Evolutionary Model of Complex Networks Based on Dynamic Space Constraints

Abstract

This invention provides a complex network evolution model based on dynamic spatial constraints, characterized by the following steps: Step S1, obtaining candidate nodes for each node based on the constraint area, and using all candidate nodes corresponding to each node as the candidate set for that node; Step S2, calculating the expected flow between each node in the two-dimensional spatial matrix and each node in the corresponding candidate set; Step S3, connecting the two nodes corresponding to the maximum value in the expected flow; Step S4, reducing the constraint area of ​​all nodes, and if the reduced constraint area is less than the minimum constraint area, then using the minimum constraint area as the reduced constraint area; Step S5, determining whether the total number of edges in the two-dimensional spatial matrix is ​​less than the upper limit E of the edges. If yes, proceed to step S1; otherwise, obtain the complex evolution network. In summary, this method can accurately simulate the node connection evolution process of a two-segment power law.

Description

Technical Field

[0001] This invention relates to the field of complex networks, and more specifically to a complex network evolution model based on dynamic spatial constraints. Background Technology

[0002] Complex networks are a new theory built upon previous research in self-organization, nonlinearity, and complexity theory. By establishing models to reveal various statistical characteristics of real-world networks, it studies network outcomes and the intrinsic mechanisms governing network evolution. As a physical model reflecting the organizational laws of real-world objects, it is widely applied in rail transit networks, language networks, forest health assessment, social security and stability maintenance, network security protection, and power grid dispatching and planning. Networks evolve based on massive amounts of data and can be viewed as a mirror model of the real world. If their evolutionary process can be efficiently improved in accordance with actual natural laws, and the constraints and causal relationships influencing the real world can be discovered, it will greatly enhance people's understanding of systems such as transportation networks, thereby effectively identifying relevant or potential hub nodes and proactively implementing various risk prevention and response measures.

[0003] In such studies, spatial constraints are often considered a key factor driving network evolution. Research shows that as spatial constraints intensify, the degree distribution of network nodes changes, gradually deviating from the initially wide power-law distribution and transforming into a narrower exponential or Gaussian distribution. This transition has a physical basis; in spatial networks, connecting nodes requires consideration of latency, construction costs, and energy dissipation density. To reduce these costs, nodes in spatial networks exhibit proximity, tending to connect with neighboring nodes, significantly reducing the number of connectable nodes and effectively avoiding the generation of nodes with high degree values. Existing network models of spatial effects indicate that structural changes can be achieved by adjusting the strength of spatial constraints, but the bisegmental power-law distribution present in aerospace networks cannot be obtained by adjusting the strength of spatial constraints. Summary of the Invention

[0004] This invention is made to solve the above problems, and aims to provide a complex network evolution model based on dynamic spatial constraints.

[0005] This invention provides a complex network evolution model based on dynamic spatial constraints. It connects N nodes with different constraint areas in a two-dimensional spatial matrix according to the upper limit E, the minimum constraint area, and the fitness of the nodes to obtain a complex evolution network. The model includes the following steps: Step S1: Based on the constraint area of ​​each node, two nodes that are not connected but are contained within each other's constraint area are designated as candidate nodes for the other node. All candidate nodes corresponding to each node are designated as the candidate set for that node. Step S2: Calculate the expected flow between each node in the two-dimensional spatial matrix and each node in the corresponding candidate set. Step S3: Connect the two nodes corresponding to the maximum value in the expected flow. Step S4: Reduce the constraint area of ​​all nodes by ΔC. If the reduced constraint area is less than the minimum constraint area, the minimum constraint area is used as the reduced constraint area. Step S5: Determine if the total number of edges in the two-dimensional spatial matrix is ​​less than the upper limit E. If yes, proceed to step S1; otherwise, the complex evolution network is obtained. Here, the constraint area is the area of ​​a circle centered at the node. The formula for calculating the expected flow is as follows: In the formula T ij Let M be the expected flow of node i and node j in the candidate set of node i, where k is a constant and M is a variable. i M represents the fitness of node i. j Let D be the fitness of node j. ij Let be the Euclidean distance between node i and node j, α be a parameter for adjusting the fitness of nodes, and γ be a parameter for adjusting distance constraints.

