Water network toughness evaluation method based on network topology structure

Abstract

The invention discloses a water network toughness evaluation method based on a network topology structure, and the method comprises the following steps: constructing a water network topology structure based on a complex network theory, abstracting a reservoir, a lake and the like as nodes, abstracting a river channel, a channel and the like as edges, and forming an undirected network; information entropy is introduced to improve traditional node importance indexes, comprehensive evaluation is carried out in combination with a weighted rank sum ratio (WRSR) method, and index weights are determined by using a CRITIC method; and constructing a toughness evaluation formula by taking the network efficiency and the maximum connected sub-graph as indexes and combining with the weight. According to the method, the adjacent node associated information is fused through the information entropy, and the conversion of the index system from isolation to system evaluation is realized; multi-dimensional indexes are fused, one-sidedness of a single index is avoided, local connection and global roles of nodes are considered, and the method is adaptive to different water networks and can be migrated. In practical application, the toughness change in water network disaster and aging scenes can be quantified, the reconstruction project is guided to preferentially supplement the shortages, the key node identification precision is improved, and the stability and the risk resistance of the water network are promoted.

Description

Technical Field

[0001] This invention relates to the field of water network system planning, and in particular to a method for assessing the resilience of water networks based on network topology. Background Technology

[0002] Water networks are complex systems composed of rivers, lakes, and pipelines, possessing both natural water system extensions and man-made engineering attributes, and undertaking functions such as water resource allocation and flood control and disaster reduction. While their interconnected nature ensures normal operation, it also increases disaster vulnerability, and climate change exacerbates the scale and uncertainty of risks, making them more difficult to predict. Water network resilience is the ability of a system to maintain basic functions and services even when some nodes fail, in the face of external shocks (natural disasters, human-caused damage) and internal changes (equipment aging, demand fluctuations). Studying water network resilience can improve the adaptability and resilience of water network systems, thereby better addressing the challenges of climate change, natural disasters, and urbanization.

[0003] Current water network resilience assessment methods mainly fall into three categories. Resilience characteristic-based assessment methods, centered on the "4Rs" (robustness, redundancy, resource availability, and speed), evaluate network resilience by analyzing one or more characteristics. While they can focus on core features and initially identify the resilience status of the water network, their assessment dimensions are relatively singular, and their coverage of the overall operational logic of the water network is limited. System performance-based assessment methods comprehensively reflect changes in system resilience through a series of indicators (such as population, per capita water resources, and total water resources), analyze the reasons affecting changes in system resilience, and provide guidance for the normal operation and resilience improvement of the system. They offer multi-dimensional assessment perspectives and provide direction for system operation optimization, but suffer from problems such as the susceptibility of indicator selection to subjective factors and insufficient consideration of the internal structural relationships of the water network. Network topology-based assessment compares changes in topological indicators (such as degree center, betweenness center, proximity center, and eigenvector center) to assess the resilience of the water network.

[0004] In network topology-based assessments, complex network theory is the core technical support. This theory starts from the perspective of the intricate relationships between nodes, constructs a water network topology, and ranks the importance of nodes by topological indicators such as associativity, betweenness, and proximity centrality. Then, it simulates disaster scenarios with attack strategies, compares the changes in topological indicators, and quantifies the water network connectivity, efficiency, and other attributes to complete the resilience assessment. At the same time, it can analyze the impact of large-scale water conservancy projects on the water network topology.

[0005] Complex network topologies are highly complex. Existing research focuses only on the attributes of individual nodes, neglecting the influence of neighboring node attributes. While it can rank nodes and assess resilience using a single topological index, the lack of integration of multiple indicators leads to incomplete assessments of node importance and biased results. Furthermore, focusing solely on node attributes while ignoring the influence of neighboring nodes fails to comprehensively represent the importance and interconnected mechanisms of nodes within the network. To address this issue, traditional node importance assessment indicators need to be improved to integrate local and global information, while also considering the dynamic changes of multiple indicators to enhance the accuracy and objectivity of the assessment. Summary of the Invention

[0006] To address the aforementioned issues, this invention proposes a water network resilience assessment method based on network topology. By introducing the concept of information entropy, it improves upon traditional indicators and proposes a node importance identification method based on information entropy. This method considers the information entropy of a node and its neighboring nodes to identify important nodes in the network, comprehensively assessing the importance of nodes in the network. The weighted rank-sum ratio (WRSR) method comprehensively considers four indicators: degree center, proximity center, betweenness center, and eigenvector center, to better identify key nodes in the network.

[0007] This invention is implemented as follows:

[0008] A method for assessing the resilience of water networks based on network topology, comprising the following steps:

[0009] Step 1, Water network topology construction:

[0010] The Space L method is used to construct a topological network, with reservoirs and lakes in the water network project as nodes and natural rivers and artificial channels in the water network project as edges, transforming it into an undirected topological network structure G =<V,e,A> Where V is the set of nodes in the water network structure, V = {v i |i∈{1,2,……,N}}, E is the set of edges between nodes in the water network structure, E={e ij =(v i ,v j Let |i,j∈{1,2,……,N}, where A represents the connection matrix of the network, A=[a ij ] N*N Where N is the total number of network nodes;

[0011] Step 2, Assessment of the Importance of Water Network Nodes:

[0012] Step 21, Evaluation Indicators for the Importance of Water Network Nodes:

[0013] The importance of water network nodes is evaluated using a multi-objective weighted rank-sum ratio method, with four evaluation indicators selected: degree center (DC), proximity center (CC), betweenness center (BC), and eigenvector center (EC).

[0014] The importance of nodes in the network is evaluated to obtain an importance result set under different indicators. The data is normalized to have a similar scale. The concept of information entropy is introduced, and the formulas for four traditional centrality indicators are improved based on information entropy to more comprehensively consider the global information of the network.

