A node evaluation method for a multi-layer complex network based on a gravity model

Abstract

The application discloses a kind of node evaluation methods of multi-layer complex network based on gravitational model, comprising, step 1, construct flow network;Step 2, according to flow network, construct force network;Step 3, according to different spatial sub-network, establish spatial network;Step 4, according to spatial network, construct distance network between nodes;Step 5, using gravitational model, derive to obtain the virtual mass calculation formula of node;Step 6, according to virtual mass calculation formula, select different two nodes to combine, to obtain the combination matrix of virtual mass calculation;Step 7, the numerical value of the force between different nodes and the distance between different nodes is substituted into the combination matrix of virtual mass formula, finally obtain the virtual mass set of node.The application combines gravitational model, carries out modeling to multi-layer complex network, and the method for evaluating node image can also be used for other multi-layer networks with similar structure.

Description

Technical Field

[0001] This invention belongs to the field of network node evaluation technology, and particularly relates to a node evaluation method for multilayer complex networks based on a gravity model. Background Technology

[0002] Complex networks are an effective mathematical tool for studying sociology, management, economics, and other disciplines. They possess a self-contained mathematical language that can be used to analyze their performance and evolution. Performance metrics include node degree, average distance, mediability, centrality, clustering coefficient, robustness, and small groups. Evolutionary processes include propagation, cooperation, and resilience. Existing complex network simulation and computation tools primarily target undirected and unweighted single-layer networks. In real life, many networks are weighted networks, with close connections between them, forming multi-layered complex networks. Especially with the development of big data collection and analysis technologies, more concrete numerical representations of the connections between nodes in complex networks have been provided.

[0003] Research on complex networks has shifted to constructing multi-layered complex networks, introducing other mathematical models, and developing new evaluation methods to keep complex network research up-to-date and solve practical problems in life, such as evaluating multi-layered complex networks that address flow-space issues.

[0004] Specifically, existing evaluation methods for multilayer complex networks have the following shortcomings:

[0005] (1) When building complex network models, most of them are single-layer and unweighted networks, and the calculation indicators are relatively simple;

[0006] (2) There is a lack of methods for analyzing multi-layered networks in real life using complex network models;

[0007] (3) There is a lack of research on how multi-layer networks are interconnected, and it is necessary to set certain boundary conditions in light of actual conditions;

[0008] (4) The research on complex networks lacks interdisciplinary applications, and it is necessary to adopt models from other disciplines for integration. Summary of the Invention

[0009] The technical problem this invention aims to solve is the evaluation of nodes in multi-layer complex networks. This invention provides a node evaluation method for multi-layer complex networks based on a gravity model, comprising the following steps:

[0010] Step 1, Build the traffic network;

[0011] Step 2: Construct the force network based on the flow network;

[0012] Step 3: Establish a spatial network based on the different spatial subnetworks;

[0013] Step 4: Based on the spatial network, construct the distance network between nodes;

[0014] Step 5: Using a gravity model, derive the formula for calculating the virtual mass of the node;

[0015] Step 6: Based on the virtual mass calculation formula, select two different nodes to combine, thereby obtaining the combination matrix for virtual mass calculation;

[0016] Step 7: Substitute the values ​​of the forces between different nodes and the distances between different nodes into the combination matrix of the virtual mass formula to finally obtain the set of virtual masses of the nodes.

[0017] Furthermore, the flow network constructed in step 1 is represented by the graph G = (V, E, F);

[0018] Where V = {v i |i=1,2,…,N} represents nodes, N represents the number of nodes, E={(v i ,v j )|v i ,v j ∈V} represents the edges connecting nodes, and the flow network matrix is ​​defined as F={f ij |(v i ,v j )∈E}, where f ij This represents the weight of the edge between nodes, i.e., the flow from node i to node j. Further, in step 2, the force network is constructed as follows: the force between any two points is defined as the sum of the forward and reverse flows between the nodes, and the force network matrix is ​​defined as T = {t...} ij} N×N , where t ij =f ij +f ji ;f ij f represents the positive flow from node i to node j. ji This represents the reverse traffic from node i to node j.