[0006] The complex network evolution model based on dynamic spatial constraints provided in this invention can also have the following feature: the fitness of nodes in the two-dimensional spatial matrix follows a power-law distribution ρ(M)~M -θ .

[0007] This invention also provides an aviation network evolution prediction method. Using any of the above-mentioned complex network evolution models based on dynamic spatial constraints, the method obtains the evolution prediction results of m air routes between n airports, including the following steps: Step T1, treating airports as nodes, and setting the corresponding nodes in a two-dimensional spatial matrix according to the actual geographical location of each airport; Step T2, setting the fitness of the nodes corresponding to the airports according to the population of the cities where the airports are located; Step T3, setting the total number of edges E to m; Step T4, based on the constraint area of ​​each node, designating two nodes that are not connected to each other but are mutually contained within each other's constraint area as each other's nodes. The candidate nodes are selected, and all candidate nodes corresponding to each node are taken as the candidate set of that node; Step T5, calculate the expected flow between each node in the two-dimensional spatial matrix and each node in the corresponding candidate set; Step T6, connect the two nodes corresponding to the maximum value in the expected flow; Step T7, reduce the constraint area of ​​all nodes by ΔC area. If the reduced constraint area is less than the minimum constraint area, then the minimum constraint area is taken as the reduced constraint area; Step T8, determine whether the total number of connected edges in the two-dimensional spatial matrix is ​​less than the upper limit of connected edges E. If yes, proceed to step T4. If no, the evolution prediction results of m air routes between n airports are obtained.

[0008] The aviation network evolution prediction method provided by this invention may also have the following feature: wherein the parameter α = 1.

[0009] The aviation network evolution prediction method provided by this invention may also have the following feature: the parameter γ takes a value range of 0.2 to 2.7.

[0010] The aviation network evolution prediction method provided by this invention may also have the following features: when the two-dimensional spatial matrix is ​​a 10*10 matrix, the radius of the initial constraint area is 500, the value range of the ΔC area is 5000 to 10000, and the minimum constraint area is 50.

[0011] The role and effect of invention

[0012] According to the complex network evolution model based on dynamic spatial constraints of the present invention, by setting fitness to reflect the connection tendency of nodes, and setting constraint area, minimum constraint area and reduction area ΔC, the connection of network nodes gradually changes from fitness-dominated to position-dominated, thereby transforming the network topology from a star structure to a road network structure. Therefore, the complex network evolution model based on dynamic spatial constraints of the present invention can accurately simulate the node connection evolution process of the two-segment power law. Attached Figure Description

[0013] Figure 1 This is a schematic diagram of the cumulative degree distribution of different minimum constraint radii in an embodiment of the present invention;

[0014] Figure 2 This is a schematic diagram of the cumulative distribution of three different budget distributions in an embodiment of the present invention;

[0015] Figure 3 This is a flowchart illustrating a complex network evolution model based on dynamic spatial constraints in an embodiment of the present invention.

[0016] Figure 4 This is a flowchart illustrating the aviation network evolution prediction method in an embodiment of the present invention;

[0017] Figure 5 This is a schematic diagram of the evolution prediction results of the China Aviation Network in an embodiment of the present invention;

[0018] Figure 6 This is a schematic diagram illustrating the degree distribution analysis of evolution prediction results in an embodiment of the present invention;

[0019] Figure 7 This is a schematic diagram illustrating the analysis of the neighboring node degree distribution of the evolution prediction results in an embodiment of the present invention;

[0020] Figure 8 This is a schematic diagram illustrating the analysis of clustering coefficients of evolution prediction results in an embodiment of the present invention. Detailed Implementation

[0021] To make the technical means, creative features, objectives and effects of this invention easier to understand, the following embodiments, in conjunction with the accompanying drawings, provide a detailed description of the complex network evolution model based on dynamic spatial constraints of this invention.

