A network resilience index multi-dimensional quantitative evaluation method

Abstract

The present application relates to the technical field of network analysis, and further relates to a network resilience index multi-dimensional quantitative evaluation method, which comprises the following steps: step 1: obtaining original topology data of a target network, and constructing a topology base resilience index of the network under the condition of no external impact; step 2: calculating a crisis exposure factor; step 3: calculating the intermediate centrality of each node; calculating the structural hole limit system of each node; combining the intermediate centrality and the information redundancy degree to generate an adaptation-recovery index for measuring the self-healing and regenerative capacity of the network; step 4: correcting the topology base resilience index and the crisis exposure factor to resist impact, then amplifying the index weight according to the adaptation-recovery index, and combining the stability correction of network resilience fluctuation to finally generate a multi-dimensional network resilience comprehensive index. The present application is helpful to realize network vulnerability identification and resilience strategy optimization based on structural attributes and dynamic historical response.

Description

Technical Field

[0001] This invention belongs to the field of network analysis technology, specifically relating to a multi-dimensional quantitative evaluation method for network resilience index. Background Technology

[0002] In current research and practical applications of complex networks, network resilience, as a multi-dimensional system characteristic, has become a hot topic of interdisciplinary attention. Network resilience is not only reflected in the network's ability to maintain function when subjected to external shocks, but also involves its rapid recovery capability and long-term stability after an impact. Existing research on network resilience largely focuses on single-dimensional or static robustness measurements, typically using basic topological indicators such as node degree distribution, average path length, and clustering coefficients as descriptive tools. For example, connectivity metrics and average path length are often used to characterize the overall robustness of a network, suggesting that high connectivity of nodes or links means the network structure is more difficult to fragment under random attacks; similarly, clustering coefficients are used to measure local clustering effects, judging the network's ability to absorb and transmit localized failures of links or nodes. Meanwhile, network degree distribution and degree correlation coefficients are also frequently used to identify network hierarchy and matching patterns, thereby explaining the network's stability and shock resistance potential from a structural characteristic perspective.

[0003] While the aforementioned topological indices and traditional measures have achieved good results in evaluating the static stability of network structures, they often fail to fully characterize the network's dynamic recovery and adaptability in the face of external shocks. This is because these static topological features lack time-dimensional information and cannot reflect the network's actual response and adjustment process under successive external shocks. Some studies have attempted to introduce dynamic models, such as propagation-based simulations, to explore changes in the network's propagation characteristics after a shock, thereby indirectly reflecting the network's resilience. However, due to the lack of actual historical shock data, these models struggle to accurately quantify the depth and duration of the shock's impact on the network's actual operational functions, leading to assessment results that often lean towards theoretical deductions rather than real network responses. Summary of the Invention

[0004] The main objective of this invention is to provide a multi-dimensional quantitative evaluation method for network resilience index, which integrates the network's topological resilience, historical crisis impact characteristics, node recovery potential, and long-term network stability. Based on the continuous fusion of multi-dimensional factors, a quantifiable, comparable, and dynamically updatable comprehensive network resilience index is generated. This index can accurately characterize the comprehensive capabilities of complex networks in resisting external shocks, achieving rapid recovery, and maintaining long-term operational stability. It helps to achieve network vulnerability identification and resilience strategy optimization based on structural attributes and dynamic historical responses.

[0005] To solve the above problems, the technical solution of the present invention is implemented as follows:

[0006] A multi-dimensional quantitative evaluation method for network resilience index, the method comprising:

[0007] Step 1: Obtain the original topology data of the target network and construct the topology resilience index of the network under the condition of no external shock. The larger the value, the healthier the network is at the inherent structural level.

[0008] Step 2: Identify all crisis stages based on historical monitoring records to obtain the total number of crisis stages; for each crisis stage, the percentage decline in network function is taken as the impact intensity; the duration of continuous decline in network function is taken as the crisis duration; the time taken for network function to recover from the lowest point to the pre-crisis baseline is taken as the recovery time; calculate the crisis exposure factor based on the total number of crisis stages, impact intensity, crisis duration, and recovery time.

[0009] Step 3: Calculate the betweenness centrality of each node to measure the bridging role of the node in the information flow of the entire network; calculate the structural hole constraint degree of each node to measure the degree of information redundancy controlled by the node; combine the betweenness centrality and the degree of information redundancy to generate the adaptation-recovery index, which measures the network's self-healing and regeneration capabilities.

[0010] Step 4: By adjusting the topological base resilience index and the crisis exposure factor for shock resistance, then amplifying the index weights based on the adaptation-recovery index, and combining the stability correction of network resilience fluctuations, a multi-dimensional comprehensive network resilience index is finally generated.

[0011] Furthermore, the original topology data includes: the degree of each node, the local clustering coefficient, the number of independent paths, and the average shortest path length to all other nodes.

[0012] Furthermore, step 1 specifically includes: at the node scale, multiplying the node degree by the local clustering coefficient to characterize the connectivity-cluster coupling effect, directly using the number of independent paths as the diversity factor, and taking the reciprocal of the average shortest path length to characterize the transmission efficiency; at the overall network scale, summing and averaging the above three types of node factors respectively, and then amplifying the connectivity-cluster coupling effect and transmission efficiency twice, and performing a logarithmic transformation on the result and the sum of the diversity factors to obtain the topological substrate resilience index under the condition of no external shock.

[0013] Furthermore, in step 2, the impact intensity and duration of each crisis stage are multiplied and summed, and the result is divided by the total sum of the impact intensity and recovery duration to obtain the crisis exposure factor; the larger the value of the crisis exposure factor, the more vulnerable the network is during the impact stage.

[0014] Furthermore, step 3 specifically includes: setting an evaluation time window and locking a baseline topology snapshot; assigning a time weight to each node in the baseline topology snapshot; calculating the standardized betweenness centrality of all nodes under the same baseline topology snapshot; simultaneously calculating the structural hole constraint degree of all nodes and performing inverse sigmoid compression on the structural hole constraint degree to make the structural hole advantage numerically present a saturation upper limit, avoiding absolute monopoly of nodes leading to unbalanced evaluation results; dividing the standardized betweenness centrality by the structural hole constraint degree node by node to obtain the bridge hole coupling rate, then using the time weight as a multiplier to perform weighted accumulation of the bridge hole coupling rate, and taking the square root of the accumulated result to form the node-level [statistic / response]. The recovery potential index is developed by assigning equal weights to the recovery potential index of each node in the network or by weighting it according to the node weights, and calculating the weighted arithmetic mean to represent the average recovery potential level of the network as a whole. Then, based on the same weighting system, the squared differences between the recovery potential index of each node and the weighted average of the network are calculated and summed to obtain the weighted squared deviation of the distribution, which reflects the uniformity of the distribution of recovery potential among different nodes. Finally, the weighted arithmetic mean and the weighted squared deviation are combined in a harmonic weighting method to form a comprehensive metric that can simultaneously reflect the average strength and distribution balance of the recovery potential among network nodes, resulting in the adaptation-recovery index.

[0015] Furthermore, the time weight is determined by the node's most recent position. The standard deviation of the fluctuation of the betweenness centrality sequence within a time period is obtained by reverse scaling, which is used to emphasize the dominant role of structurally stable nodes in the recovery process; the standardized betweenness centrality is obtained by a triple transformation of quartile removal, 0-1 linear stretching and power law reduction, to ensure that the bridge effect can maintain the distribution difference and is not masked by extreme values.

