CN113190939A - Large sparse complex network topology analysis and simplification method based on polygon coefficient - Google Patents
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Abstract
The invention discloses a large sparse complex network topology analysis and simplification method based on polygon coefficients, which comprises the following steps: 1) defining network polygon coefficients; 2) calculating the coefficient of the network polygon; 3) calculating network distribution coefficients; 4) analyzing the network topology; 5) the network topology is simplified. The method is used for analyzing and simplifying the topological structure of the large-scale sparse wide area network, reducing the time consumption of a large number of traversal algorithms, heuristic algorithms and approximation algorithms based on the network topology, and reducing the number of nodes and links contained in the network on the basis of not changing the original topological basic architecture.
Description
Technical Field
The invention relates to a network topology analysis and simplification technology, in particular to a large sparse complex network topology analysis and simplification method based on polygon coefficients.
Background
With the development of 5G and internet of things technologies, communication and computer networks are accessing more and more terminals with different intelligence levels to the networks and providing services and services with different service qualities. In order to meet the quality of service requirements of different types of services and services, current communication and computer networks require flexible and intelligent management and optimization of network resources.
This type of flexible, intelligent network management and optimization is typically achieved through various traversal, heuristic, learning, and approximation algorithms based on the network topology. The basic idea is to find a specific node and link that meets the requirement by traversing all nodes and links of the network, or selectively traversing some nodes and links, and to achieve the goal of resource management and optimization by performing special processing on these nodes and links. The network topology abstracts the physical and logical relationships of nodes and links in the network. The scale of the network topology, namely the number of nodes and links in the network and how they are interconnected, affects the calculation accuracy and time consumption of the algorithm, and directly affects the feasibility and the feasibility of the application based on the algorithm.
Classified by geographic scope, current communication and computer networks can be simply divided into local area networks and wide area networks. The local area network is responsible for interconnecting hosts in an organization with a small geographic range into an internal network, and the wide area network is responsible for interconnecting the internal networks together to form an interconnected network. Therefore, in communication and computer networks, the local area network is usually located at the edge of the network, and the topology is usually a regular tree or an extended type based on the tree; the wide area network is located at the core of the network and is responsible for connecting the local area network at the edge of the network and the access network to form a network with a larger geographic range. Wide area networks are typically comprised of a backbone of large telecommunication service providers, internet service providers. Since these backbones are usually selected to be deployed in the central city of the administrative area, the wide area networks are usually sparse in nodes and have large node distances. Meanwhile, due to long-term evolution of the backbone network, the topological structure of the backbone network is greatly influenced by geographical conditions, economic conditions and national policies, and the backbone network is complex in structure and difficult to describe by a conventional method. When the number of nodes of a Wide Area Network (WAN for short) reaches a certain number, the time overhead of various traversal algorithms based on Network topology is very large. To reduce this time overhead, the usual approach is to simplify the network topology. However, since the wide area network has a complex topology structure which is difficult to describe by a conventional method, the description and simplification of the network topology are difficult, so that the management and optimization application of the network are difficult to be applied and deployed in practice.
Complex networks (Complex networks) generally refer to networks whose network structure is difficult to describe in a conventional manner, such as human brain organization networks, social networks, virus propagation networks, and the like. Denseness is one way to characterize complex network topologies. The consistency is usually described by Clustering coefficient (Clustering), Density coefficient (Density), polygon community coefficient (Clique Communities), and the like. The clustering coefficient calculates the proportion of the edges forming the triangle in the total number of the edges in the network; calculating the proportion of the formed edges in the network to the edges which are most likely to be formed by the density coefficient; the polygon community coefficients account for the number of all nested polygons present in the network given the number of polygon edges, k. Although these coefficients analyze the dense characteristics of the network from different angles, they are not suitable for describing the wide area network, because the wide area network is very sparse, resulting in very small Clustering coefficients (Clustering), Density coefficients (Density), and polygon community coefficients, so that the network topology of the wide area network is difficult to analyze and identify by using these coefficients. The cooperative coefficients calculate Pearson correlation among nodes and can reflect the topological characteristics of the network to a certain extent. For example, a network with a positive coequal coefficient indicates that nodes of the network with similar degrees are typically connected to each other; a network with a negative paring number generally means that nodes of the network that are very different are connected together, that is, a network with a positive paring number may have a linear or ring network topology, and a network with a negative paring number may have a star network topology, but when two networks have a paring number of the same sign, the difference in network topology is still unrecognizable.