[0022] Networks in daily life, such as aviation networks, undergo complex evolution processes. In order to predict the evolution of these networks, it is necessary to construct dynamic constraint models that conform to these networks.

[0023] A spatial network model based on node energy constraints is constructed within a two-dimensional space [0,10]. From the perspective of budget and cost, each node is assigned energy B, which is similar to the population capacity of a city or the number of friends in a social circle. t It is the maximum length that nodes can be connected after being subject to spatial constraints, l t This refers to the actual edge length. In spatial networks, making long-distance connections incurs a cost proportional to the edge length. Furthermore, a minimum edge length r0 is introduced; connections shorter than this do not incur any cost. Based on this, node energy consumption is primarily driven by long-edge connections. Therefore, a new node can only connect to other nodes with budget margins or those located nearby. The modeling mechanism under this setting is as follows:

[0024] Step R1, within each time interval t, a constrained radius Lt and energy B t A new node t enters the system and is assigned a position according to random properties.

[0025] Step R2, when node t is making edge connections, it is selected from the candidate node set G. t Internal compliance probability Select node i to connect, where k i It refers to the degree value of node i, G. t The set of all nodes in the spatial network whose distance to node t is less than r0, or located at the maximum length L of node t. t It consists of a set of nodes that are internal, have a budget, and are not connected to node t. r0

[0026] Step R3, during the loop, newly created edges l t When the length of the edge is greater than r0, the energy consumption of the two nodes connected by the edge will be proportional to the energy consumption of the edge. The energy of the two nodes is updated to the original energy of the node minus the energy of the edge l. t The value after the length. When the energy budget of a node is less than 0 after several loop connections, its constraint radius is set to the minimum r0. If this node is connected in the next loop, no energy cost is required. After completing one edge connection, return to step T1 and repeat the entire process.

[0027] The above modeling mechanism is constructed using homogeneous and heterogeneous constraints, and the specific process is as follows:

[0028] I. Homogeneity Constraint:

[0029] The initial budget assigned to each node is B. i =B, where the corresponding maximum upper bound of the connected edges Li = L, that is, the influence area of ​​node i in the early stage of evolution can be taken as S. i =πL 2 In principle, a sufficient budget in the initial stage allows it to access any node within a distance of L; once the budget is exhausted, i.e., B... i <0, edges can only be generated within the range of r0. In this case, connecting nodes does not incur any cost, and the area affected by the node is S. i =πr0 2 This leads to abrupt changes in both the model's connection probabilities and dynamic formulas at the critical point. To assess when this critical event occurs, we assume r0 is small; therefore, when B... i When r > 0, the probability that the new edge is shorter than r0 can be determined by r0. 2 / L 2 The computation is generally negligible. That is, all links created at this stage are essentially remote connections. Therefore, when node i reaches transition point k... c At that time, a critical event will occur, in which The dynamic formula for the degree value of node i is obtained as follows:

[0030]

[0031] Monte Carlo simulation was used to set the total number of nodes N=15000, the initial budget B=20, the maximum constraint radius of the nodes L=3, and the minimum constraint radius r0=0.5, 1, and 2 respectively for simulation.

[0032] Figure 1 This is a schematic diagram of the cumulative degree distribution of different minimum constraint radii in an embodiment of the present invention.

[0033] like Figure 1 As shown, the horizontal axis represents the degree value k of the node, the vertical axis represents the cumulative degree distribution value P(k), and the vertical dashed line represents the estimated k. c To better visualize the location, all data points were moved one unit to the right, data points with r0 = 0.5 were moved upward by the same distance, and data points with r0 = 1.5 were moved downward by the same distance, thus making them more clearly distinguishable from data points with r0 = 1. This resulted in the minimum constraint radius exhibiting a bisegmental power law curve for different values.