[0016] Furthermore, step 3 also includes: in parallel, traversing all possible pairwise combinations between nodes, counting the number of independent parallel paths between each pair of nodes, and calculating the ratio of this ratio to the theoretical maximum number of parallel paths between two nodes to quantify the link diversity of the network; simultaneously, calculating the modularity coefficient of the Louvain community partitioning results of the network, and converting it into a decay factor using a reciprocal shift to reflect the positive contribution of partition isolation to diffusion suppression; multiplying the primary recovery coefficient by the link diversity ratio, and then adding the decay factor additively to obtain the comprehensive recovery potential without scale compression; if the comprehensive recovery potential exceeds the theoretical resilience upper limit, then using a hyperbolic tangent function for mapping compression to ensure consistency of the index range; to offset the index comparability bias caused by differences in network size, introducing a logarithmic term for the number of nodes as a denominator adjustment, so that larger and more dispersed networks will not have their recovery potential overestimated due to the large number of nodes, thus obtaining the scale-standardized result; performing numerical normalization processing based on the scale-standardized result, and finally outputting the adaptation-recovery index, which is continuously monotonic between 0 and the theoretical maximum value, and maintains a first-order positive correlation with bridge-cavity synergy, link diversity, and partition isolation.

[0017] Furthermore, the process of numerical normalization based on the scale-normalized results includes: using the scale-normalized results as a benchmark, performing a Box-Cox transformation on them and multiplying by a constant. The numerical stretching is performed, and the final output is the adaptation-recovery index.

[0018] Furthermore, step 4 specifically includes: using the topological substrate toughness index divided by the crisis exposure factor plus one as the structure-impact correction term; setting the index weight of the structure-impact correction term as the ratio of the adaptation-recovery index to the adaptation-recovery index plus one, to reflect the amplifying effect of recovery capacity on overall toughness, thus obtaining the correction adjustment result; selecting the most recent Using a year as the time window, the standard deviation of all topological base resilience indices within that time window is calculated to represent the long-term fluctuation range of base resilience. The standard deviation of the negative topological base resilience index of the natural exponential function is used as a stability correction term. The correction result is multiplied by the stability correction term to obtain a multi-dimensional network resilience comprehensive index. The larger the value of the network resilience comprehensive index, the stronger the network resilience.

[0019] This invention provides a multi-dimensional quantitative evaluation method for network resilience index, which has the following advantages: it enables comprehensive quantification of network resilience across multiple dimensions, overcoming the limitations of traditional methods based on a single topology or propagation model. This method fully considers the local structural attributes of nodes, such as connectivity, clustering coefficient, number of independent paths, and global transmission efficiency. Simultaneously, it introduces time-series information based on the impact intensity, duration, and recovery time of historical crisis events, effectively offsetting the network's inherent structural strength against historical external shocks. By introducing time-weighted bridge coupling rate and structural hole limitation correction, it further integrates the bridge effect and redundancy information control capabilities at the node level. Combined with link diversity ratio and modularity coefficient to measure redundant paths and community isolation effects, it achieves a three-dimensional characterization of adaptability and recovery capabilities. Through the fusion of an index weight amplification mechanism and a stability correction term, it ensures that the comprehensive resilience index can sensitively capture recovery potential and long-term fluctuation characteristics, with monotonic and continuous values, strong comparability, and suitability for dynamic tracking. In summary, the method not only improves the accuracy of network response identification when facing external shocks, but also provides operable indicators to support resilience strategy optimization, providing a strong theoretical basis and practical tool for multi-level, full-cycle resilience enhancement management of complex networks. Attached Figure Description

[0020] Figure 1 This is a flowchart illustrating a multi-dimensional quantitative evaluation method for network resilience index provided in an embodiment of the present invention.

[0021] Figure 2 This is a scatter plot showing the relationship between the intermediation centrality and the structural hole constraint degree in an embodiment of the present invention.

[0022] Figure 3 This is a distribution diagram of node degree and local clustering coefficient in an embodiment of the present invention. Detailed Implementation

[0023] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0024] Example 1: Reference Figure 1 A multi-dimensional quantitative evaluation method for network resilience index, the method comprising:

[0025] Step 1: Obtain the original topology data of the target network and construct the topology resilience index of the network under the condition of no external shock. The larger the value, the healthier the network is at the inherent structural level.

[0026] Step 2: Identify all crisis stages based on historical monitoring records to obtain the total number of crisis stages; for each crisis stage, the percentage decline in network function is taken as the impact intensity; the duration of continuous decline in network function is taken as the crisis duration; the time taken for network function to recover from the lowest point to the pre-crisis baseline is taken as the recovery time; calculate the crisis exposure factor based on the total number of crisis stages, impact intensity, crisis duration, and recovery time.

[0027] Step 3: Calculate the betweenness centrality of each node to measure the bridging role of the node in the information flow of the entire network; calculate the structural hole constraint degree of each node to measure the degree of information redundancy controlled by the node; combine the betweenness centrality and the degree of information redundancy to generate the adaptation-recovery index, which measures the network's self-healing and regeneration capabilities.

[0028] Step 4: By adjusting the topological base resilience index and the crisis exposure factor for shock resistance, then amplifying the index weights based on the adaptation-recovery index, and combining the stability correction of network resilience fluctuations, a multi-dimensional comprehensive network resilience index is finally generated.

[0029] Specifically, at the structural layer, the method abstracts any real network into a topological object composed of nodes and edges, arguing that the connectivity between nodes, the degree of local clustering, the information transmission distance, and the number of parallel paths jointly determine the normal functional level that the network can maintain when undisturbed. At this level, the method aggregates the global connectivity strength, cluster compactness, transmission efficiency, and redundancy path richness of the network and categorizes them into a topological basis resilience index, thereby using a single value to map the "buffer space" provided by the inherent structure of the network to the system.

[0030] Secondly, in the impact layer, the method introduces the concept of exposure from risk science, treating historically observed external crises as repeated "experiments." By statistically analyzing four factors—impact intensity, duration, recovery time, and the total number of crisis stages—the entire process of each functional decline and recovery is transformed into a measurable time risk trajectory. These trajectories are then aggregated and converted into a crisis exposure factor, thereby revealing the composite risk load and cumulative vulnerability faced by the network at different times. The essence of this factor is a joint characterization of impact frequency, magnitude, and recovery rate, which compresses the temporal impact history into a spatially comparable scale.

[0031] Then, at the adaptation layer, the method follows organizational behavior regarding the importance of "bridge nodes," "information control nodes," and "redundant paths" for rapid recovery. It measures the information hub efficiency represented by betweenness centrality, the resource exclusivity represented by structural hole constraint, the redundancy flow richness represented by link diversity ratio, and the community isolation strength reflected by modularity coefficient, respectively. These are mapped to the adaptation-recovery index through nonlinear coupling to express the network's regenerative potential of self-organizing to form alternative pathways and to what extent it can block cascading failures after functional impairment. At this level, the bridging effectiveness and information redundancy of nodes are regarded as micro-driving forces, while multipath and partition isolation are regarded as macro-safety valves. Both are transformed into individual units after scale unification.

[0032] Finally, at the integration layer, the method considers that the contributions of the structural layer, shock layer, and adaptation layer are not linearly added, but rather have significant amplification or offsetting effects. Therefore, it proposes a three-step fusion logic of shock resistance correction, exponential weight amplification, and stability correction: shock resistance correction uses the topological base resilience index to deduct the crisis exposure factor, so that the inherent buffer and external load are offset within the same framework; exponential weight amplification uses the adaptation-recovery index to amplify the correction result exponentially within the normalization interval of zero to one, emphasizing the multiplier effect of self-healing potential on overall resilience; stability correction introduces the structural base fluctuation amplitude of a sliding window, maps the inverse weight of long-term volatility to the exponential space, and suppresses the score of networks with short-term peaks but long-term instability.