Disclosure of Invention
The invention aims to provide a large sparse complex network topology analysis and simplification method based on polygon coefficients aiming at the defects of the prior art. The method is used for analyzing and simplifying the topological structure of the large-scale sparse wide area network, reducing the time consumption of a large number of traversal algorithms, heuristic algorithms and approximation algorithms based on the network topology, and reducing the number of nodes and links contained in the network on the basis of not changing the original topological basic architecture.
The technical scheme for realizing the purpose of the invention is as follows:
a large sparse complex network topology analysis and simplification method based on polygon coefficients comprises the following steps:
1) definition of network polygon coefficients: given a network, abstracting the network topology into an undirected graph G (V, E), where V represents a set of nodes of the network, | V | represents the total number of nodes included in the network, E represents a set of links of the given network, links in the set of links refer to network paths connecting two adjacent nodes, and if I represents a set of non-nested polygons included in the given network and Ci represents a non-nested polygon in I, the polygon coefficient of the given network is (Σ)i∈ICi)/|V|；
2) And (3) calculating the network polygon coefficient: if the network has N ═ V | nodes, the adjacency matrix A is an N multiplied by N square matrix, N is the total node number of the network, the value of the element Aij of the square matrix is 1 or 0, 1 is taken to represent that the nodes i and j are directly connected with each other, 0 is taken to represent that no link is connected with each other, N is calculated through the adjacency matrix A, N is the denominator of the polygon coefficient, the non-nested polygon set is obtained through calculation of a built-in polygon class of Matlab, then all the node numbers contained in the non-nested polygon set, namely the numerator of the polygon coefficient, and finally the polygon coefficient of the network is obtained;
3) calculating the network distribution coefficient: calculating Pearson correlation coefficients among node degrees of different nodes by using the same distribution coefficients, inputting the adjacency matrix A of the known network in the step 2) into Matlab, and directly calculating the same distribution coefficients of the network through an Assertility function built in the Brain Connectivity Toolbox of the Matlab;
4) analyzing the network topology: networks are classified into the following 7 types according to polygon coefficients and parity coefficients: (1) the polygon coefficient is 0 and has a positive homogement coefficient, (2) the polygon coefficient is 0 and has a negative homogement coefficient, (3) the polygon coefficient is less than 1 and has a positive homogement coefficient, (4) the polygon coefficient is less than 1 and has a negative homogement coefficient, (5) the polygon coefficient is 1 and has a positive homogement coefficient, (6) the polygon coefficient is 1 and has a negative homogement coefficient, and (7) the polygon coefficient is greater than 1, wherein (1) the type network corresponds to a linear topology, (2) the type network corresponds to a star-shaped structure, (3) the topology corresponding to the type network has a linear backbone, part of branches have a ring-shaped structure, (4) the topology corresponding to the type network has a star-shaped backbone, part of branches have a ring-shaped structure, the type network is less, (5) the type network has a ring-shaped backbone, (6) the type network has a mesh-shaped structure, and (7) the type network has a mesh-shaped structure;
5) and (3) simplifying the network topology: according to the network type, the network is simplified by adopting a comprehensive strategy of longest path segmentation and small-area non-nested polygon combination, the simplification requirement is to reduce nodes of the network on the premise of not changing the main topological characteristics of the network, so that a topology-based algorithm is operated on the simplified topology, the time consumption is reduced on the premise of not losing the network characteristics, and the specific simplification method of various network types is as follows: (1) the class and (3) class networks directly find the longest path, and the longest path is cut off to obtain a plurality of sub-networks to realize the network segmentation; (2) and (4) the network only needs to consider the central node, namely the nodes with high node degree and the links between the nodes form the backbone of the network; (5) the class network has a ring topology, and the topology ring has high order and small quantity and is not simplified; (6) class networks are rare, and topology structure and a simplified method are difficult to determine; (7) class networks have a large number of lower order rings and a large number, and the simplified approach is to perform a multi-edge merge.
The calculation process of the network polygon coefficient in the step 2) is as follows;
(1) inputting an adjacency matrix A of a network;
(2) calculating a node degree matrix D, wherein the node degree of each node is the direct connection of the node and the number of nodes;
(3) calculating the total node number N;
(4) simplifying a node degree matrix D, and deleting nodes with the node degree smaller than 2 from the D, wherein the nodes cannot form a polygon;
(5) calculating a set of non-nested polygons for the network;
(6) calculating the number n of all nodes contained in the nesting-free polygon set;
(7) calculating the polygon coefficient which is N/N;
(8) and outputting the polygon coefficients.