[0015] Step 22, multi-index fusion calculation based on weighted rank sum ratio (WRSR):

[0016] Define the evaluation objects and indicators, construct an initial sample matrix Z, and use the integer rank method to compile the rank R of each evaluation object under each indicator. ij The weight of each evaluation indicator is calculated using the CRITIC method, and the weighted rank sum ratio (WRSR) is calculated based on the weights of each indicator. i Then, the distribution of WRSR is determined sequentially, and a fitting analysis is performed using regression equations to finally obtain the ranking results; Step 3, Water network resilience assessment:

[0017] Step 31, Water Network Resilience Assessment Method:

[0018] Sort all nodes in descending order of importance, and perform "failure" operations according to the sorting order, removing all "failed" nodes and their connected edges; calculate the resilience of the newly obtained water network, and repeat the above steps until the water network has only one node left.

[0019] Step 32, Water Network Resilience Assessment Indicators:

[0020] A combination of network efficiency Q and the relative size S of the largest connected subgraph is used as the evaluation index R for resilient water networks. Further, in step 1, the connection matrix of the topological network structure... ij The formula for calculating the numerical value is as follows:

[0021]

[0022] Furthermore, the degree center DC mentioned in step 21 is calculated based on the degree of the node and is an indicator of the density of node connections. i The degree center is denoted as DC i The calculation formula is as follows:

[0023]

[0024] Where, k i It is related to node v i The number of edges connected in one step, N-1 represents the number of edges connected to node v.i The number of edges connected to other nodes;

[0025] The near-center CC i It is node v i The reciprocal of the sum of the shortest distances to other nodes is used to measure the importance of a node. The formula is as follows:

[0026]

[0027] Where, d ij Represents node v i The shortest distance to all other points;

[0028] The betweenness center BC identifies the importance of a node based on the frequency with which it acts as a bridge or intermediary in the network. The formula for calculating the betweenness center of a node is as follows:

[0029]

[0030] in, This indicates that the process has passed through node v. i The number of shortest paths, g st This represents the number of shortest paths connecting points s and t;

[0031] The eigenvector center EC is calculated based on the strength of the connection between a node and its neighboring nodes. The calculation formula is as follows:

[0032]

[0033] Where c is a proportionality constant, x i and x j They represent node v respectively i and v j Importance, x = [x1, x2, x3, ..., x N ] T When it reaches stability after multiple iterations, it can be written in the form x = cAx, where x is the eigenvalue c of matrix A. -1 eigenvectors.

[0034] Furthermore, in step 21, the calculation formula for the normalization method is as follows:

[0035]

[0036] Where I(i) represents the evaluation node v of index I. i Importance, max(P) I ) and min(P I ) represent the maximum and minimum values ​​obtained by evaluating all nodes using index I, respectively. b(i) represents the result after normalization.

[0037] Furthermore, in step 21, the formulas for the four traditional centrality indices based on information entropy are improved, and the calculation formulas are as follows:

[0038]

[0039] Among them, V (i) Represents all nodes v i The set of adjacent nodes, I e (i) represents node v under the evaluation of index I. i The ratio of the importance of a node to the sum of the importance of all nodes, I E (i) represents node v after improvement based on information entropy under the evaluation of index I. i Importance.

[0040] Furthermore, in step 22, the initial sample matrix Z is calculated using the following formula:

[0041]

[0042] Among them, z ij The sample data (i = 1, 2…m; j = 1, 2…n) are used, where m is the evaluation object and n is the evaluation index.

[0043] Furthermore, in step 22, the rank R of each evaluation object under each indicator... ij The calculation formula is as follows:

[0044]

[0045] Furthermore, in step 22, the weighted rank sum ratio WRSR i The calculation formula is as follows:

[0046]

[0047] Among them, WRSR i ω is the weighted rank sum ratio of the i-th evaluation object; j R represents the weight value of the j-th evaluation indicator. ij Let be the rank of the i-th evaluation object under the j-th indicator.

[0048] Furthermore, in step 32, the network efficiency Q is calculated using the following formula:

[0049]

[0050] Where e is the global network efficiency, which is the average of the sum of the reciprocals of the shortest distances between nodes, and is calculated using the following formula:

[0051]

[0052] Where, d ij This represents the shortest distance between two nodes;

[0053] The relative size of the largest connected subgraph is S, and it is calculated using the following formula:

[0054]

[0055] Where L is the maximum connected subgraph, calculated as follows:

[0056] L = max(N) i '), i∈V

[0057] Where, N′ i Let i be the set of nodes in the connected subgraph after node i is destroyed.

[0058] Furthermore, in step 32, the resilience R of the water network is calculated using the following formula:

[0059]

[0060] in, β represents the weight of the water network's ability to defend against floods and droughts; β represents the weight of the ability to optimize water resource allocation.

[0061] The beneficial effects of the invention are: a water network resilience assessment method based on network topology, which can accurately identify key nodes and scientifically assess resilience, providing core support for water network operation and maintenance.

[0062] By introducing information entropy to introduce the correlation information of adjacent nodes, the indicator system is shifted from isolated evaluation to system evaluation. Combined with the weighted rank sum ratio (WRSR) method, multi-dimensional indicators are integrated, which not only avoids the one-sidedness of single indicators, but also takes into account the local connectivity characteristics of nodes and global information, adapts to different water network structures, and provides a transferable solution for the identification of key nodes of other infrastructure.

[0063] In practical applications, this method can quantitatively characterize the resilience changes of water networks under scenarios such as disasters and aging, providing a basis for structural optimization. For example, in renovation projects, it can assess the extent to which the resilience of the proposed solution is improved, guiding the priority renovation of weak nodes and reducing ineffective costs; the weighting system can dynamically adapt to regional needs.