[0033] The multi-dimensional network resilience index generated by the product of these three factors simultaneously grasps the four main lines of "inherent strength," "historical load," "acquired recovery," and "long-term stability," achieving a quantitative expression of the resilience of complex networks throughout their entire lifecycle using a single scale. The core innovation of this principle lies in two aspects: First, it extends the traditional static robustness based solely on connectivity or average path length to a dynamic resilience that considers the timing of shocks and functional reconstruction. Second, by introducing multi-scale coupling of community isolation and information hubs, it overcomes the shortcomings of simply using global or local indicators for unified measurement. Furthermore, the modular architecture in the computational framework allows the topological base resilience index, crisis exposure factor, adaptation-recovery index, and comprehensive index to be independent yet interconnected, ensuring flexible substitution during method implementation and guaranteeing the comparability and accumulative nature of the indicator system. In practical implementation, the system first records node relationships and timestamp-marked functional indicators at the data storage layer using a graph database or sparse matrix. At the computation layer, it sequentially calls the structure extraction service, historical impact analysis service, self-healing potential assessment service, and index fusion service, with intermediate results transmitted between services via a unified data interface. At the presentation layer, the system projects the multi-dimensional network resilience comprehensive index and its sub-indices onto an interactive time-series panel and structural heatmap, helping users observe resilience evolution simultaneously from both temporal and structural dimensions. At the application layer, the index can drive risk warning, vulnerability location, and resource scheduling decisions, and can be linked with a simulation platform to verify the real-time effectiveness of intervention strategies in improving overall resilience. In summary, this method's framework is based on the topological host of complex networks, incorporates the time-series exposure concept of impact chains, combines a multi-scale self-healing mechanism of nodes, paths, and communities, and then converts it into individual indicators through multi-factor coupling, ultimately achieving a quantitative assessment of the entire network resilience landscape.

[0034] Furthermore, the original topology data includes: the degree of each node, the local clustering coefficient, the number of independent paths, and the average shortest path length to all other nodes.

[0035] The original topology data is defined as a set of basic structural quantities that can fully characterize the local connectivity features and global transmission characteristics of nodes. Specifically, it includes four components: First, the degree of each node, which quantifies the number of direct connections a node has and reflects its basic connectivity potential in the network; Second, the local clustering coefficient of each node, which measures the probability of forming closed triangles among the node's neighbors and reveals the cluster compactness around the node; Third, the number of independent paths, which counts the number of parallel, conflict-free paths between node pairs and reflects the overall redundancy reserves of the network; Fourth, the average shortest path length to all other nodes, which characterizes the node's radiation radius and information transmission efficiency on a global scale.

[0036] See Figure 2This figure illustrates the distribution of the relationship between network node betweenness centrality and structural hole constraint. The horizontal axis represents standardized betweenness centrality, ranging from 0 to 1.6, and the vertical axis represents structural hole constraint, ranging from 0 to 0.8. Each solid black circle in the figure represents a node in the network, with a total of 20 node samples. Analysis of these node data reveals a negative correlation between standardized betweenness centrality and structural hole constraint; that is, nodes with higher betweenness centrality tend to have lower structural hole constraint. This indicates that key nodes that act as bridges in network information flow typically have stronger information control capabilities and lower structural constraints. The dashed line in the figure represents the trend fitting line of the data points, extending from the lower left to the upper right, further verifying the negative correlation between the two. This distribution provides an important data foundation for the subsequent calculation of the bridge-hole coupling rate, which is obtained by dividing the standardized betweenness centrality by the structural hole constraint node by node and is used to measure the potential contribution of a node in the network recovery process.

[0037] See Figure 3 This figure illustrates the distribution relationship between network node degree and local clustering coefficient. The horizontal axis represents node degree, ranging from 0 to 16, and the vertical axis represents the local clustering coefficient, ranging from 0 to 0.8. Each solid black circle in the figure represents a node in the network, with a total of 16 node samples. The scatter distribution shows a negative correlation between node degree and local clustering coefficient; that is, nodes with higher node degrees have relatively lower local clustering coefficients, which aligns with typical characteristics in complex network theory. The label in the lower right corner of the figure clearly explains the calculation method for the connectivity-clustering coupling effect, which involves multiplying the node degree by the local clustering coefficient. This coupling effect is a crucial component in constructing the topology basis resilience index, characterizing the synergistic effect of network connectivity and clustering at the node scale. At the overall network scale, by summing and averaging the connectivity-clustering coupling effect of all nodes, and combining transmission efficiency and diversity factors, after secondary amplification and logarithmic transformation, the final topology basis resilience index under the condition of no external shocks is formed. This index reflects the health of the network at the intrinsic structural level.

[0038] Furthermore, step 1 specifically includes: at the node scale, multiplying the node degree by the local clustering coefficient to characterize the connectivity-cluster coupling effect, directly using the number of independent paths as the diversity factor, and taking the reciprocal of the average shortest path length to characterize the transmission efficiency; at the overall network scale, summing and averaging the above three types of node factors respectively, and then amplifying the connectivity-cluster coupling effect and transmission efficiency twice, and performing a logarithmic transformation on the result and the sum of the diversity factors to obtain the topological substrate resilience index under the condition of no external shock.

[0039] The method multiplies the degree of each node with its local clustering coefficient, interpreting the product as a connectivity-cluster coupling effect to simultaneously capture the number of direct associations between nodes and the compactness of neighborhood clusters. The method regards the number of independent paths as a diversity factor, believing that the more parallel and non-interfering paths there are, the richer the redundancy support the network has in the event of local failure. The method takes the reciprocal of the average shortest path length from a node to all other nodes and interprets it as a transmission efficiency indicator, because the shorter the average distance, the faster information or resources can spread in the network.

[0040] After completing the three calculations at the node scale, the method proceeds to the overall network scale processing: First, the connectivity-cluster coupling effect, diversity factor, and transmission efficiency index of all nodes are summed, and then divided by the total number of network nodes to obtain the arithmetic mean of the three indices. Next, the method performs a second amplification operation on the average of the connectivity-cluster coupling effect and the average of the transmission efficiency. This amplification emphasizes the contribution of structural strength to resilience in the numerical space, giving the advantages of high connectivity-high clustering and high-speed transmission quadratic weights. Immediately afterward, the method sums the amplified result with the average of the diversity factor, converging the information from the three dimensions into a single scalar. A natural logarithmic transformation is then applied to this scalar to compress the numerical values ​​output by networks of different sizes and ensure that the result distribution is closer to a normal distribution. This continuous transformation chain starts from the dual-feature coupling at the local node level, proceeds through global average merging, nonlinear weight amplification, and logarithmic contraction, and finally outputs the calibrated value of the topological basis resilience index. A larger value indicates that the network can provide a larger buffer space by relying on its own structure in the absence of external shocks, because high connectivity-aggregation can reduce the split caused by node detachment, high diversity can provide alternative paths for failed nodes, high transmission efficiency can shorten the crawl time required for rewiring or information rerouting, and logarithmic compression ensures that the exponents of networks with different numbers of nodes are comparable.

[0041] The implementation process requires first traversing all nodes at the data acquisition level to generate four mapping tables in real-time or offline: node degree, local clustering coefficient, number of independent paths, and average shortest path length. Then, in the computing service, connectivity-cluster coupling effect, diversity factor, and transmission efficiency are derived at the node level. Next, the aggregation service performs network-wide summation and averaging. Finally, the index fusion module performs square amplification, summation, and logarithmic transformation to obtain the standardized topological base resilience index. This index is then written into the evaluation database for subsequent use by crisis exposure factors and adaptation-recovery indices, thus playing the role of an inherent structural health benchmark in the subsequent steps of the multi-dimensional quantitative evaluation method for network resilience.

[0042] Specifically, the topological substrate toughness index for:

[0043] ;

[0044] in, This represents the total number of network nodes. For nodes The degree; For nodes The local clustering coefficient; For nodes The number of independent paths to other nodes; For nodes The average shortest path length to all other nodes.

[0045] Furthermore, in step 2, the impact intensity and duration of each crisis stage are multiplied and summed, and the result is divided by the total sum of the impact intensity and recovery duration to obtain the crisis exposure factor; the larger the value of the crisis exposure factor, the more vulnerable the network is during the impact stage.