[0034] To assess the quantitative accuracy of the analysis, the power-law exponents γ1 and γ2 and the transition degree k were selected. c With r0 = 1.5, the network size reaches 1.5 × 10⁻⁶. 4 There are 3 nodes, each data point runs independently more than 20 times on average, L = 3, B ∈ [20, 70], and the network size is 1.5 × 10⁻⁶. 4 γ2, γ1 and L 2 The theoretical relationship between them is a relationship k c Replace with The derivation shows good agreement with the measurement results, and the measurement transition degree k c The linear relationship with budget B proves this. The effectiveness.

[0035] II. Heterogeneous Constraints:

[0036] Generally, the higher the energy level of a node, the wider its influence range, so it is directly proportional to Li. Although their long-range connection sizes differ, the fraction B... i / L i It has the same value for all nodes. The edge length of the node and k i The dynamics of (t) at the transition degree k c ~B i / L iA sudden change will occur at that point. Therefore, equation (1) can be specifically written as:

[0037]

[0038]

[0039] make Let f(t) = Ct. Before solving equation (2), note that under the current heterogeneous constraints, the calculation of the first term in f(t) should be performed for all possible L. i 2 Take the average value, where L i 2 Obey ρ(L) i 2 ),and

[0040] Substituting f(t)=Ct into equation (2), the first equation is the same as the dynamics of the fitness model, and its power-law exponent γ1 of the degree distribution is... and The boundary is the power-law exponent corresponding to the second equation in (2). It is greater than the upper limit of γ1. Therefore, as long as the cost constraint causes a sudden change in the edge length, the double power-law distribution can occur in a very general case.

[0041] Set maximum budget B max =40, Minimum Budget B min =15, with scaling factor B / L=7 and r0=0.5, the network size reaches 1.5×10 4 For each node, three different budget distributions are selected: exponential distribution, average distribution, and power-law distribution, to obtain the cumulative distribution of different budget distributions.

[0042] Figure 2 This is a schematic diagram of the cumulative distribution of three different budget distributions in an embodiment of the present invention.

[0043] like Figure 2 As shown, the horizontal axis represents the degree value k of the node, the vertical axis represents the cumulative degree distribution value P(k), and the vertical dashed line represents the estimated k. c To avoid the boundary, all data points were moved one unit to the right. Data points in the power-law distribution were moved up by the same distance, and data points in the exponential distribution were moved down by the same distance, thus making a clearer distinction from the data points in the average distribution. The curves of the three different budget distributions all showed a bisegment power-law distribution.

[0044] Based on the reason for the existence of the two-segment power law in the above dynamic constraints, by selecting the corresponding variables and designing the corresponding constraint conditions, we can realize the dynamic constraint of the spatial network and thus obtain the evolution model of the complex network.

[0045] Figure 3 This is a flowchart illustrating a complex network evolution model based on dynamic spatial constraints in an embodiment of the present invention.

[0046] like Figure 3 As shown, a complex network evolution model based on dynamic spatial constraints is used to connect N nodes with different constraint areas in a two-dimensional spatial matrix according to the upper limit E of the connection, the minimum constraint area, and the fitness of the nodes to obtain a complex evolution network. The constraint area is the area of ​​a circle centered at the node, and it has the following characteristics:

[0047] Step S1: Based on the constraint area of ​​each node, two nodes that are not connected to each other but are contained within each other's constraint area are respectively regarded as candidate nodes of the other node, and all candidate nodes corresponding to each node are regarded as the candidate set of that node.

[0048] Step S2: Calculate the expected flow between each node in the two-dimensional spatial matrix and each node in the corresponding candidate set.

[0049] The formula for calculating the expected flow is as follows:

[0050]

[0051] In the formula T ij Let M be the expected flow of node i and node j in the candidate set of node i, where k is a constant and M is a variable. i M represents the fitness of node i. j Let D be the fitness of node j. ij Let be the Euclidean distance between nodes i and j. In actual networks, the edges between nodes need to consider the influence of geographical constraints. α is a parameter that adjusts the fitness of a node, and γ is a parameter that adjusts the distance constraint. The fitness of a node is directly proportional to its degree value and can be defined as M ~ k. λ Combined with the flow formula between nodes w ij ~(k) i k j ) θ Therefore, we get w. ij ~(M) i M j ) θ The fitness of nodes in a two-dimensional spatial matrix follows a power-law distribution ρ(M)~M -θ .