The simplification process of the (7) type network in the step 6) by adopting polygon merging is as follows:
(1) inputting an adjacency matrix A and a polygon area threshold st of a network;
(2) calculating a node degree matrix D;
(3) simplifying a node degree matrix D, and deleting nodes with the node degree smaller than 2 from the D, wherein the nodes cannot form a polygon;
(4) calculating a set of non-nested polygons for the network;
(5) calculating the area of each polygon in the non-nested polygon set;
(6) merging the polygon with the area smaller than the threshold st with the adjacent polygon;
(7) calculating an adjacency matrix A of the merged network;
(8) and outputting A.
The method is used for analyzing and simplifying the topological structure of the large-scale sparse wide area network, reducing the time consumption of a large number of traversal algorithms, heuristic algorithms and approximation algorithms based on the network topology, and reducing the number of nodes and links contained in the network on the basis of not changing the original topological basic architecture.
Drawings
FIG. 1 is an undirected graph used to abstract a network topology in an embodiment;
FIG. 2 is a schematic diagram illustrating a process for calculating polygon coefficients in an embodiment;
FIG. 3 is a diagram illustrating a network topology according to an embodiment;
FIG. 4 is a schematic diagram of an embodiment in which the network topology is a star;
FIG. 5 is a diagram illustrating an embodiment in which a network topology is a linear backbone + a small number of rings;
FIG. 6 is a diagram illustrating an embodiment in which the network topology is a ring backbone;
FIG. 7 is a diagram illustrating a mesh network topology according to an embodiment;
FIG. 8 is a flow chart illustrating polygon merging in an embodiment;
fig. 9 is a comparison diagram before and after merging of network topologies in the embodiment.
In the figure, the abscissa represents the dimension, and the ordinate represents the longitude.
Detailed Description
The invention will be further elucidated with reference to the drawings and examples, without however being limited thereto.
Example (b):
the computer communication wide area network is generally complex in node sparse structure, but the scale and density of a ring structure formed by local network determine the main topological characteristic of the network, and because the conventional network analysis index cannot describe the characteristic, a polygon coefficient is designed, and the scale and density of the ring structure in the network are described by calculating the ratio of the number of nodes contained in all non-nested polygons in the network to the total number of nodes in the network.
A large sparse complex network topology analysis and simplification method based on polygon coefficients comprises the following steps:
1) definition of network polygon coefficients: given a network, abstracting the network topology into an undirected graph G (V, E), where V represents a set of nodes of the network, | V | represents the total number of nodes included in the network, E represents a set of links of the given network, links in the set of links refer to network paths connecting two adjacent nodes, and if I represents a set of non-nested polygons included in the given network and Ci represents a non-nested polygon in I, the polygon coefficient of the given network is (Σ)i∈ICi)/|V|；
In this example, points in the undirected graph represent network devices, edges represent network links, edges have no directionality, and the performance of surface forward and reverse links is consistent, as shown in fig. 1, a 37-node network undirected graph is provided;
2) and (3) calculating the network polygon coefficient: if the network has N ═ V | nodes, the adjacency matrix A is an N multiplied by N square matrix, N is the total node number of the network, the value of the element Aij of the square matrix is 1 or 0, 1 is taken to represent that the nodes i and j are directly connected with each other, 0 is taken to represent that no link is connected with each other, N is calculated through the adjacency matrix A, N is the denominator of the polygon coefficient, the non-nested polygon set is obtained through calculation of a built-in polygon class of Matlab, then all the node numbers contained in the non-nested polygon set, namely the numerator of the polygon coefficient, and finally the polygon coefficient of the network is obtained;
the adjacency matrix a in this example is a 37 × 37 square matrix, and when i is not equal to j, Aij represents the node degree, and since it is an undirected graph, the square matrix is a symmetric matrix, where Aij ═ Aji, which means that when i is different from j, the element values in i row and j column are equal to the element values in j row and i column, Aij is 0 or 1, 0 means that there is no link connection between nodes i and j, and 1 means that there is a link directly connected, and the adjacency matrix of the subgraph formed by nodes 1 to 11 in fig. 1 is as follows:
3) calculating the network distribution coefficient: calculating Pearson correlation coefficients among node degrees of different nodes by using the same distribution coefficients, inputting the adjacency matrix A of the known network in the step 2) into Matlab, and directly calculating the same distribution coefficients of the network through an Assertility function built in the Brain Connectivity Toolbox of the Matlab;
4) analyzing the network topology: networks are classified into the following 7 types, as shown in table 1, according to the polygon coefficients and the coequal coefficients: (1) the polygon coefficient is 0 and has a positive congruent coefficient, (2) the polygon coefficient is 0 and has a negative congruent coefficient, (3) the polygon coefficient is less than 1 and has a positive congruent coefficient, (4) the polygon coefficient is less than 1 and has a negative congruent coefficient, (5) the polygon coefficient is 1 and has a positive congruent coefficient, (6) the polygon coefficient is 1 and has a negative congruent coefficient, and (7) the polygon coefficient is greater than 1, wherein (1) the network corresponds to a linear topology, as shown in fig. 3, (2) the network corresponds to a star-shaped architecture, as shown in fig. 4, (3) the network corresponds to a topology having a linear trunk, and a part of branches has a ring-shaped structure, as shown in fig. 5, (4) the network corresponds to a topology having a star-shaped trunk, and a part of branches has a ring-shaped structure, as shown in fig. 6, (6) the network rarely has a ring-shaped trunk, as shown in fig. 5, (7) The class network has a mesh structure, as shown in FIG. 7;
table 1: network classification
；
5) And (3) simplifying the network topology: according to the network type, the network is simplified by adopting a comprehensive strategy of longest path segmentation and small-area non-nested polygon combination, the simplification requirement is to reduce nodes of the network on the premise of not changing the main topological characteristics of the network, so that a topology-based algorithm is operated on the simplified topology, the time consumption is reduced on the premise of not losing the network characteristics, and the specific simplification method of various network types is as follows: (1) the class and (3) class networks directly find the longest path, and the longest path is cut off to obtain a plurality of sub-networks to realize the network segmentation; (2) and (4) the network only needs to consider the central node, namely the nodes with high node degree and the links between the nodes form the backbone of the network; (5) the class network has a ring topology, and the topology ring has high order and small quantity and is not simplified; (6) class networks are rare, and topology structure and a simplified method are difficult to determine; (7) the class network has a large number of low-order rings and a large number, the simplified method is to perform multi-edge merging, as shown in fig. 9, the size of the threshold value affects the number of nodes after merging, and selecting an appropriate threshold value is a key for balancing accuracy and time consumption.
As shown in fig. 2, the calculation process of the network polygon coefficients in step 2) is as follows;
(1) inputting adjacency matrixes A to Matlab of the network;
(2) calculating a node degree matrix D, wherein the node degree of each node is the direct connection of the node and the number of nodes;
(3) calculating the total node number N;
(4) simplifying a node degree matrix D, and deleting nodes with the node degree smaller than 2 from the D, wherein the nodes cannot form a polygon;
(5) calculating a set of non-nested polygons for the network;
(6) calculating the number n of all nodes contained in the nesting-free polygon set;
(7) calculating the polygon coefficient which is N/N;
(8) and outputting the polygon coefficients.
As shown in fig. 8, the simplified process of the class (7) network in step 6) using polygon merging is as follows:
(1) inputting an adjacency matrix A and a polygon area threshold st of a network to Matlab;
(2) calculating a node degree matrix D;
(3) simplifying a node degree matrix D, and deleting nodes with the node degree smaller than 2 from the D, wherein the nodes cannot form a polygon;
(4) calculating a set of non-nested polygons for the network;
(5) calculating the area of each polygon in the non-nested polygon set;
(6) merging the polygon with the area smaller than the threshold st with the adjacent polygon;
(7) calculating an adjacency matrix A of the merged network;
(8) and outputting A.