[0019] Furthermore, the establishment of the spatial network mentioned in step 3 refers to establishing the spatial network according to rules; the rules are that if any two nodes are adjacent on the network, then the two nodes establish a proximity relationship on the spatial network.

[0020] The spatial network matrix is ​​defined as S = {S ij} N×N S ij It represents the boundary value that connects any two nodes in space.

[0021] Furthermore, the distance network between nodes described in step 4, the distance network matrix is ​​defined as D = {d ij} N×N d ij It represents the shortest distance between any two nodes on a spatial network.

[0022] Furthermore, in step 5, the formula for calculating the virtual mass of a node is derived using a gravity model, as follows:

[0023] Assuming the gravitational model is Where t ij Let G be the force between nodes i and i', k be the gravitational coefficient, and G' be the force between nodes i and i'. i G represents the virtual mass of node i. j Let d be the virtual mass of node j. ij The distance between node i and node j;

[0024] The derivation yields

[0025] Taking the logarithm of both sides of the equation yields the virtual mass calculation formula InG. i +InG j =-Ink+Int ij +2Ind ij +ε ij , ε ij This is the error.

[0026] Furthermore, in step 6, the combined matrix for the virtual quality calculation is obtained, denoted as R = {r ij (InG i ,InG j )} N×N , where i≠j.

[0027] Furthermore, step 7 specifically involves substituting the inter-node forces and inter-node distances into the combination matrix of the virtual mass formula to calculate {InG}. i} i＝1,…N Furthermore, the virtual mass set {G} of the nodes is obtained. i} i＝1,…N .

[0028] Furthermore, the flow in the flow network refers to passenger flow, product flow, trade flow, or cooperative relationships; the spatial network refers to a transportation network, geographical network, trade network, or cooperative network; and the nodes refer to people, stations, groups, enterprises, or countries.

[0029] Beneficial effects: This invention provides a node evaluation method for multi-layer complex networks based on a gravity model. This method, which combines a gravity model to model multi-layer complex networks and evaluate node characteristics, can also be applied to other multi-layer networks with similar structures. Flow networks reflect the exchange relationships of flow between nodes, while spatial networks reflect the spatial proximity relationships between nodes.

[0030] (1) In the prior art, when building complex network models, most of them are single-layer and unweighted networks with relatively simple calculation indicators; the present invention adopts a multi-layer network composed of a flow network and a distance network, uses flow and distance as weighting values, and uses the virtual quality of nodes as evaluation indicators.

[0031] (2) The single-layer network established by the existing technology cannot reflect the actual situation to a large extent; the present invention proposes a method for constructing and integrating multi-layer networks, which can be extended to the construction of other multi-layer networks;

[0032] (3) Existing technologies lack descriptions of the boundary conditions associated with multi-layer networks; this invention, in light of practical situations, proposes the boundary conditions involved in the construction of multi-layer networks.

[0033] (4) Existing technologies lack research on the interdisciplinary application of complex networks; this invention applies the gravity model from physics and economics to multi-layer networks of flow-distance, and can be applied to the analysis of related disciplinary models.

[0034] This invention derives a formula for calculating the virtual mass of nodes based on a gravity model. The virtual mass of a node considers the interaction forces and distance factors between nodes, and can serve as a comprehensive evaluation index of the importance of a node. This invention expands the research approach for multilayer complex networks. Attached Figure Description

[0035] Figure 1 It is based on the sub-network to establish the spatial network structure diagram. Detailed Implementation

[0036] This invention addresses multi-layered complex networks based on flow-space relationships. The flow network is a directed weighted globally coupled network. The spatial network reflects the spatial connections between nodes and is a directed weighted local network. The distance between any two nodes can be derived from the spatial network. A formula for calculating the virtual mass of nodes is obtained by inversely extrapolating from the gravity model, with input parameters including the flow and distance between any two nodes. The importance of nodes is evaluated based on their virtual mass.