[0064] Improved water network node importance assessment enhances identification accuracy. During disasters, critical nodes can be quickly located, shortening water outage duration; in daily operations, bottleneck nodes can be targeted for identification, providing maintenance recommendations and effectively improving water network stability and resilience.

[0065] The present invention will be further explained in detail below with reference to the accompanying drawings and specific embodiments. Attached Figure Description

[0066] Figure 1 This is a flowchart of the water network resilience assessment method based on network topology of the present invention;

[0067] Figure 2 This is a schematic diagram of the distribution of water network nodes in southern Hebei Province, as shown in Example 2.

[0068] Figure 3 This is a network structure diagram of the existing and planned water network in southern Hebei Province, as shown in Example 2.

[0069] Figure 4 This is a comparison chart of the indicators of the water network in southern Hebei before and after the improvement in Example 2;

[0070] Figure 5 This is a graph showing the changes in the node importance indicators of the existing and planned water networks in southern Hebei Province, as described in Example 2.

[0071] Figure 6 This is a comparison diagram of the importance of nodes in the existing and planned water networks of southern Hebei Province, as shown in Example 2.

[0072] Figure 7 This is a graph showing the changes in resilience indicators of the existing and planned water networks in southern Hebei Province, as described in Example 2.

[0073] Figure 8 This is Example 2, a diagram showing the changes in resilience of the existing and planned water networks in southern Hebei under different weight combinations. Detailed Implementation

[0074] Example 1:

[0075] This embodiment presents a water network resilience assessment method based on network topology, such as... Figure 1 As shown, the method steps are as follows: Step 1, Water network topology construction:

[0076] The Space L method is used to construct a topological network, treating reservoirs and lakes in the water network project as nodes in a complex network, and abstracting the natural waterways and artificial channels of the water network project as edges of the topological network, thus transforming it into an undirected topological network structure G =<V,E,A> Where V is the set of nodes in the water network structure, V = {v i |i∈{1,2,……,N}}, E is the set of edges between nodes in the water network structure, E={e ij =(v i ,v j Let |i,j∈{1,2,……,N}, where A represents the connection matrix of the network, A=[a ij ] N*N Where N is the total number of network nodes, a ij =0 or a ij=1, the calculation formula is as follows:

[0077]

[0078] When nodes i and j in a complex network are directly connected, a ij =1, if nodes i and j are not directly connected, then a ij =0. This embodiment focuses on the importance of engineering nodes. Factors such as water flow direction, node capacity, and water conveyance capacity have a relatively small impact on the importance of nodes and are not included in the consideration for the time being. At the same time, only the topology of the water network is considered, and the actual distance factor is not involved. Therefore, the water network is set as an unweighted network, and the length of all edges in the network is uniformly assigned a value of 1.

[0079] Step 2, Assessment of the Importance of Water Network Nodes:

[0080] Step 21, Evaluation Indicators for the Importance of Water Network Nodes:

[0081] The importance of nodes in a water network is evaluated using a multi-objective weighted rank-sum ratio method. Four evaluation indicators are selected: degree centrality (DC), closeness centrality (CC), betweenness centrality (BC), and eigenvector centrality (EC). Degree centrality reflects local attributes, closeness centrality reflects global attributes, betweenness centrality reflects propagation attributes, and eigenvector centrality reflects the importance and influence of nodes in the network.

[0082] I. Traditional node importance assessment indicators:

[0083] 1. Degree Center (DC)

[0084] DC is calculated based on the degree of a node. A higher degree centrality value indicates that the node has more connections in the network, and thus has a greater impact on information transmission and the overall function of the network. i The degree center is denoted as DC i The expression is as follows:

[0085]

[0086] In the formula, k i It is an existing node v i The number of edges connected in one step, N-1 represents the number of edges connected to node v. i The number of edges connected to other nodes.

[0087] 2. Near the center (CC)

[0088] Near the center CC i It is node v i The reciprocal of the sum of the shortest distances to other nodes is used to measure the importance of a node. The higher the CC value, the shorter the distance between the node and other nodes, and the more efficient the information transmission and influence propagation. The expression is as follows:

[0089]

[0090] In the formula, d i Represents node v i The shortest distance to all other points.

[0091] 3. Betweenness Center (BC)

[0092] Betweenness center (BC) identifies the importance of a node based on how frequently it acts as a bridge or intermediary in the network. It measures the number of times a node is traversed by shortest paths between other nodes in the network. Specifically, it calculates the ratio of the number of shortest paths passing through a node to the total number of shortest paths, thus measuring the degree to which a node acts as an intermediary. The expression for the betweenness center of a node is as follows:

[0093]

[0094] In the formula: This indicates that the process has passed through node v. i The number of shortest paths, g st This represents the number of shortest paths connecting points s and t.

[0095] 4. Eigenvector Center (EC)

[0096] Eigenvector centrality (EC) calculates a node's importance based on the strength of its connections with its neighboring nodes. In eigenvector centrality calculation, a node's importance is related to the importance of its neighboring nodes. If a node is connected to other important nodes, its eigenvector centrality will increase accordingly, as shown in the following expression:

[0097]

[0098] In the formula: c is a proportionality constant, x i and x j They represent node v respectively i and v j The importance of x = [x1, x2, x3, ..., x N ] T When it reaches stability after multiple iterations, it can be written in the form x = cAx, where x is the eigenvalue c of matrix A. -1 eigenvectors.

[0099] II. Improvements to the node importance index based on information entropy:

[0100] The importance of nodes in the network is evaluated to obtain an importance result set under different indicators. The normalization method is used to make the data have similar scales. The concept of information entropy is introduced to improve the formulas of four traditional centrality indicators and to more comprehensively consider the global information of the network.