[0046] The method draws heavily on the concepts of risk accumulation and time weighting in constructing crisis exposure factors, treating each external shock as a load segment of a resilience test. It unifies the "force" and "quantity" of the shock within the same evaluation framework using two descriptive quantities: the magnitude of functional loss and the duration of impact. The method first identifies all identifiable crisis phases within the monitoring period on a timeline. The division of crisis phases is automatically captured based on the continuous decline trajectory of network function indicators. Whenever network function experiences a sustained decline over multiple periods and its absolute value falls below the previous baseline, the system marks a new shock event and records the start time, lowest point, and endpoint of recovery to the baseline. For each confirmed crisis phase, the method extracts three core pieces of information: First, the impact intensity, defined as the percentage decrease in network function at the lowest point relative to the pre-crisis baseline. This measure characterizes the proportion of instantaneous loss to network function caused by external disturbances. Second, the crisis duration, defined as the length of time it takes for network function to decline from the start to the lowest point. This measure reflects the time dimension of the impact maintaining high pressure. Third, the recovery time, defined as the length of time it takes for network function to climb back up from the lowest point and touch the pre-crisis baseline. This time period measures the time cost required for the network to spontaneously repair itself and indirectly reflects the efficiency of the network's internal repair mechanism. Using these three indicators in conjunction can simultaneously depict the depth and breadth of the impact and the speed of network recovery.

[0047] After obtaining the impact intensity and duration of all crisis phases, the method couples them by product, interpreting the product of impact depth and duration as the contribution of a single crisis to the cumulative damage of the network. This is because a higher magnitude of loss over a longer duration means greater functional weakening and resource dissipation. The method then sums the cumulative damage contributions of all crisis phases, accumulating all impact energies over historical periods to form the numerator. Simultaneously, the method also sums the impact intensity and recovery time of each crisis phase, combining them into the denominator. The physical meaning of this step is to compare damage contribution and repair cost on the same scale: when the network suffers significant losses in some phases but recovers quickly, the recovery time in the denominator reduces the increase in overall vulnerability due to damage accumulation; conversely, if the network suffers relatively minor losses in some phases but the duration is extremely long or the recovery is extremely slow, the numerator will be rapidly amplified over a long period, while the repair time in the denominator cannot effectively offset this amplification effect due to continuous increases, leading to an overall increase in the crisis exposure factor, thus reflecting the network's vulnerability weighting in the time dimension. The ratio designed in this way can be regarded as an energy density measure with a time factor in an algebraic sense. The damage impulse of each impact is accumulated with time weighting, and then normalized with the repair consumption with the same time attribute. The result will adaptively amplify the comprehensive vulnerability characteristics of long-term, high-intensity and slow recovery.

[0048] In practical implementation, the system first constructs a time-resolution adjustable functional surface using a continuously sampled sequence of functional indicators. This surface exhibits a fluctuating pattern on the time axis, with each local minimum and adjacent local maximum considered a potential shock event. The algorithm enters crisis mode upon detecting a continuous downward trend, accumulating the decline amplitude in real time and observing the lowest point. Once the functional curve rebounds and re-intersects the baseline before the decline after several periods, a crisis phase is declared over. All identified crisis phases are added to a diagnostic list in chronological order, with each record containing the start time, end time, lowest point time, and corresponding three indicator values. To mitigate the impact of observation errors and data noise on the assessment of shock intensity, the system employs a sliding median filter to eliminate occasional spikes in the functional decline amplitude and time window smoothing to eliminate critical point jitter in the duration and recovery time. Then, when calculating the crisis exposure factor, the system multiplies and sums all smoothed shock intensities with their durations, then simultaneously sums all shock intensities with their recovery times, and finally divides the former by the latter to output the calibrated value of the crisis exposure factor. Because impact intensity and duration are accumulated through a product, both short, intense impacts and long, gradual impacts are given appropriate weights, resulting in a "damage volume" that combines temporal and loss depths after comprehensive aggregation. Similarly, the sum of the denominator's repair time and impact intensity couples the recovery rate with the loss magnitude before summing them all, ensuring that the denominator's growth rate generally follows the network's spontaneous repair capability. When the network exhibits a high recovery rate, the denominator increases faster, and the crisis exposure factor decreases; when the network recovers slowly, the denominator increases slower, and the crisis exposure factor increases. Thus, the direction of numerical change perfectly corresponds to the intuitive meaning of "vulnerability," that is, the larger the value, the more vulnerable the network.

[0049] In terms of its properties, the crisis exposure factor is a dimensionless, non-negative quantity with theoretically no upper limit. For networks with no historical impact records, the numerator is zero, and the factor value is naturally zero. For networks with extremely short impacts but extremely rapid recovery, the numerator is not zero, but the denominator grows very rapidly, and the factor value approaches zero. For networks subjected to prolonged impacts and slow recovery, the ratio of the numerator to the denominator increases rapidly, and the factor value climbs to a higher range. This property allows the crisis exposure factor to be regarded as a risk density indicator based on the cumulative damage and repair capacity over time. It is also the direct numerator for the impact correction in the subsequent multi-dimensional network resilience comprehensive index. When the topological base resilience index is corrected by the crisis exposure factor, it can present the offsetting results of "structural buffer" and "historical load" on the same dimension, further laying a discriminative foundation for the weight amplification of the recovery index in the index fusion process.

[0050] To ensure cross-network comparability of indicators, the system employs a unified sampling frequency and crisis judgment threshold when evaluating different networks. This ensures consistency in the measurement of impact intensity and duration, while maintaining the same recovery baseline definition to preserve the comparability of recovery duration. When real-time updates to the danger situation are needed, the system can continuously import the latest monitoring data in a rolling window, redraw the crisis stage according to the latest functional curve, incrementally update the impact intensity, duration, and recovery time, and then recalculate the crisis exposure factor online. This makes the phased risk load indicator traceable in real time over time. In summary, the crisis exposure factor, through a dual time weighting mechanism of multiplicative product and additive normalization, integrates the depth and breadth of external shocks and the network's self-healing speed into a single-value metric. Furthermore, by ensuring consistency between the value range direction and vulnerability level, it becomes an indispensable core quantitative component in the multi-dimensional quantitative assessment method of network resilience index for evaluating shock resistance.

[0051] Furthermore, step 3 specifically includes: setting an evaluation time window and locking a baseline topology snapshot; assigning a time weight to each node in the baseline topology snapshot; calculating the standardized betweenness centrality of all nodes under the same baseline topology snapshot; simultaneously calculating the structural hole constraint degree of all nodes and performing inverse sigmoid compression on the structural hole constraint degree to make the structural hole advantage numerically present a saturation upper limit, avoiding absolute monopoly of nodes leading to unbalanced evaluation results; dividing the standardized betweenness centrality by the structural hole constraint degree node by node to obtain the bridge hole coupling rate, then using the time weight as a multiplier to perform weighted accumulation of the bridge hole coupling rate, and taking the square root of the accumulated result to form the node-level [statistic / response]. The recovery potential index is developed by assigning equal weights to the recovery potential index of each node in the network or by weighting it according to the node weights, and calculating the weighted arithmetic mean to represent the average recovery potential level of the network as a whole. Then, based on the same weighting system, the squared differences between the recovery potential index of each node and the weighted average of the network are calculated and summed to obtain the weighted squared deviation of the distribution, which reflects the uniformity of the distribution of recovery potential among different nodes. Finally, the weighted arithmetic mean and the weighted squared deviation are combined in a harmonic weighting method to form a comprehensive metric that can simultaneously reflect the average strength and distribution balance of the recovery potential among network nodes, resulting in the adaptation-recovery index.