[0037] Example 1

[0038] This invention discloses a node evaluation method for multilayer complex networks based on a gravity model. In this embodiment, the evaluation is performed on nodes of a flow-space multilayer complex network, and includes the following steps.

[0039] S1: Building a Traffic Network

[0040] A traffic network is both a directed weighted network and a globally coupled network, meaning that there is traffic between any two nodes, only the amount of traffic differs.

[0041] A flow network can be represented by a graph G = (V, E, F), where V = {v i |i=1,2,…,N} represents nodes, N represents the number of nodes, E={(v i ,v j )|v i ,v j ∈V} represents the edges connecting nodes, and the flow network matrix is ​​defined as F={f ij |(v i ,v j )∈E}, where f ij This represents the weight of the edge between nodes, i.e., the flow from node i to node j.

[0042] S2: Construct a force network based on the flow network;

[0043] The force between any two points is defined as the sum of the forward and reverse flows between the nodes. The force network matrix is ​​defined as T = {t} ij} N×N , where t ij =f ij +f ji f ij f represents the positive flow from node i to node j. ji This represents the reverse traffic from node i to node j.

[0044] S3: Based on different spatial subnetworks, establish spatial networks according to certain projection rules.

[0045] Figure 1 This demonstrates the rules for constructing the Space L model for complex networks. If any two nodes are adjacent on a network, then these two nodes establish a proximity relationship on the Space L model.

[0046] The spatial network matrix is ​​defined as S = {S ij} N×N S ij It represents the boundary value that connects any two nodes in space.

[0047] S4: Based on the spatial network, construct the distance network between nodes.

[0048] The distance network matrix is ​​defined as D = {d} ij} N×N d ij It represents the shortest distance between any two nodes on a spatial network.

[0049] S5: Using a gravity model, the formula for calculating the virtual mass of a node is derived.

[0050] Assuming the gravitational model is Where t ij Let G be the force between nodes i and i', k be the gravitational coefficient, and G' be the force between nodes i and i'. i G represents the virtual mass of node i. j Let d be the virtual mass of node j. ij Let be the spatial distance from node i to node j.

[0051] The derivation yields

[0052] Taking the logarithm of both sides of the equation yields the virtual mass calculation formula InG. i +InG j =-Ink+Int ij +2Ind ij +ε ij , ε ij This is the error.

[0053] S6: Based on the virtual mass calculation formula, different values ​​are taken for node i and node j to obtain the combination matrix for virtual mass calculation, i.e., R = {r ij (InG i ,InG j )} N×N , where i≠j. This is an overdetermined system of equations.

[0054] S7: Substitute the inter-node forces and inter-node distances into the combination matrix of the virtual mass formula to calculate {InG}. i} i＝1,…N Furthermore, the virtual mass set {G} of the nodes is obtained. i} i＝1,…N .

[0055] S8: Evaluate the virtual quality indicators of the nodes based on the specific circumstances.

[0056] Specific examples are as follows: Traffic can be passenger flow, product flow, trade flow, cooperative relationships, etc.; spatial networks can be transportation networks, geographical networks, trade networks, cooperative networks, etc. Nodes can be people, stations, groups, enterprises, countries, etc.

[0057] Example 2

[0058] The method of this invention is used to evaluate passenger flow networks, and the steps are as follows:

[0059] Step 1: Build a customer flow network;

[0060] The passenger flow network can be represented by the graph G = (V, E, F), where V = {v i |i=1,2,…,N} represents station nodes, N represents the number of station nodes, E={(v i ,v j )|v i ,v j ∈V} represents the edges connecting station nodes, and the passenger flow network matrix is ​​defined as F={f ij |(v i ,v j )∈E}, where f ij This represents the weight of the edge between station nodes, i.e., the flow from station node i to node j;

[0061] Step 2: Construct an action network based on the passenger flow network; the action force between any two stations is defined as the sum of the forward and reverse flow between the stations.