[0052] Step S3: Connect the two nodes corresponding to the maximum value in the expected flow.

[0053] Step S4: Reduce the constraint area of ​​all nodes by ΔC. If the reduced constraint area is less than the minimum constraint area, then the minimum constraint area is taken as the reduced constraint area.

[0054] Step S5: Determine whether the total number of edges in the two-dimensional spatial matrix is ​​less than the upper limit of edges E. If yes, proceed to step S1; otherwise, obtain the complex evolutionary network.

[0055] Based on the two-segment power-law characteristics of complex network evolution models with dynamic spatial constraints, the evolution of aviation networks that also exhibit two-segment power-law characteristics can be predicted.

[0056] Figure 4 This is a flowchart illustrating the aviation network evolution prediction method in an embodiment of the present invention.

[0057] like Figure 4 As shown, the aviation network evolution prediction method, based on a complex network evolution model with dynamic spatial constraints, obtains the evolution prediction results of m air routes between n airports, including the following steps:

[0058] Step T1: Treat the airports as nodes and set the corresponding nodes in a two-dimensional spatial matrix according to the actual geographical location of each airport.

[0059] Step T2: Set the fitness of the corresponding airport nodes based on the population of the city where the airport is located.

[0060] Step T3: Set the total number of connected edges E to m.

[0061] Step T4: Based on the constraint area of ​​each node, two nodes that are not connected to each other but are contained within each other's constraint area are respectively regarded as candidate nodes of the other node, and all candidate nodes corresponding to each node are regarded as the candidate set of that node.

[0062] Step T5: Calculate the expected flow between each node in the two-dimensional spatial matrix and each node in the corresponding candidate set.

[0063] The formula for calculating the expected flow is as follows:

[0064]

[0065] In this embodiment, parameter α = 1, and parameter γ ranges from 0.2 to 2.7.

[0066] Step T6: Connect the two nodes corresponding to the maximum value in the expected flow.

[0067] Step T7: Reduce the constraint area of ​​all nodes by ΔC. If the reduced constraint area is less than the minimum constraint area, then the minimum constraint area is taken as the reduced constraint area.

[0068] Step T8: Determine whether the total number of edges in the two-dimensional spatial matrix is ​​less than the upper limit of the number of edges E. If yes, proceed to step T4; otherwise, obtain the evolution prediction results of m air routes between n airports.

[0069] In this embodiment, the two-dimensional spatial matrix is ​​set to a 10*10 matrix. Initially, the radius of the constraint area is 500, the value range of the ΔC area is 5000~10000, the minimum constraint area is 50, the parameter α=1, the parameter γ=1.5, and 121 airport stations are selected as nodes for the China Aviation Network. 686 routes are selected to generate, that is, the total number of connected edges E=686.

[0070] Figure 5 This is a schematic diagram of the evolution prediction results of the China Aviation Network in an embodiment of the present invention.

[0071] like Figure 5 As shown, there are pivotal nodes in the network, corresponding to the actual geographical locations of Beijing, Shanghai, Guangzhou, Chengdu, Wuhan, Harbin, and Urumqi. The predicted results are largely consistent with the air routes of the core cities in the real Chinese aviation network.

[0072] Figure 6 This is a schematic diagram illustrating the degree distribution analysis of evolution prediction results in an embodiment of the present invention. Figure 7 This is a schematic diagram illustrating the analysis of the neighboring node degree distribution of the evolution prediction results in an embodiment of the present invention. Figure 8 This is a schematic diagram illustrating the analysis of clustering coefficients in the evolution prediction results in an embodiment of the present invention. The horizontal axis represents the degree value k of each node, and the vertical axis represents the cumulative degree distribution P(k) and the average degree K of neighboring nodes, respectively. nn (k) and cluster degree relationship coefficient C(k).