[0052] The construction of the Adaptive-Recovery Index is considered a key link between structural resilience and historical shocks. Only by accurately characterizing the network's internal self-healing and regeneration potential can the "innate buffer" of the structure and the "acquired load" of shocks be effectively coupled at the comprehensive index level. The calculation of this index does not rely on external macroscopic information but is entirely based on the network's own endogenous characteristics in its microscopic connectivity and temporal evolution trajectory; therefore, it possesses both topological interpretability and temporal sensitivity. To achieve this goal, the method first introduces the concept of an evaluation time window, strictly limiting the observation range of adaptive-recovery potential to a clearly defined time interval. This aims to avoid semantic conflicts caused by superimposing structural changes from different time periods onto the same projection plane. The upper and lower limits of this time window can be set according to the network's business cycle, topology update frequency, or the strategy evaluation period that managers are concerned with. Locking the baseline topology snapshot involves arbitrarily selecting a representative time segment within the selected time window, freezing the nodes and connections at that time, so that subsequent calculations of node attributes are based on the same structural description, thus eliminating errors caused by structural drift. This forms a "time-topology" dual-fixed analysis framework: the time direction limits the observation depth by the upper and lower boundaries of the evaluation time window, and the structural direction limits the skeleton of node interactions by the baseline topological snapshot. Together, they provide the starting point for evaluation.

[0053] On a fixed baseline topology snapshot, the first step is to assign time weights to each node. The design of these time weights aims to give higher contributions to subsequent index aggregation for nodes that maintain stable bridging roles and consistently play a crucial transitive role in information flow throughout the evaluation time window, thus truly reflecting the long-term reliable recovery support value of these nodes. The method uses the node's betweenness centrality sequence over the entire evaluation time window as input, calculates its standard deviation of volatility, and performs an inverse mapping on the standard deviation: smaller volatility results in higher weights, and larger volatility results in lower weights. This inverse mapping can use linear, logarithmic, or other monotonically decreasing transformations, but it must ensure that the weights fall within a closed interval of zero to one so that they can be directly multiplied with the normalized betweenness centrality discussed later. Through this mechanism, the method makes nodes with persistent and stable bridging roles the main focus of recovery potential assessment, while compressing the contribution of nodes with occasional peaks but drastic overall fluctuations, avoiding the amplified impact of short-term volatility on the index.

[0054] After time weighting is completed, the system calculates the standardized betweenness centrality for all nodes on the same baseline topology snapshot. The standardization process consists of two steps: First, quartile removal is used to truncate outliers beyond a certain distance from the upper quartile and lower quartile, eliminating extreme high or low values ​​caused by observation noise and data entry anomalies. Then, the remaining values ​​are mapped to a unified interval using a zero-to-one linear transformation to ensure comparability of betweenness centralities between networks of different sizes and densities, while also ensuring that any subsequent weighting or division operations are performed within the same numerical space. This standardized betweenness centrality can be viewed as a pure functional measure of a node's role as a bridge; a higher value indicates that the node is more frequently on the shortest path and can provide alternative pathways during information rerouting.

[0055] Next, the system synchronously calculates the structural hole constraint degree of all nodes. The structural hole constraint degree measures the redundancy of a node within its adjacent subgraph, or in other words, the potential advantage of a node in acquiring non-repetitive information. After the calculation, to prevent a few nodes from infinitely amplifying their contribution in subsequent coupling processes due to their absolute monopoly, the method performs inverse sigmoid compression on the original structural hole constraint degree. Inverse sigmoid compression essentially maps the original non-negative value to the zero-to-one interval and assigns a smooth upper limit. When the original value reaches an extremely high level, the mapped value gradually converges to one and will not continue to rise, thus allowing the structural hole advantage to have a numerical capping effect. This compression process not only prevents a single super-monopolistic node from raising the expected recoverability of the entire network, but also ensures that when comparing different networks, high redundancy and ultra-high redundancy no longer increase linearly, but maintain diminishing marginal contributions, which is more in line with the empirical law of diminishing returns in actual networks under extreme redundancy conditions.

[0056] The method then divides the standardized betweenness centrality of a node by the compressed structural hole constraint on a node-by-node basis to obtain the bridge hole coupling ratio. This ratio is proposed to quantify the degree to which a node simultaneously possesses information hub efficiency and redundant information control: if a node has high betweenness centrality but low structural hole constraint, it indicates that while it is an information hub, its redundancy is insufficient, and its own failure will lead to severe disruption; conversely, if it has high structural hole constraint but low betweenness centrality, it controls redundant resources but is not on the critical transmission link, and the probability of it being called during the recovery process is limited. Only when both conditions are met will the bridge hole coupling ratio reach a high value, thus determining that the node has high substitution value in post-disaster rerouting scenarios. After obtaining the bridge hole coupling ratio, the system uses time weight as a multiplier to perform weighted accumulation, combining the node's stability across time with the single-point bridge hole coordination capability; then, the square root of the accumulated result is taken to suppress the nonlinear surge risk caused by excessively high time weight or abnormally high bridge hole coupling ratio, and the square root result is defined as a node-level recovery potential index. The node-level recovery potential index thus encompasses both static bridge coupling and dynamic stability contribution, possessing a "three-dimensional" meaning: strong bridge, strong redundancy, and long-term stability.

[0057] After obtaining the recovery potential indices of all nodes in the network, the system first calculates a weighted arithmetic mean under a unified node weight system to characterize the average recovery potential level of the network as a whole. Node weights can all be set to one, or they can be assigned based on node importance, such as capacity weights or asset weights, as long as consistency is maintained throughout the evaluation process. A high weighted arithmetic mean indicates that the network possesses a high self-healing capability in a general sense, but this mean cannot identify whether the recovery potential is evenly distributed among nodes. Therefore, the system proceeds to the next step of calculating the weighted squared deviation. Under the same weight system, the system subtracts the network mean from the recovery potential index of each node, squares the result, multiplies it by the corresponding weight, and sums the results to obtain the dispersion of the recovery potential distribution. A larger deviation indicates that the recovery potential is highly concentrated in a few key nodes, while the vast majority of nodes contribute little. From a resilience perspective, this is a potential vulnerability, because once a key node fails, it is difficult to replace it with other nodes. A smaller deviation indicates that the recovery potential is dispersed, and the network simultaneously possesses bridge-to-bridge coordination capabilities at multiple node levels, exhibiting high redundancy.

[0058] To integrate the average level and dispersion into a single index, the system uses a harmonic weighting method. The idea behind harmonic weighting is to treat the weighted arithmetic mean and weighted squared deviation as positive and negative factors, respectively, and to negatively penalize the deviation term through a reciprocal or fractional structure. Specifically, the method maintains the linear positive contribution of the average level to the exponent, while introducing the deviation term as a denominator or exponential decay term, so that the exponent decreases overall when the deviation increases. When the average level and deviation are simultaneously input into the exponential generation function, the system ultimately obtains a dimensionless and monotonically increasing adaptation-recovery exponent. This exponent continuously varies between zero and the theoretical upper limit; a larger value indicates better network performance in rapid recovery and regeneration, and this advantage comes from multiple nodes sharing the same value, without drastic decay due to single-point failure within the same time window. Conversely, when the network's average recovery potential is high but the deviation is extremely large, the exponent is significantly suppressed by the penalty term, indicating that the network's self-healing ability is highly dependent on a few nodes, requiring timely redundancy compensation at the structural stability level.

[0059] At the implementation level, to address the high computational complexity brought about by large-scale networks, the system stores node bridge coupling rates and time weights in a columnar in-memory table. Vectorized instructions are used to perform weight multiplication, cumulative summation, and square root operations in a single pass. Subsequently, high-performance aggregation functions are used to calculate the average and deviation in a single pass. Since harmonic weighting involves fractional forms, the system employs high-precision floating-point numbers to ensure numerical stability and prevent the risk of division by zero due to excessively small deviations. After generating the index, the system not only records the historical trajectory of the adaptation-recovery index in the index library but also records node-level recovery potential indicators, providing micro-level support for subsequent strategy analysis. Administrators can identify weak links lacking recovery potential based on node rankings and implement link additions or redundancy optimizations. They can also monitor the index's changing trend within a rolling time window to identify early signs of structural fault formation and intervene promptly in situations where the crisis exposure factor rises sharply, improving overall resilience. When linked with the simulation module, the adaptation-recovery index can also serve as an objective function to evaluate the immediate gains of different fault injection or repair strategies on network resilience, helping to form a rapid closed loop. Through this precise, fine-grained construction logic that combines temporal and topological perspectives, this invention effectively fills the gap in traditional resilience assessments that only focus on static structure or average level while ignoring distribution uniformity, providing a new, quantifiable, traceable, and intervention-friendly tool for improving network resilience throughout its entire lifecycle.