[0101] 1. Node identification:

[0102] Based on node importance indices, the importance of nodes in the network is evaluated, resulting in importance result sets under different indices, P. I = [I(1), I(2), ..., I(n)], where I = DC, BC, CC, EC, representing the identification of node importance using different indicators.

[0103] 2. Data normalization:

[0104] Different indicators have different units of measurement and need to be normalized to allow for more effective data processing. Normalization also eliminates unit differences between data points, making them more comparable. Therefore, before improving indicators based on information entropy, the data for different indicators should be normalized first. The calculation formula is as follows:

[0105]

[0106] Where I(i) represents the evaluation node v of index I. i Importance, max(P) I ) and min(P I ) represent the maximum and minimum values ​​obtained by evaluating all nodes using index I, respectively. b (i) represents the result after normalization.

[0107] 3. Improvements to the node importance index based on information entropy:

[0108] By introducing the concept of information entropy, we can consider the global information of the network more comprehensively. This embodiment improves four traditional centrality indicators based on information entropy, and the calculation formula is as follows:

[0109]

[0110] Among them, V (i) Represents all nodes v i The set of adjacent nodes, I e (i) represents node v under the evaluation of index I. i The ratio of the importance of a node to the sum of the importance of all nodes, I E (i) represents node v after improvement based on information entropy under the evaluation of index I. i Importance.

[0111] Step 22, multi-index fusion calculation based on weighted rank sum ratio (WRSR):

[0112] Define the evaluation objects and indicators, construct an initial sample matrix Z, and use the integer rank method to compile the rank R of each evaluation object under each indicator. ij The weight of each evaluation indicator is calculated using the CRITIC method, and the weighted rank sum ratio (WRSR) is calculated based on the weights of each indicator. i Then, the distribution of WRSR is determined sequentially, and a fitting analysis is performed using regression equations to finally obtain the ranking results;

[0113] The Weighted Rank Ratio (WRSR) is a comprehensive evaluation method that assigns different weights to different indicators and combines this with the concept of the rank ratio to conduct a comprehensive evaluation. The specific method is as follows:

[0114] In a matrix consisting of m evaluation objects and n evaluation indicators, the dimensionless statistic RSR is obtained through rank transformation to assess the importance of the nodes. The weight of each evaluation indicator is calculated using the CRITIC method, thus obtaining the weighted rank sum ratio (WRSR) for each indicator.

[0115] Suppose there are m evaluation objects and n evaluation indicators, and the initial sample matrix is ​​Z = [z ij ] m*n , z ij The sample data (i = 1, 2…m; j = 1, 2…n).

[0116]

[0117] For matrix Z = [z ij ] m*n Re-rank the indicators and determine the rank R of each evaluation object under each indicator. ij The selected degree center, proximity center, betweenness center, and eigenvector center are all maximal indices, calculated using the following formulas:

[0118]

[0119] When the values ​​of the same evaluation index are the same, the average rank is calculated.

[0120] The CRITIC method is an objective weighting method that determines indicator weights using contrast strength and conflict. Contrast strength is represented by the standard deviation of the indicator data, while conflict is represented by the correlation coefficient between indicators. The larger the standard deviation of the indicator data and the smaller the correlation coefficient between indicators, the greater the indicator weight. The specific calculation process is as follows: calculate the information content e of the j-th evaluation indicator. j and weight w j .

[0121]

[0122] Where, σ j Let r be the standard deviation of the j-th evaluation index data; jt Let be the correlation coefficient between the j-th and t-th evaluation indicators.

[0123]

[0124] The weighted rank sum ratio is calculated based on the weights of each evaluation indicator, using the following formula:

[0125]

[0126] Among them, WRSR i ω is the weighted rank sum ratio of the i-th evaluation object; j R represents the weight value of the j-th evaluation indicator. ij Let be the rank of the i-th evaluation object under the j-th indicator.

[0127] The node importance assessment based on the weighted rank sum ratio (WRSR) first uses the integer rank method to assign a corresponding rank to all nodes for each assessment index in the water network (such as degree center DC, proximity center CC, betweenness center BC, and eigenvector center EC). When different nodes have the same value under the same index, the average rank is calculated to quantify the relative performance of each node under each index.

[0128] The weight of each evaluation indicator is calculated using the CRITIC method. The rank of each node under each indicator is multiplied by its corresponding weight, and all products are summed to obtain the weighted rank-sum ratio (WRSR) for each node. The distribution characteristics of the WRSR are then clarified, and a univariate linear regression equation is constructed for fitting analysis (the equation is not fixed and has variability). Finally, the ranking results are obtained. For example, when the DC indicator in a water network uses the number of nodes as the independent variable, the regression equation can be y = -0.0099x + 0.5934 (the goodness of fit R of the regression model is...). 2 =0.9595, where x is the number of nodes and y is the DC value. All regression equations use the number of nodes as the independent variable and the observed values ​​of the corresponding indicators (DC / CC / BC / EC) as the dependent variable. The equation parameters (slope, intercept) have no fixed standard and are adjusted entirely according to the analysis scenario, indicator type, and core variables. A higher WRSR fit value indicates a more important node.

[0129] Step 3, Water Network Resilience Assessment:

[0130] Step 31, Water Network Resilience Assessment Method:

[0131] When a water network is attacked and some nodes "fail," the remaining nodes can still maintain connectivity and ensure the normal operation of the network, indicating that the water network has strong resilience. In order to evaluate the impact of node "failure" on the resilience of the water network, nodes are removed one by one based on the importance ranking of water network nodes assessed by the WRSR method, and the impact of node "failure" on the overall resilience of the water network is evaluated.

[0132] The specific steps are as follows: Sort all nodes in descending order of importance, perform a "failure" operation according to the sorting order, and remove the "failed" nodes and all edges connected to them; calculate the resilience of the newly obtained water network. Repeat the above steps until only one node remains in the water network.