Claims (3)
1. A large sparse complex network topology analysis and simplification method based on polygon coefficients is characterized by comprising the following steps:
1) definition of network polygon coefficients: given a network, abstracting the network topology into an undirected graph G (V, E), where V represents a set of nodes of the network, | V | represents the total number of nodes included in the network, E represents a set of links of the given network, links in the set of links refer to network paths connecting two adjacent nodes, and if I represents a set of non-nested polygons included in the given network and Ci represents a non-nested polygon in I, the polygon coefficient of the given network is (Σ)i∈ICi)/|V|；
2) And (3) calculating the network polygon coefficient: knowing an adjacent matrix A of a network, wherein the adjacent matrix A is an N multiplied by N square matrix, N is the total number of nodes of the network, the Aij value of an element of the square matrix is 1 or 0, 1 is taken to represent that the nodes i and j are directly connected through a link, 0 is taken to represent that no link is connected, N is calculated through the adjacent matrix A, N is the denominator of a polygon coefficient, a non-nested polygon set is obtained through calculation of a polygon class built in Matlab, then all the number of the nodes contained in the non-nested polygon set, namely the numerator of the polygon coefficient, and finally the polygon coefficient of the network is obtained;
3) calculating the network distribution coefficient: calculating Pearson correlation coefficients among node degrees of different nodes by using the same distribution coefficients, inputting the adjacency matrix A of the known network in the step 2) into Matlab, and directly calculating the same distribution coefficients of the network through an Assertility function built in the Brain Connectivity Toolbox of the Matlab;
4) analyzing the network topology: networks are classified into the following 7 types according to polygon coefficients and parity coefficients: (1) the polygon coefficient is 0 and has a positive homogement coefficient, (2) the polygon coefficient is 0 and has a negative homogement coefficient, (3) the polygon coefficient is less than 1 and has a positive homogement coefficient, (4) the polygon coefficient is less than 1 and has a negative homogement coefficient, (5) the polygon coefficient is 1 and has a positive homogement coefficient, (6) the polygon coefficient is 1 and has a negative homogement coefficient, and (7) the polygon coefficient is greater than 1, wherein (1) the type network corresponds to a linear topology, (2) the type network corresponds to a star-shaped structure, (3) the topology corresponding to the type network has a linear backbone, part of branches have a ring-shaped structure, (4) the topology corresponding to the type network has a star-shaped backbone, part of branches have a ring-shaped structure, the type network is less, (5) the type network has a ring-shaped backbone, (6) the type network has a mesh-shaped structure, and (7) the type network has a mesh-shaped structure;
5) and (3) simplifying the network topology: according to the network type, the network is simplified by adopting a comprehensive strategy of longest path segmentation and small-area non-nested polygon combination, the simplification requirement is to reduce the nodes of the network on the premise of not changing the main topological characteristics of the network, and the specific simplification method of various network types comprises the following steps: (1) the class and (3) class networks directly find the longest path, and the longest path is cut off to obtain a plurality of sub-networks to realize the network segmentation; (2) and (4) the network only needs to consider the central node, namely the nodes with high node degree and the links between the nodes form the backbone of the network; (5) the class network has a ring topology, and the topology ring has high order and small quantity and is not simplified; (6) class networks are rare, and topology structure and a simplified method are difficult to determine; (7) class networks have a large number of lower order rings and a large number, and the simplified approach is to perform a multi-edge merge.
2. The method for analyzing and simplifying the topology of the large sparse complex network based on the polygon coefficients as claimed in claim 1, wherein the calculation process of the network polygon coefficients in step 2) is as follows;
(1) inputting an adjacency matrix A of a network;
(2) calculating a node degree matrix D, wherein the node degree of each node is the direct connection of the node and the number of nodes;
(3) calculating the total node number N;
(4) simplifying a node degree matrix D, and deleting nodes with the node degree smaller than 2 from the D;
(5) calculating a set of non-nested polygons for the network;
(6) calculating the number n of all nodes contained in the nesting-free polygon set;
(7) calculating the polygon coefficient which is N/N;
(8) and outputting the polygon coefficients.
3. The method for analyzing and simplifying the topology of the large sparse complex network based on the polygon coefficients as claimed in claim 1, wherein the simplification process of the (7) type network by polygon merging in step 6) is as follows:
(1) inputting an adjacency matrix A and a polygon area threshold st of a network;
(2) calculating a node degree matrix D;
(3) simplifying a node degree matrix D, and deleting nodes with the node degree smaller than 2 from the D;
(4) calculating a set of non-nested polygons for the network;
(5) calculating the area of each polygon in the non-nested polygon set;
(6) merging the polygon with the area smaller than the threshold st with the adjacent polygon;
(7) calculating an adjacency matrix A of the merged network;
(8) and outputting A.
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