[0060] Furthermore, the time weight is determined by the node's most recent position. The standard deviation of the fluctuation of the betweenness centrality sequence within a time period is obtained by reverse scaling, which is used to emphasize the dominant role of structurally stable nodes in the recovery process; the standardized betweenness centrality is obtained by a triple transformation of quartile removal, 0-1 linear stretching and power law reduction, to ensure that the bridge effect can maintain the distribution difference and is not masked by extreme values.

[0061] Specifically, for each node, the system retrieves the betweenness centrality observations for τ consecutive time periods from the monitoring data lake and calculates the sample standard deviation to obtain the structural fluctuation amplitude. To avoid mapping zero fluctuation to infinite weights and to prevent nodes with extremely high fluctuations from being assigned almost zero weights, the method employs an inverse monotonic function with both lower and upper bound constraints to map the standard deviation to between zero and one. For example, a commonly used transformation is the first reciprocal of the standard deviation plus one as the denominator, followed by logarithmic or hyperbolic tangent compression, ensuring both monotonically decreasing weights and continuous boundaries. After mapping, the weights are directly multiplied by the results of the node in the bridge coupling rate calculation, emphasizing the time contribution of the steady-state bridge. Since the weight definition is based on the fluctuation standard deviation rather than the mean, nodes with short-term spikes but long-term instability will be automatically downweighted due to high fluctuations, thus allowing nodes that continuously provide hub services over long periods to dominate the overall recovery potential.

[0062] The triple transformation process of standardized betweenness centrality aims to shape the bridging effect on a numerical scale, preserving relative diversity while suppressing excessive dependence on extreme values. This process first performs quartile de-extremum removal, systematically calculating the upper and lower quartiles of the network's betweenness centrality distribution. These are then multiplied by common 1x or 1x5 range thresholds to define outlier regions. All observations above the upper threshold or below the lower threshold are truncated to the threshold boundary, eliminating sampling noise and outlier effects from anomalous structures at the source. The de-extremum-removed sequence then undergoes zero-to-one linear stretching, mapping the minimum to zero, the maximum to one, and the remaining values ​​to a proportional linear distribution. This ensures that betweenness centrality distributions across different network sizes or densities fall within the same interval, facilitating subsequent connections with structural holes. Numerical interactions across dimensions such as system and time weights avoid dimensional inconsistencies. Finally, power-law reduction is used to progressively compress the stretched high interval. Specifically, each value in the zero-to-one interval is raised to a power with a positive exponent less than one, typically between 0.5 and 0.8. This gradually narrows the differences in the mid-to-high intervals, while maintaining the original linear slope in the low intervals. This approach preserves the order between nodes while limiting the exponential amplification of extreme high values ​​in the accumulation and multiplication stages, preventing a few super-hubs from becoming unbalanced in the bridge hole coupling rate or node-level recovery potential. When the standardized betweenness centrality obtained after the triple transformation is divided by the compressed structural hole constraint, the numerical fluctuation of the numerator is within a controllable range, and the denominator does not exhibit excessive scaling due to the inverse Sigmoid saturation mechanism. Therefore, the overall distribution of the bridge hole coupling rate is smoother. Combined with the continuous suppression of time weights, nodes with high stability, high bridges, and high redundancy are endowed with significant recovery potential, while the single-point contribution of extreme nodes is weakened to a reasonable level by a series of nonlinear compression functions. Through this time-scale dual calibration, the method ensures that the adaptation-recovery index not only measures strength but also accurately reflects the spatial distribution of strength among nodes. In principle, this avoids the distortion caused by the "super node" hijacking the recovery potential and truly connects with the core mechanism of network self-organization and regeneration.

[0063] Furthermore, step 3 also includes: in parallel, traversing all possible pairwise combinations between nodes, counting the number of independent parallel paths between each pair of nodes, and calculating the ratio of this ratio to the theoretical maximum number of parallel paths between two nodes to quantify the link diversity of the network; simultaneously, calculating the modularity coefficient of the Louvain community partitioning results of the network, and converting it into a decay factor using a reciprocal shift to reflect the positive contribution of partition isolation to diffusion suppression; multiplying the primary recovery coefficient by the link diversity ratio, and then adding the decay factor additively to obtain the comprehensive recovery potential without scale compression; if the comprehensive recovery potential exceeds the theoretical resilience upper limit, then using a hyperbolic tangent function for mapping compression to ensure consistency of the index range; to offset the index comparability bias caused by differences in network size, introducing a logarithmic term for the number of nodes as a denominator adjustment, so that larger and more dispersed networks will not have their recovery potential overestimated due to the large number of nodes, thus obtaining the scale-standardized result; performing numerical normalization processing based on the scale-standardized result, and finally outputting the adaptation-recovery index, which is continuously monotonic between 0 and the theoretical maximum value, and maintains a first-order positive correlation with bridge-cavity synergy, link diversity, and partition isolation.

[0064] After completing the bridge coupling rate and node-level recovery potential metrics, a parallel process is simultaneously initiated to measure the network's regeneration potential at the macro-topological redundancy and community isolation levels. The two processes are then connected in numerical space to form a continuous recovery potential metric chain. The parallel process first iterates through all possible pairwise combinations of nodes against a baseline topology snapshot, without distinguishing between directed and undirected pairs; any node pair with connection potential under the current structure is included in the scan. For each node pair, the system calls a multi-source shortest path and traffic allocation algorithm to enumerate all topologically independent and parallelizable paths between the two nodes, excluding any path combinations that cross or conflict at the transit node or transit edge level, retaining truly non-interfering redundant paths, and then recording their number. After the traversal, the system holds a sparse matrix with node pairs as keys and the number of independent parallelizable paths as values. The number of paths for each record is then divided by the theoretically maximum number of parallel paths that the node pair can reach in the entire graph, yielding a ratio between zero and one. This ratio is defined as a local entry for that node pair in the link diversity ratio. The link diversity ratio of the entire network is obtained by averaging or weighted averaging all local entries across the network. Since the theoretical maximum number of parallel paths increases rapidly with the size and density of the network, this ratioization step can automatically eliminate the scale effect initially, ensuring that the link redundancy measurement focuses on "utilization" rather than "absolute number of paths".

[0065] Simultaneously, the system performs Louvain community partitioning on the same baseline topology snapshot. The Louvain community partitioning algorithm aims to maximize modularity, alternating between local greedy merging and macroscopic hierarchical compression to divide the network into several densely connected, sparsely distributed sub-communities. After the algorithm outputs community labels, the system immediately calculates the modularity coefficient based on the same topology. A larger coefficient indicates a clearer community structure, weaker coupling between communities, and a greater ability to prevent cascading failures by leveraging community boundaries. To translate the positive effects of strong community isolation into readily apparent numerical contributions, the system converts the modularity coefficient into a decay factor through a reciprocal shift: a larger coefficient results in a smaller factor after the reciprocal shift, but this factor is subsequently additively added in the next step. Therefore, a larger coefficient means a weaker penalty to the overall recovery potential, in other words, a more positive contribution. A smaller coefficient indicates a looser community structure and poorer isolation; the larger the factor obtained after the reciprocal shift, the greater the burden on the indicator when added to the overall recovery potential, thus reducing the self-healing evaluation. The purpose of the reciprocal shift is to allow the advantages of partition isolation to be expressed in a way that reduces the burden, rather than multiplying by additional weights to cause scaling amplification, thereby maintaining the link diversity ratio and node-level recovery potential in the same positive scale semantics.