[0133] Step 32, Water Network Resilience Assessment Indicators:

[0134] To better describe resilience, this embodiment uses network efficiency and maximum connected subgraph as two indicators for calculating resilience. The maximum connected subgraph characterizes the network's ability to optimize water resource allocation; a larger maximum connected subgraph value indicates a stronger ability to optimize water resource allocation. Network efficiency characterizes the water network's resistance to disasters; a larger global network efficiency value indicates a stronger ability to cope with disasters.

[0135] 1. Global network efficiency:

[0136] Global network efficiency is the average of the sum of the reciprocals of the shortest distances between nodes, usually denoted by e, and its expression is:

[0137]

[0138] Where n is the number of nodes in the topology; d ij This represents the shortest distance between two nodes.

[0139] Network efficiency is represented by Q, and its expression is:

[0140]

[0141] Where e is the initial efficiency value of the network, and e′ is the efficiency value after the water network is damaged.

[0142] 2. Maximum connected subgraph

[0143] A maximum connected subgraph represents a network that, after being attacked, is divided into two or more sub-networks. The subgraph with the most nodes among all subgraphs is usually denoted by L, and its expression is:

[0144] L = max(N) i '), i∈V (16)

[0145] In the formula: N′ iLet i be the set of nodes in the connected subgraph after node i is destroyed.

[0146] The relative size of the largest connected subgraph is S, and its expression is:

[0147]

[0148] In the formula: N is the total number of network nodes.

[0149] 3. Toughness Calculation

[0150] The purpose of water network construction is to improve water resource utilization efficiency and better resist floods and droughts. This method combines network efficiency and maximum connected subgraph to propose an evaluation method for resilient water networks. The resilience of the water network is denoted as R, and its expression is:

[0151]

[0152] in, β represents the weight of the water network's ability to defend against floods and droughts; β represents the weight of the ability to optimize water resource allocation. In this method β is always taken as 0.5.

[0153] Example 2:

[0154] This embodiment is an example of resilience assessment of the water network in southern Hebei Province, such as... Figure 2 As shown, a water network is essentially a network composed of nodes and edges, which can be abstracted into a topological structure using graph theory and complex network theory. Constructing the topological structure diagram of a water network is an important tool for studying its resilience. It not only helps identify key nodes and weak links but also provides a scientific basis for optimization design and resilience improvement. Network structure diagrams of the existing and planned water networks in southern Hebei are constructed as follows: Figure 3 As shown. The existing water network is numbered according to the principle of "reservoir first, canal later". The existing water network is numbered 1-41, and the planned water network will add nodes to the existing water network, bringing the total to 60.

[0155] I. Comparative Analysis of Indicators Before and After Improvement:

[0156] When assessing the importance of nodes in the Hebei-Central-Southern Water Network, four indicators were selected: Degree Center (DC), Proximity Center (CC), Betweenness Center (BC), and Eigenvector Center (EC). Traditional indicators only consider the influence of a node itself, neglecting the influence of neighboring nodes, leading to some bias in the assessment results. To address this issue, this invention introduces the concept of information entropy to improve the traditional indicators. A comparison of the results before and after the indicator improvement is shown below. Figure 4 As shown.

[0157] Depend on Figure 4It is evident that the variation range of the improved indicators based on information entropy is significantly larger than that of the traditional indicators. For example, the variation range of the traditional DC indicator is 0.01 to 0.1, while the variation range of the improved indicator is 0.01 to 0.8. Furthermore, there are no instances where two or more nodes have the same indicator, making the evaluation results more accurate. Moreover, the p-values ​​of the four indicators—degree center (DC), proximity center (CC), betweenness center (BC), and eigenvector center (EC)—are all less than 0.05, indicating significant differences and statistical significance. This demonstrates that the improved indicators based on information entropy significantly outperform the traditional indicators, more effectively identifying critical nodes that have a significant impact on the network structure. Therefore, by protecting critical nodes from damage during network attacks or failures, the improved indicators better maintain the normal operation of the network.

[0158] II. Improved Water Network Node Importance Assessment Based on Information Entropy:

[0159] This embodiment selects four indicators—Degree Center (DCBE), Proximity Center (CCBE), Betweenness Center (BCBE), and Eigenvector Center (ECBE)—based on improved information entropy, and uses a single indicator to evaluate the importance of water network nodes. Then, from the perspective of the overall network, the Weighted Rank-Sum Ratio (WRSR) is used to consider the above four indicators, and the CRITIC method is used to assign weights to each indicator to comprehensively evaluate the importance of nodes. The results are as follows: Figure 5 As shown.

[0160] Depend on Figure 5 As shown, in the current water network, the values ​​of several indicators are relatively close in the initial stage. Among them, the values ​​of WRSR, CCBE, BCBE, and ECBE are all greater than 1, with ECBE having the largest initial value and indicating higher importance, while the value of BCBE is slightly less than 1. The changing trends of DCBE, CCBE, BCBE, ECBE, and WRSR are all steep at first and then slow down, with WRSR showing a relatively gentler change. Regression fitting analysis was performed on the five indicators, and the R-squared values ​​of WRSR, DCBE, CCBE, BCBE, and ECBE were... 2 The initial values ​​for WRSR, CCBE, and ECBE were 0.9627, 0.909, 0.8655, 0.8085, and 0.8233, respectively, with WRSR showing the best model fit. In the planned water network, the initial values ​​of WRSR and BCBE were greater than 1, with WRSR's initial value being higher than BCBE's, indicating the highest importance. The initial values ​​of DCBE, CCBE, and ECBE were relatively close, all between 0.7 and 0.8, with ECBE having the largest initial value. The trends of DCBE, CCBE, BCBE, ECBE, and WRSR were all steep initially followed by a gradual decrease, while WRSR's change was relatively gentler. Regression fitting analysis was performed on the five indicators, and the R-values ​​for WRSR, DCBE, CCBE, BCBE, and ECBE were... 2The R values ​​were 0.9606, 0.9595, 0.9157, 0.8125, and 0.8789, respectively, with WRSR showing the best model fit. A comparison of the R values ​​for the existing and planned water networks was made. 2 The value shows that the planned water network is better than the existing water network as a whole, which indicates that the more nodes in the water network, the better the importance assessment results.