[0073] like Figure 6 , Figure 7 and Figure 8 As shown, the predicted Chinese aviation network has a cumulative degree distribution P(k) and an average degree K of neighboring nodes. nn Both the predicted Chinese aviation network and the cluster degree coefficient C(k) exhibit the curve characteristics of a two-segment power law, showing a hierarchical organizational structure. This closely resembles the characteristics of the actual Chinese aviation network, indicating that the predicted Chinese aviation network has a good match with the actual Chinese aviation network.

[0074] The role and effect of the embodiments

[0075] According to the complex network evolution model based on dynamic spatial constraints involved in this embodiment, by setting a fitness that conforms to a power-law distribution to reflect the connection tendency of nodes, and then setting a constraint area, a minimum constraint area, and a reduction area ΔC, the connection of network nodes gradually shifts from fitness-dominated to location-dominated, thereby transforming the network topology from a star structure to a road network structure. Therefore, the complex network evolution model based on dynamic spatial constraints of the present invention can accurately simulate the node connection evolution process of a two-segment power law.

[0076] The aviation network evolution prediction method described in this embodiment uses the city population as the fitness parameter of nodes, and then sets constraint area, minimum constraint area, and shrinkage area ΔC to reflect the role of geographical constraints in route evolution, so that the connection of airport nodes conforms to the actual aviation network layout, achieving accurate prediction of airport routes in the aviation network. In summary, this method can predict the route evolution results of airports relatively accurately.

[0077] The above embodiments are preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention.

Claims

1. A method for predicting the evolution of aviation networks, characterized in that: A complex evolutionary network is obtained by connecting N nodes with different constraint areas in a two-dimensional spatial matrix according to the upper limit E of the connection, the minimum constraint area, and the fitness of the nodes. Based on the complex evolutionary network, the evolution prediction results of m air routes between n airports are obtained, including the following steps: Step T1: Using the airport as the node, set the corresponding node in the two-dimensional spatial matrix according to the actual geographical location of each airport; Step T2: Set the fitness of the node corresponding to the airport based on the population of the city where the airport is located; Step T3: Set the upper limit E of the connecting edge to m; Step T4: Based on the constraint area of ​​each node, two nodes that are not connected to each other and are contained within each other's constraint area are respectively regarded as candidate nodes of the other node, and all candidate nodes corresponding to each node are regarded as the candidate set of the node, wherein the constraint area is the area of ​​the circle with the node as the center. Step T5: Calculate the expected flow between each node in the two-dimensional spatial matrix and each node in the corresponding candidate set; Step T6: Connect the two nodes corresponding to the maximum value in the expected flow; Step T7: Reduce the constraint area of ​​all nodes by ΔC area. If the reduced constraint area is less than the minimum constraint area, then the minimum constraint area is used as the reduced constraint area. Step T8: Determine whether the total number of edges in the two-dimensional spatial matrix is ​​less than the upper limit of the number of edges E. If yes, proceed to step T4; otherwise, obtain the evolution prediction results of the m air routes between the n airports.

2. The aviation network evolution prediction method according to claim 1, characterized in that: in, The formula for calculating the expected flow is as follows: In the formula T ij Let M be the expected flow of node i and node j in the candidate set of node i, where k is a constant and M is a variable. i M represents the fitness of node i. j Let D be the fitness of node j. ij Let be the Euclidean distance between node i and node j, α be a parameter for adjusting the fitness of nodes, and γ be a parameter for adjusting distance constraints.

3. The aviation network evolution prediction method according to claim 2, characterized in that: in, The fitness of the nodes in the two-dimensional spatial matrix follows a power-law distribution ρ(M)~M -θ .

4. The aviation network evolution prediction method according to claim 2, characterized in that: in, The parameter α = 1.

5. The aviation network evolution prediction method according to claim 2, characterized in that: in, The parameter γ has a value range of 0.2 to 2.

7.

6. The aviation network evolution prediction method according to claim 1, characterized in that: in, When the two-dimensional spatial matrix is ​​a 10*10 matrix, the initial radius of the constraint area is 500, the value range of the ΔC area is 5000 to 10000, and the minimum constraint area is 50.

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