[0066] Once the link diversity ratio and the primary recovery coefficient are ready, the system multiplies them to obtain a composite value that integrates microscopic bridge-cavity collaborative capabilities and injects macroscopic redundant path potential. The attenuation factor, obtained by shifting the inverse of the modular coefficients, is then additively added to this composite value to derive the comprehensive recovery potential without scale compression. Numerically, the order of multiplication followed by addition is used because both node-level recovery potential and link diversity ratio exhibit positive multiplier effects; the higher the coupling between them, the smoother the overall network redundancy rerouting. Partition isolation, on the other hand, suppresses diffusion; its essence is not to amplify transmission efficiency but to weaken cascading failures. Therefore, introducing it as an additive addition after multiplication better reflects the meaning of "burden reduction." If the comprehensive recovery potential exceeds the preset theoretical resilience upper limit, to avoid the index losing resolution in the extremely high value region, the system uses the hyperbolic tangent function for monotonic mapping compression. The hyperbolic tangent has inherent upper and lower bound characteristics, enabling asymptotic convergence as the value approaches extremes, while maintaining a near-linear response in the intermediate segment. Therefore, it can avoid large value overflow without destroying monotonicity. After mapping, the overall recovery potential falls into a compressed range between zero and one. However, a potential problem remains: the number of nodes varies greatly across networks of different sizes. The product of node-level recovery potential and link diversity ratio is often underestimated in large-scale sparse networks and overestimated in large-scale dense networks. To offset this comparability bias caused by scale differences, the method introduces a logarithmic term for the number of nodes as a denominator adjustment. That is, the total number of nodes is incremented by one, the natural logarithm is taken, and then the compressed overall recovery potential is divided by this logarithmic term. The logarithmic term grows slower than linearly but faster than a constant, which can gently balance the numerical inflation caused by scale expansion and ensure that extremely small networks do not obtain unreasonably high values ​​due to an excessively small denominator.

[0067] After the scaling results are completed, the system performs another numerical normalization process. This process can be based on historical or theoretical extreme values ​​using zero-one normalization, or mean-variance scaling; regardless of the method, consistency is required only when comparing across networks. The normalized result is formally named the adaptation-recovery index and is hard-coded to be continuously monotonic between zero and the theoretical maximum value. This ensures that any input that logically increases bridge-cave synergy, link diversity, or partition isolation will lead to a monotonically increasing index; if any one of the three dimensions decreases while the others remain unchanged, the index will necessarily decrease, achieving a first-order positive correlation. During index persistence, the system simultaneously records the overall recovery potential before compression, the scaling results, and the normalization weight parameters, providing a basis for subsequent traceability and sensitivity analysis. The entire process achieves a three-dimensional coupling of microstructural stability, macro-redundancy richness, and community isolation resilience at the algorithm design level. At the numerical level, multiple compression and scale correction are introduced to ensure cross-network comparability. The final output adaptation-recovery index can reflect both the average recovery strength and its balanced distribution among nodes, thus providing a high-confidence self-healing potential characterization for the comprehensive index fusion of multi-dimensional quantitative evaluation methods for network resilience index.

[0068] Furthermore, the process of numerical normalization based on the scale-normalized results includes: using the scale-normalized results as a benchmark, performing a Box-Cox transformation on them and multiplying by a constant. The numerical stretching is performed, and the final output is the adaptation-recovery index.

[0069] Furthermore, step 4 specifically includes: using the topological substrate toughness index divided by the crisis exposure factor plus one as the structure-impact correction term; setting the index weight of the structure-impact correction term as the ratio of the adaptation-recovery index to the adaptation-recovery index plus one, to reflect the amplifying effect of recovery capacity on overall toughness, thus obtaining the correction adjustment result; selecting the most recent Using a year as the time window, the standard deviation of all topological base resilience indices within that time window is calculated to represent the long-term fluctuation range of base resilience. The standard deviation of the negative topological base resilience index of the natural exponential function is used as a stability correction term. The correction result is multiplied by the stability correction term to obtain a multi-dimensional network resilience comprehensive index. The larger the value of the network resilience comprehensive index, the stronger the network resilience.

[0070] First, a structure-impact correction term is introduced. By dividing the topological base resilience index by the crisis exposure factor plus one, the inherent structural health of the network and the historical cumulative impact load are placed within the same fractional framework. This fractional construction can be seen as a direct offset between "total buffer capacity and impact density." When the topological base resilience index is much greater than the crisis exposure factor, the correction term is higher than one, indicating that the structural buffer is still sufficient. Conversely, when the two are close or the crisis exposure factor rises rapidly, the correction term will fall to a level close to or even lower than one, reflecting that the impact has almost exhausted the structural buffer. Next, to allow the resilience to have a multiplier effect on the overall resilience, the method applies the ratio of the adaptation-recovery index to the adaptation-recovery index plus one as an exponential weight to the structure-impact correction term. The selection of the exponential weights leverages the concave increasing characteristic of the exponential function within the interval greater than one. This means that when the adaptation-recovery exponent is high, the structure-impact correction term is exponentially amplified, reflecting the strong recovery potential's ability to multiply the impact load. Conversely, when the adaptation-recovery exponent is low, the exponential weights approach one, resulting in very weak or even negligible amplification of the correction term, indicating insufficient recovery potential to significantly alleviate impact pressure. This amplification mechanism transforms node-level bridge-hole collaboration, link diversity, and community isolation into a comprehensive resilience accelerator, providing a nonlinear benefit to highly resilient networks.

[0071] However, resilience is not merely about instantaneous buffering and self-healing speed; it must also consider the ability to maintain stable performance over a longer period. Therefore, the method further incorporates a long-term fluctuation penalty logic. The system selects the most recent y years as a time window, extracts the entire topological base resilience index sequence within the window, calculates the standard deviation of this sequence, and uses it as a statistical scale for the fluctuation of base resilience over time. A larger standard deviation indicates that the inherent structure of the network has oscillated violently in recent years, and the structure can be used to buffer instability; a smaller standard deviation indicates that the structural strength remains in a relatively flat range, providing more predictable protection for risk management and resource allocation. After obtaining this fluctuation amplitude, the system feeds it as the negative exponential variable of the exponential function into the natural exponential function and outputs a stability correction term. Since the natural exponent decays rapidly on the negative half-axis, once the standard deviation of structural fluctuations increases, the stability correction term shrinks rapidly, directly reducing the exponentially amplified correction result; when the standard deviation approaches zero, the correction term approaches one, producing almost no compression on the comprehensive index. By using this externally parameter-free exponential decay design, the method incorporates long-term structural stability into resilience evaluation, ensuring that networks that rely solely on short-term peaks for resilience but oscillate continuously overall are appropriately penalized, while networks that are stable in the long term and steadily improving receive full exponential bonuses.

[0072] In the final step, the system multiplies the correction result with the stability correction term to obtain a multi-dimensional network resilience comprehensive index, which is defined as a unified non-negative dimensionless scale. The correction result interweaves structural buffering, impact load, and recovery potential into a positive multiplier, while the stability correction term, as a multiplier factor with an absolute value not greater than one, lowers the overall score, making the upper bound of the comprehensive index limited to the theoretical extreme value and the lower bound zero. Furthermore, it monotonically increases when structural buffering is improved, impact mitigation is achieved, recovery potential is enhanced, or volatility decreases. The numerical logic of the comprehensive index thus achieves a fusion of four physical meanings: when the topological base resilience index rises while the crisis exposure factor remains unchanged or decreases, the index climbs, reflecting inherent structural strengthening; when the adaptation-recovery index increases, the index is non-linearly amplified by the index weight, reflecting acquired recovery enhancement; when long-term volatility is compressed, the index is exponentially decayed, retaining more of its original value, reflecting steady-state operation advantages; if any dimension shows an inverse change, the index is immediately adjusted downwards accordingly, ensuring monotonic consistency. When the system persists the multi-dimensional network resilience comprehensive index, it also simultaneously writes the structure-impact correction term, index weight, stability correction term, and topology base resilience index sequence within the y-year window into the analysis repository, providing underlying support for subsequent sensitivity analysis and scenario simulation. On the real-time monitoring panel, the comprehensive index curve and its sub-curves of the three major components are displayed in parallel, which can help operation and maintenance or decision-making personnel identify the source of index changes and promptly carry out targeted interventions for structural reinforcement, recovery acceleration, or fluctuation control.