[0161] Compared to evaluating node importance using a single metric, WRSR's multi-metric evaluation has the following three advantages: ① Comprehensiveness: Degree centrality only considers the number of connections a node has, proximity centrality considers the distance from a node to other nodes, betweenness centrality considers the frequency of a node's occurrence in the shortest path, and eigenvector centrality considers the importance of a node's neighbors. WRSR avoids the one-sidedness of single-metric methods by comprehensively considering information from multiple metrics, providing a more comprehensive evaluation. ② Stability and Reliability: Figure 5 It can be seen that the WRSR curve is relatively smooth with small fluctuations, and the evaluation results are relatively stable. This indicates that when evaluating the importance of nodes, WRSR will not fluctuate significantly due to drastic changes in a single indicator, demonstrating good stability and reliability. In contrast, single-indicator evaluation methods can experience significant fluctuations due to local changes in the network structure. ③ Comprehensive Ranking: Single-indicator methods can only provide importance information for one aspect and cannot comprehensively consider the influence of multiple aspects; WRSR can comprehensively rank nodes, taking into account the weights of multiple indicators, and can more objectively reflect the actual importance of nodes in the entire network.

[0162] Based on the weighted rank sum ratio (WRSR) method, the 10 most important nodes of the existing water network and the planned water network were obtained respectively. Figure 6 It is known that in the existing water network, nodes 19, 31, 20, 25, 24, 30, 18, 9, 32, and 35 are the 10 most important nodes. In the planned water network, the 10 most important nodes are 19, 20, 49, 14, 31, 9, 21, 55, 56, and 32. Among them, nodes 9, 19, 20, 27, 31, and 32 are of extremely high importance in both the existing and resilient water networks. The importance of nodes 11, 14, and 28 varies considerably between the existing and planned networks, with node 28 being the most important. In the existing water network, node 28 is only connected to node 29. In the planned water network, node 50 is connected to node 28 through newly planned projects. Simultaneously, node 56 is also connected to node 28. In the planned water network, the degree center and proximity center indicators of node 28 change, thus increasing its importance.

[0163] By comparing the importance of nodes in the existing water network and the planned water network, the rationality of the water network plan can be assessed. If the importance of key nodes in the planned water network is significantly increased, it indicates that the plan may have played a positive role in optimizing the water network structure and improving its efficiency. More maintenance resources or upgrades are needed at these important nodes to ensure the overall performance of the water network.

[0164] III. Water Network Resilience Assessment:

[0165] Figure 7 This study demonstrates how the resilience of a water network changes as nodes continuously "fail." The results show that, given the same number of node failures, methods exhibiting greater resilience changes are more accurate in identifying critical nodes in complex networks, thus making them more effective. Figure 7 It can be seen that ECBE exhibits the worst performance in both existing and planned water networks, but has high overall resilience and failed to accurately identify key nodes.

[0166] In the existing water network, the ECBE index performs best when the first 14.5% of nodes are "failed"; DCBE performs best when the number of nodes is between 14.5% and 17.1%; BCBE outperforms the WRSR index in the range of 19.5% to 43.9% of nodes being "failed"; and in all other cases, the WRSR index performs best, specifically when 58.5% of nodes are "failed". In the planned water network, CCBE outperforms the WRSR index in the range of 8.3% to 15% of nodes being "failed"; DCBE outperforms the WRSR index in the range of 25% to 30% of nodes being "failed"; CCBE outperforms the WRSR index in the range of 50% to 65% of nodes being "failed"; and in all other cases, the WRSR index performs best, specifically when 73.3% of nodes are "failed".

[0167] Furthermore, in the existing water network, when the node failure rate reaches 15%, the network resilience decreases from 1 to approximately 0.41; in the planned water network, when the node failure rate reaches 15%, the network resilience decreases from 1 to approximately 0.35. This indicates that the failure of a few key nodes has a significant impact on the network resilience. Attacking nodes 42-60 sequentially in the planned water network, causing them to fail, transforms the planned water network into the existing network, and the network resilience decreases from 1 to 0.83. The results show that compared to the existing water network, the planned water network resilience is improved by 20.5%.

[0168] The results show that the more complex the water network structure and the more nodes there are, the stronger the resilience of the water network. The planned water network has more nodes and a more complex structure than the existing water network, making it more resilient and effectively improving water resource allocation efficiency, drought resistance and disaster reduction capabilities, aquatic ecosystem protection capabilities, and water supply capacity and security.

[0169] Further investigation The effect of different values ​​of β on toughness was calculated, and the WRSR index was used to obtain the ranking number of node importance. The results are as follows: Figure 8 As shown. By Figure 8 It can be seen that as β increases, meaning the proportion of the largest connected subgraph in the weights increases, the resilience of the water network continuously improves. When β is 0.9, the resilience change after a node "failure" is lower than when β is at other values, demonstrating stronger resilience. Due to the uncertainty of complex networks, in order to comprehensively capture the resilience of the network under different states, this paper adopts a balanced weight allocation strategy. The weights for β are all set to 0.5. This weighting provides a balanced assessment of resilience under different network conditions, thus better adapting to network changes.