[0062] Step 3: Establish a spatial network based on different spatial sub-networks, wherein the spatial network is a transportation network;

[0063] Step 4: Based on the transportation network, construct a distance network between nodes, where each node is a station;

[0064] Step 5: Using a gravity model, derive the formula for calculating the virtual mass of the node;

[0065] Step 6: According to the virtual quality calculation formula, take different values ​​for any two station nodes and combine them to obtain the combination matrix for virtual quality calculation;

[0066] Step 7: Substitute the forces between stations and the distances between stations into the combination matrix of the virtual mass formula to finally obtain the virtual mass set of the nodes.

[0067] Obviously, the embodiments described in this invention are only a part of the embodiments of this application, and not all of the embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without inventive effort are within the scope of protection of this application.

Claims

1. A node evaluation method for multilayer complex networks based on a gravity model, characterized in that, Includes the following steps: Step 1, Build the traffic network; Step 2: Based on the flow network, construct the force network; specifically, the force between any two points is defined as the sum of the forward and reverse flows between the nodes, and the force network matrix is ​​defined as follows: ,in ; Represents a node To the node Positive traffic, Represents a node To the node Reverse traffic; Step 3: Establish a spatial network based on the different spatial subnetworks; Step 4: Based on the spatial network, construct the distance network between nodes; Step 5: Using a gravity model, the formula for calculating the virtual mass of a node is derived; the details are as follows: Assuming the gravitational model is ;in For nodes With nodes The force in the middle, The gravitational coefficient, For nodes virtual quality, For nodes virtual quality, For nodes To the node Spatial distance; The derivation yields ; Taking the logarithm of both sides of the equation yields the formula for calculating virtual mass. , For error; Step 6: Based on the virtual mass calculation formula, select two different nodes to combine, thereby obtaining the combination matrix for virtual mass calculation; Step 7: Substitute the values ​​of the forces between different nodes and the distances between different nodes into the combination matrix of the virtual mass formula to finally obtain the set of virtual masses of the nodes. The traffic in the traffic network refers to passenger flow, product flow, trade flow, or cooperative relationships. The spatial network is a transportation network, a geographical network, a trade network, or a cooperation network; The nodes can be people, stations, groups, businesses, or countries.

2. The node evaluation method for a multilayer complex network based on a gravity model according to claim 1, characterized in that, The traffic network diagram constructed in step 1 express; in Represents a node. Indicates the number of nodes. The edges representing connecting nodes are defined as follows: The flow network matrix is ​​defined as follows: ,in This represents the weight of the edge between nodes, i.e., the weight of the node. To the node Traffic.

3. The node evaluation method for a multilayer complex network based on a gravity model according to claim 1, characterized in that, The establishment of a spatial network in step 3 refers to establishing a spatial network according to rules; the rules are that if any two nodes are adjacent on the network, then the two nodes establish a proximity relationship on the spatial network. Spatial network matrix is ​​defined as , It represents the boundary value that connects any two nodes in space.

4. The node evaluation method for a multilayer complex network based on a gravity model according to claim 3, characterized in that, The distance network between nodes described in step 4 is defined as follows: , This represents the shortest distance between any two nodes on a spatial network.

5. The node evaluation method for a multilayer complex network based on a gravity model according to claim 1, characterized in that, The combination matrix obtained in step 6 for the virtual quality calculation is represented as follows: ,in .

6. The node evaluation method for a multilayer complex network based on a gravity model according to claim 5, characterized in that, Step 7 specifically involves substituting the inter-node forces and inter-node distances into the combination matrix of the virtual mass formula to calculate the results. This further yields the virtual mass set of the nodes. .

7. The node evaluation method for a multilayer complex network based on a gravity model according to claim 1, characterized in that, Step 7 specifically involves substituting the inter-node forces and inter-node distances into the combination matrix of the virtual mass formula to calculate the results. This further yields the virtual mass set of the nodes. .

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