[0073] Specifically, the multi-dimensional network resilience comprehensive index for:

[0074] ;

[0075] in:

[0076] ;

[0077] Here, It is the topological substrate toughness index; As a crisis exposure factor; For the adaptation-recovery index; For the most recent Standard deviation of all topological substrate toughness indices over the years; for Topological substrate toughness index at 1 year; For the most recent The mean of all topological substrate toughness indices for the year; This is the starting year of the time window.

[0078] The above-described embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A multi-dimensional quantitative evaluation method for network resilience index, characterized in that, The method includes: Step 1: Obtain the original topology data of the target network and construct the topology resilience index of the network under the condition of no external shock. The larger the value, the healthier the network is at the inherent structural level. Step 2: Identify all crisis stages based on historical monitoring records to obtain the total number of crisis stages; for each crisis stage, the percentage decline in network function is taken as the impact intensity; the duration of continuous decline in network function is taken as the crisis duration; the time taken for network function to recover from the lowest point to the pre-crisis baseline is taken as the recovery time; calculate the crisis exposure factor based on the total number of crisis stages, impact intensity, crisis duration, and recovery time. Step 3: Calculate the betweenness centrality of each node to measure the bridging role of the node in the information flow of the entire network; calculate the structural hole constraint degree of each node to measure the degree of information redundancy controlled by the node; combine the betweenness centrality and the degree of information redundancy to generate the adaptation-recovery index, which measures the network's self-healing and regeneration capabilities. Step 4: By adjusting the topological base resilience index and the crisis exposure factor for shock resistance, then amplifying the index weights based on the adaptation-recovery index, and combining the stability correction of network resilience fluctuations, a multi-dimensional comprehensive network resilience index is finally generated. Step 3 specifically includes: setting an evaluation time window and locking a baseline topology snapshot; assigning time weights to each node in the baseline topology snapshot; calculating the normalized betweenness centrality of all nodes under the same baseline topology snapshot; simultaneously calculating the structural hole constraint degree of all nodes and performing inverse sigmoid compression on the structural hole constraint degree to make the structural hole advantage numerically present a saturation upper limit, avoiding absolute monopoly of nodes leading to unbalanced evaluation results; dividing the normalized betweenness centrality and the structural hole constraint degree node by node to obtain the bridge hole coupling rate, then using the time weight as a multiplier to perform weighted accumulation of the bridge hole coupling rate, and taking the square root of the accumulated result to constitute a node-level recovery potential index; and then... Each node's recovery potential index is assigned the same weight or weighted according to node weights, and a weighted arithmetic mean is calculated to represent the average recovery potential level of the entire network. Next, based on the same weighting system, the squared difference between each node's recovery potential index and the network's weighted average is calculated and summed to obtain the weighted squared deviation of the recovery potential index distribution among the network's nodes, which reflects the uniformity of the recovery potential distribution among different nodes. Finally, the weighted arithmetic mean and the weighted squared deviation are combined in a harmonic weighting method to form a comprehensive metric that can simultaneously reflect the average strength and distribution balance of the recovery potential among network nodes, resulting in the adaptation-recovery index.

2. The multi-dimensional quantitative evaluation method for network resilience index as described in claim 1, characterized in that, The original topology data includes: the degree of each node, the local clustering coefficient, the number of independent paths, and the average shortest path length to all other nodes.

3. The multi-dimensional quantitative evaluation method for network resilience index as described in claim 2, characterized in that, Step 1 specifically includes: at the node scale, multiplying the node degree by the local clustering coefficient to characterize the connectivity-cluster coupling effect, directly using the number of independent paths as the diversity factor, and taking the reciprocal of the average shortest path length to characterize the transmission efficiency; at the overall network scale, summing and averaging the connectivity-cluster coupling effect, the diversity factor, and the transmission efficiency respectively, and then amplifying the connectivity-cluster coupling effect and the transmission efficiency twice, and performing a logarithmic transformation on the sum of the results and the diversity factor to obtain the topological substrate resilience index under the condition of no external shock.

4. The multi-dimensional quantitative evaluation method for network resilience index as described in claim 3, characterized in that, In step 2, the impact intensity and duration of each crisis stage are multiplied and summed, and then the result is divided by the total sum of the impact intensity and recovery duration to obtain the crisis exposure factor; the larger the value of the crisis exposure factor, the more vulnerable the network is during the impact stage.

5. The multi-dimensional quantitative evaluation method for network resilience index as described in claim 4, characterized in that, Time weights are determined by the node's most recent position. The standard deviation of the fluctuation of the betweenness centrality sequence within a time period is obtained by reverse scaling, which is used to emphasize the dominant role of structurally stable nodes in the recovery process; the standardized betweenness centrality is obtained by a triple transformation of quartile removal, 0-1 linear stretching and power law reduction, to ensure that the bridge effect can maintain the distribution difference and is not masked by extreme values.

6. The multi-dimensional quantitative evaluation method for network resilience index as described in claim 5, characterized in that, Step 3 also includes: in parallel, traversing all possible pairwise combinations between nodes, counting the number of independent parallel paths between each pair of nodes, and calculating the ratio of this ratio to the theoretical maximum number of parallel paths between two nodes to quantify the link diversity of the network; simultaneously, calculating the modularity coefficient of the Louvain community partitioning results of the network, and converting it into a decay factor using a reciprocal shift to reflect the positive contribution of partition isolation to diffusion suppression; multiplying the primary recovery coefficient by the link diversity ratio, and then adding the decay factor additively to obtain the comprehensive recovery potential without scale compression; if the comprehensive recovery potential exceeds the theoretical resilience upper limit, then using a hyperbolic tangent function for mapping compression to ensure consistency of the index range; to offset the index comparability bias caused by differences in network size, introducing a logarithmic term for the number of nodes as a denominator adjustment, so that larger and more dispersed networks will not have their recovery potential overestimated due to the large number of nodes, thus obtaining the scale-standardized result; performing numerical normalization processing based on the scale-standardized result, and finally outputting the adaptation-recovery index, which is continuously monotonic between 0 and the theoretical maximum value, and maintains a first-order positive correlation with bridge-cavity synergy, link diversity, and partition isolation.

7. The multi-dimensional quantitative evaluation method for network resilience index as described in claim 6, characterized in that, The process of numerical normalization based on the scale-normalized results includes: using the scale-normalized results as a benchmark, performing a Box-Cox transformation on them and multiplying by a constant. The numerical stretching is performed, and the final output is the adaptation-recovery index.

8. The multi-dimensional quantitative evaluation method for network resilience index as described in claim 7, characterized in that, Step 4 specifically includes: using the topological substrate toughness index divided by the crisis exposure factor plus one as the structure-impact correction term; setting the index weight of the structure-impact correction term as the ratio of the adaptation-recovery index to the adaptation-recovery index plus one, to reflect the amplification effect of recovery capacity on overall toughness, thus obtaining the correction and adjustment result; selecting the most recent Using a year as the time window, the standard deviation of all topological base resilience indices within that time window is calculated to represent the long-term fluctuation range of base resilience. The standard deviation of the negative topological base resilience index of the natural exponential function is used as a stability correction term. The correction result is multiplied by the stability correction term to obtain a multi-dimensional network resilience comprehensive index. The larger the value of the network resilience comprehensive index, the stronger the network resilience.

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