[0170] IV. Conclusion:

[0171] 1. The Weighted Rank-Sum Ratio (WRSR) method, improved based on information entropy, demonstrates significant advantages in identifying key nodes in the application of the Hebei Central and Southern Water Network. From the perspective of complex network theory, traditional single centrality indicators (such as degree center and betweenness center) can only reflect local or certain aspects of a node's attributes. However, water networks, as typical complex systems, require comprehensive consideration of multiple dimensions of node importance, including local connectivity, global reachability, information transmission mediation, and the influence of neighboring nodes. This study introduces the correlation information of adjacent nodes through information entropy, shifting the indicator system from "isolated evaluation" to "system evaluation." WRSR achieves multi-indicator fusion, solving the problem of single-indicator bias. For example, the accurate identification of nodes 19 and 31 in the existing water network and nodes 19 and 20 in the planned water network verifies the universality of this method under different network structures. Theoretically, this method overcomes the limitation of "single indicator" in traditional complex network evaluation, providing a transferable methodological framework for identifying key nodes in complex systems.

[0172] 2. The constructed water network resilience assessment method, through quantitative analysis, confirms that the planned water network exhibits a 20.5% improvement in resilience compared to the existing network. This result aligns closely with the theory that "structure determines function" in complex networks. From a topological perspective, the method adds 19 nodes to form a "four vertical, seven horizontal, and multiple reservoirs" network. The exponential expansion of connectivity paths resulting from the increase in the number of nodes provides quantitative support for structural optimization and resilience enhancement. Regarding the weighting system design, the method uses a β=0.5 balance to allocate weights for the largest connected subgraph (representing water resource allocation function) and network efficiency (representing disaster resistance function). This reflects a synergistic consideration of the dual functions of the water network and, through sensitivity analysis showing optimal resilience at β=0.9, verifies the crucial impact of weight allocation on the assessment results. The proposed "dynamic weighting" approach (e.g., strengthening the β value in water-scarce areas and emphasizing efficiency weights in flood-prone areas) enhances the model's adaptability to different regions. However, the specific construction of its dynamic adjustment mechanism still requires further research and refinement. Overall, this method possesses both scientific validity and practical guiding value in the quantification of structural complexity and the assessment of functional balance.

[0173] 3. The constructed topological resilience assessment model achieved the expected results in the application of the Hebei-Central-Southern Water Network. However, from a theoretical and methodological perspective, the following limitations still exist: First, the model assumes the network is unweighted and does not consider the weighting factors such as river transport capacity and node storage capacity in the actual water network, which may lead to bias in the identification of key nodes. Second, the current model only focuses on the impact of topology on resilience, and its core assumption is that the disruption of physical connectivity is the main factor affecting the resilience of the water network. However, the behavior of the actual water network under disturbance is profoundly affected by complex hydraulic processes (such as transient flow, pressure fluctuations, and reverse flow) and water quality migration processes (such as pollutant diffusion and reaction decay). Because these dynamic factors are not included, this model can only describe "whether water can reach," but cannot simulate "how water reaches" (hydraulic dynamics) and "whether the water quality is safe" (water quality dynamics). Future research can be expanded in two directions: First, by introducing the concept of a "weighted network," engineering parameters such as flow direction and water conveyance capacity can be incorporated into the topological model to construct a more realistic weighted network resilience assessment method. Second, by exploring a multi-level resilience assessment model construction method that integrates topology, hydraulics, and water quality. This model uses topological analysis as the first level (rapidly identifying key structural points) to drive the second level of hydraulic simulation (assessing pressure / flow distribution) and the third level of water quality modeling (quantifying pollution risk), achieving a synergistic "structure-function" resilience assessment to provide a more comprehensive resilience evaluation. Furthermore, considering the real-world need for addressing the frequent occurrence of extreme disasters under climate change, conducting research on the evolution of water network resilience under multiple disaster scenarios is also an important direction for enhancing the practical value of the theory.

[0174] This paper proposes a resilience assessment model based on network topology to evaluate the resilience of water network systems. The applicability and feasibility of the model are verified through its application in the resilience assessment of the water network in southern Hebei Province. Based on this, the main conclusions are as follows:

[0175] (1) Model Construction and Methodological Innovation: A water network resilience assessment model based on network topology was proposed. By introducing information entropy, the traditional node importance index was improved, and the bias problem of single index not considering the influence of adjacent nodes was solved. By combining multiple indicators such as degree center and proximity center in the weighted rank sum ratio method, the comprehensiveness and accuracy of node importance assessment were improved.

[0176] (2) The model was applied to the water network in southern Hebei Province. The results showed that: ① The importance of nodes evolved dynamically. In the current water network, key nodes are nodes 24 and 31. After the planned water network added nodes 50 and 56, nodes 20 and 49 entered the ranks of important nodes due to improved connectivity. The six key nodes shared by the current and planned water networks verified the core position of the water conveyance backbone network. ② The resilience enhancement effect: The planned water network formed a "four vertical, seven horizontal, and multiple reservoirs" network structure by adding 19 nodes. Its global network efficiency increased by 18.7% compared to the current situation, the maximum connected subgraph increased by 22.3%, and the water network resilience increased by 20.5%. This shows that the coordinated layout of inter-basin water transfer projects and storage reservoirs can significantly enhance the water resource optimization and disaster resistance capabilities. ③ Weight Sensitivity Analysis: When the weight β of the largest connected subgraph increases from 0.1 to 0.9, the resilience of the planned water network under the 15% node failure scenario improves from 0.35 to 0.62. Among them, the synergistic protection effect of network efficiency and connectivity is most significant when β = 0.9, verifying the balanced weight. Adaptability in uncertainties of complex networks.

