomly selecting network nodes in the production line parameter associated network structure, with a random attack ratio τ. RA This is the ratio of the initial number of network nodes in a random attack to the total number of network nodes.

[0155] To further explain, in S4, the intent attack IA refers to attacking network nodes with high out-degree or high in-degree after sorting them by their out-degree or in-degree. The intent attack ratio τ IA This is the ratio of the initial number of network nodes intended for an attack to the total number of network nodes.

[0156] By analyzing the random attack ratio τ RA Ratio of Intended Attacks τ IA The impact of parameter association network structure on the robustness of mobile phone production assembly line was investigated. Intentional and random attacks were performed on five network structures in the parameter association network structure of the production line, namely tree, mesh, hybrid, star and ring. It was found that the mesh network structure has the greatest impact on the robustness of the parameter association network structure of the production line.

[0157] To further illustrate, in S5, the normalized avalanche magnitude S(t) is used to quantify the infection rate β and recovery rate γ of network nodes at time t, and the impact of changes in the network structure related to production line parameters on the robustness of multilayer networks is analyzed.

[0158] The expression for the magnitude of the avalanche, S(t), is:

[0159]

[0160] Where S(t) represents the normalized avalanche magnitude caused by network node i in the infected state I in the production line parameter association network structure after cascading propagation, and N represents the total number of network nodes in the production line parameter association network structure.

[0161] Considering the dynamic process of cascading propagation, the infection rate β and recovery rate γ of the nodes at time t are used to evaluate the robustness effect of changes in the parameter association network structure, and the normalized avalanche size S(t) is used to quantify the robustness of the interdependent parameter association network structure to cascading failures.

[0162] In summary, this method can obtain the propagation process curve of the epidemic model in the parameter-related network structure of a multi-layer network in a mobile phone production line during design changes. Furthermore, it analyzes the relationship between relevant parameters and the robustness of the production line parameter-related network structure.

[0163] 1. By analyzing the combination of μ and λ, and taking into account the variation of the propagation threshold parameter R0, the relationship between β and γ on the robustness of the production line parameter-related network structure can be obtained. In the epidemic model, parameters β and γ determine the total protection cost of the interdependent parameter-related network structure; that is, the larger the value of β and the smaller the value of γ, the higher the protection cost, and the stronger the robustness of the interdependent network parameter-related network structure against cascading failures.

[0164] 2. The performance of interdependent parameter-related network structures was measured using values ​​β and γ, as well as the normalized avalanche size S, providing some guidance for balancing the costs of initial investment and subsequent maintenance or replacement.

[0165] 3. By analyzing the random attack ratio τ RA Ratio of Intended Attacks τ IA The impact of the network structure on the robustness of the production line parameter association network was investigated, and it was determined that the mesh network structure has the greatest impact on the robustness of the production line parameter association network structure.

[0166] The technical principles of the present invention have been described above with reference to specific embodiments. These descriptions are merely for explaining the principles of the invention and should not be construed as limiting the scope of protection of the invention in any way. Based on this explanation, those skilled in the art can readily conceive of other specific embodiments of the invention without inventive effort, and these equivalent variations or substitutions are all included within the scope defined by the claims of this application.< / t> < / k>

Claims

1. A method for robustness analysis of the propagation of changes in the network structure of production line parameters, characterized in that, Includes the following steps: S1. The design parameters of the production line in four dimensions—configuration, dynamics, control, and optimization—are abstracted into network nodes, and the interactions between parameters are mapped into network topology edges. This constructs a production line parameter association network structure that includes five multi-layer network structures: tree, mesh, hybrid, star, and ring. S2. Define the network node states as susceptible state S, infected state I, recovered state R, and reinfected state A, and define the network node state probabilities as infection rate β, recovery rate γ, reinfection rate μ, and reinfection rate λ. Based on the change propagation process of the epidemic model under attack, the state transition of network nodes is simulated by infection rate β, recovery rate γ, reinfection rate μ and reinfection rate λ, so as to realize the change propagation process of the production line parameter associated network structure under cascading failure, and form a change propagation model of the production line parameter associated network structure. S3, Construct the propagation threshold According to the propagation threshold Determine the outbreak or subsidence of infection changes propagation in network nodes within the network structure change propagation model associated with production line parameters; S4. Simulate the propagation process of changes in the production line parameter-related network structure under two attack strategies: random attack (RA) and intentional attack (IA). Calculate the random attack ratio. and intentional attack ratio ; S5. Normalize avalanche size. As a robustness metric, we analyze the impact of changes in the network structure associated with production line parameters on the robustness of multi-layer network structures.

2. The robustness analysis method for the propagation of changes in the network structure of production line parameters according to claim 1, characterized in that, In S1, the five multilayer network structures include: The tree-like network structure formed between the two dimensions of configuration and dynamics; The mesh network structure formed between the dynamic and control dimensions; The hybrid network structure formed between the two dimensions of control type and optimization type; The star-shaped network structure formed between the internal components of the control dimension; The ring network structure formed between the internal components of the optimal dimension.

3. The robustness analysis method for the propagation of changes in the network structure of production line parameter associations according to claim 1, characterized in that, In S2, simulating network node state transitions using infection rate β, recovery rate γ, reinfection rate μ, and reinfection rate λ includes: S21. Set the initial network node state of all network nodes in the production line parameter association network structure to the susceptible state S. S22. A network node in a susceptible state S is attacked and then becomes an infected state I with an infection rate β. S23. Network nodes in infected state I that transition to recovered state R with recovery rate γ. S24. A network node in the recovery state R can become a network node in the susceptible state S again, and be reinfected as a network node in the reinfection state A with a reinfection rate μ. S25. Network nodes in the reinfection state A are transformed into the recovery state R with a reinfection rate λ. S26. Repeat S22-S25 until no network node in the production line parameter association network structure is infected due to a susceptible state S.

4. The robustness analysis method for the propagation of changes in the network structure of production line parameters according to claim 1, characterized in that, In S4, the random attack The random attack ratio is used to randomly select network nodes in the production line parameter association network structure. This is the ratio of the initial number of network nodes in a random attack to the total number of network nodes.

5. The robustness analysis method for the propagation of changes in the network structure of production line parameters according to claim 1, characterized in that, In S4, the intent attack The intent attack ratio is to attack network nodes with high out-degree or high in-degree after sorting them by their out-degree or in-degree. This is the ratio of the initial number of network nodes intended for an attack to the total number of network nodes.

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