[0177] Finally, it should be noted that the above is only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention (such as the application of various formulas, the order of steps, etc.) without departing from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for assessing the resilience of water networks based on network topology, characterized in that, The evaluation method steps are as follows: Step 1, Water network topology construction: The Space L method is used to construct a topological network, with reservoirs and lakes in the water network project as nodes and natural rivers and artificial channels in the water network project as edges, transforming it into an undirected topological network structure G =<V,E,A> Where V is the set of nodes in the water network structure, V = {v i |i∈{1,2,……,N}}, E is the set of edges between nodes in the water network structure, E={e ij =(v i ,v j Let |i,j∈{1,2,……,N}, where A represents the connection matrix of the network, A=[a ij ] N*N Where N is the total number of network nodes; Step 2, Assessment of the Importance of Water Network Nodes: Step 21, Evaluation Indicators for the Importance of Water Network Nodes: The importance of water network nodes is evaluated using a multi-objective weighted rank-sum ratio method, with four evaluation indicators selected: degree center (DC), proximity center (CC), betweenness center (BC), and eigenvector center (EC). The importance of nodes in the network is evaluated to obtain an importance result set under different indicators. The data is normalized to have a similar scale. The concept of information entropy is introduced, and the formulas for four traditional centrality indicators are improved based on information entropy to more comprehensively consider the global information of the network. Step 22, multi-index fusion calculation based on weighted rank sum ratio (WRSR): Define the evaluation objects and indicators, construct an initial sample matrix Z, and use the integer rank method to compile the rank R of each evaluation object under each indicator. ij The weight of each evaluation indicator is calculated using the CRITIC method, and the weighted rank sum ratio (WRSR) is calculated based on the weights of each indicator. i Then, the distribution of WRSR is determined sequentially, and a fitting analysis is performed using regression equations to finally obtain the ranking results; Step 3, Water Network Resilience Assessment: Step 31, Water Network Resilience Assessment Method: Sort all nodes in descending order of importance, and perform "failure" operations according to the sorting order, removing all "failed" nodes and their connected edges; calculate the resilience of the newly obtained water network, and repeat the above steps until the water network has only one node left. Step 32, Water Network Resilience Assessment Indicators: The network efficiency Q and the relative size S of the largest connected subgraph are combined as the evaluation index R for resilient water networks.

2. The water network resilience assessment method based on network topology according to claim 1, characterized in that, In step 1, a in the connection matrix of the topology network structure ij The formula for calculating the value is as follows:

3. The water network resilience assessment method based on network topology according to claim 1, characterized in that, The degree center DC mentioned in step 21 is calculated based on the degree of the node and is an indicator of the density of node connections. Node v i The degree center is denoted as DC i The calculation formula is as follows: Where, k i It is related to node v i The number of edges connected in one step, N-1 represents the number of edges connected to node v. i The number of edges connected to other nodes; The proximity center CC i It is node v i The reciprocal of the sum of the shortest distances to other nodes is used to measure the importance of a node. The formula is as follows: Where, d ij Represents node v i The shortest distance to all other points; The betweenness center BC identifies the importance of a node based on the frequency with which it acts as a bridge or intermediary in the network. The formula for calculating the betweenness center of a node is as follows: in, This indicates that the process has passed through node v. i The number of shortest paths, g st This represents the number of shortest paths connecting points s and t; The eigenvector center EC is calculated based on the strength of the connection between a node and its neighboring nodes. The calculation formula is as follows: Where c is a proportionality constant, x i and x j They represent node v respectively i and v j Importance, x = [x1, x2, x3, ..., x N ] T When it reaches stability after multiple iterations, it can be written in the form x = cAx, where x is the eigenvalue c of matrix A. -1 eigenvectors.

4. The water network resilience assessment method based on network topology according to claim 1, characterized in that, In step 21, the calculation formula for the normalization method is as follows: Where I(i) represents the evaluation node v of index I. i Importance, max(P) I ) and min(P I ) represent the maximum and minimum values ​​obtained by evaluating all nodes using index I, respectively. b (i) represents the result after normalization.

5. The water network resilience assessment method based on network topology according to claim 1, characterized in that, In step 21, the formulas for the four traditional centrality indices are improved based on information entropy, and the calculation formulas are as follows: Among them, V (i) Represents all nodes v i The set of adjacent nodes, I e (i) represents node v under the evaluation of index I. i The ratio of the importance of a node to the sum of the importance of all nodes, I E (i) represents node v after improvement based on information entropy under the evaluation of index I. i Importance.

6. The water network resilience assessment method based on network topology according to claim 1, characterized in that, In step 22, the initial sample matrix Z is calculated using the following formula: Among them, z ij The sample data (i = 1, 2…m; j = 1, 2…n) are used, where m is the evaluation object and n is the evaluation index.

7. The water network resilience assessment method based on network topology according to claim 1, characterized in that, In step 22, the rank R of each evaluation object under each indicator is determined. ij The calculation formula is as follows:

8. The water network resilience assessment method based on network topology according to claim 1, characterized in that, In step 22, the weighted rank sum ratio WRSR i The calculation formula is as follows: Among them, WRSR i ω is the weighted rank sum ratio of the i-th evaluation object; j R represents the weight value of the j-th evaluation indicator. ij Let be the rank of the i-th evaluation object under the j-th indicator.

9. The water network resilience assessment method based on network topology according to claim 1, characterized in that, In step 32, the network efficiency Q is calculated using the following formula: Where e is the global network efficiency, which is the average of the sum of the reciprocals of the shortest distances between nodes, and is calculated using the following formula: Where, d ij This represents the shortest distance between two nodes; The relative size of the largest connected subgraph is S, and it is calculated using the following formula: Where L is the maximum connected subgraph, calculated as follows: L<max(N i '),i∈V Where, N′ i Let i be the set of nodes in the connected subgraph after node i is destroyed.

10. The water network resilience assessment method based on network topology according to claim 1, characterized in that, In step 32, the toughness R of the water network is calculated using the following formula: in, β represents the weight of the water network's ability to defend against floods and droughts; β represents the weight of the ability to optimize water resource allocation.

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