CN101114967A - Method for constructing complex network quotient space model - Google Patents
Method for constructing complex network quotient space model Download PDFInfo
- Publication number
- CN101114967A CN101114967A CNA200710131393XA CN200710131393A CN101114967A CN 101114967 A CN101114967 A CN 101114967A CN A200710131393X A CNA200710131393X A CN A200710131393XA CN 200710131393 A CN200710131393 A CN 200710131393A CN 101114967 A CN101114967 A CN 101114967A
- Authority
- CN
- China
- Prior art keywords
- network
- quotient space
- quotient
- weight
- edge
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Pending
Links
Images
Landscapes
- Management, Administration, Business Operations System, And Electronic Commerce (AREA)
Abstract
The invention discloses a construction method of a complex network quotient space model, and is characterized in that the equivalence relation of a quotient space theory is used for gradually classifying the particle size coarsening of a network, and a hierarchical quotient space chain is formed; furthermore, according to the position of each node in different quotient spaces on the hierarchical quotient space chain, different nodes are provided with hierarchical numbers to form the complex network quotient space model. According to the hierarchical numbers of any two nodes in the complex network quotient space model constructed by the invention, the path distribution of 'the optimal path' between the two nodes can be found visually; further 'the optimal path' between any two nodes can be found according to the hierarchical quotient space chain. According to the model, the communication path between two nodes can be searched for from the quotient space with the coarsest particle size to the quotient space with the finest particle size after gradual refinement, thus quickly finding 'the optimal path' between any two nodes. Computers can also be used for constructing the complex network quotient space model and quickly searching for 'the optimal path' in the complex network.
Description
The technical field is as follows:
the invention belongs to the technical field of Complex Networks, and particularly relates to a quotient space model of a non-directional weighted network and a method for constructing the quotient space model of the Complex network by adopting quotient space granularity calculation classification and hierarchical hierarchy.
Background art:
in recent years, qualitative analysis of complex networks has been a focus of research, and from the viewpoint of granularity calculation, the analysis only gives some macroscopic properties of the complex networks at the coarsest granularity, and for problem solving at a given target, only the direction has no method, and no substantial progress is made.
According to the introduction of the complex network (Shanghai science and technology education Press, guo Lei, xu Xue Ming Dynasty, 11.1 st edition 2006, pages 271-275), when solving the shortest path problem of the complex network, the commonly used methods are a Dqjkstra algorithm and a Floyd algorithm, the Dijkstra algorithm is suitable for solving the shortest path problem of a single source point, and the Floyd algorithm has a good effect of solving the shortest path between all points, but both the methods belong to heuristic search algorithms. Although the parallel algorithm idea of the shortest path of the existing large-scale network can process a large-scale complex network, the calculation amount is not reduced, and the workload and the resource consumption of message transmission and data transmission are increased from the overall view of the workload and the resource consumption.
For the path search of the undirected network, the prior art is also Dijkstra algorithm and Floyd algorithm and the deformation thereof on the basis, and due to the bottleneck of network data storage and the problem of calculation amount, the method is only suitable for the case of small network scale and the search is time-consuming.
In a road traffic network, the process of searching for the optimal driving route of a vehicle is a specific application of the shortest path problem in a vehicle navigation system, the Dijkstra algorithm is the classic at present, the algorithm is also a theoretical basis of some other related algorithm designs, and the method aims to reduce the time complexity of the search algorithm theoretically by methods of a computer data structure, operation research and the like, but often neglects road characteristics in an actual urban road network and the tendency that most drivers want to walk on roads with good walking conditions as much as possible in the driving process, and the search algorithms are only embodied in the driving route search in the city, and the search problem of the optimal driving route between cities and in the city is not considered yet.
Similarly, in the electric power transportation of the electric power system and the routing search of the computer network, a heuristic search algorithm is still used so far to solve the problem by using the idea of the shortest path, and a high-voltage low-loss transmission route and a route composed of high-flow network paths in the computer network in the electric power transportation process cannot be searched yet.
According to the introduction of the theory and application of problem solution, namely the theory and application of computation of quotient space granularity (2 nd edition) (Qinghua university Press, zhang Ling, zhang cymbal, 2 nd edition 3 month 2007, pages 1-6, 12-14, 27-36, 38-39, 90-105), the conventional quotient space theory uses triplets (X, F, T) to represent the domain X, the attribute F, and the structure T of the research, uses the triplets ([ X ], [ F ], [ T ]) corresponding to the quotient set [ X ] to represent different granularities, namely quotient space, and the model can represent not only the attributes of the objects, but also the structural relationship between the objects, and can describe the relationships of transfer, transformation, synthesis, and decomposition among various granularities, and the like among the world, and can fully represent the ability of analyzing the problem from coarse to fine, from table and interior, from multiple sides and multiple levels. But the equivalence relation proposed in the quotient space theory is only a general form in the mathematical field, and does not relate to an expression form aiming at a specific complex network; the quotient space theory has not yet applied the granular hierarchical idea to a concrete representation method of building a model of a complex network.
Until now, no usable specific model representation method of a large-scale complex network exists, and no specific application of the quotient space theory in the complex network exists.
The invention content is as follows:
the invention aims to provide a construction method of a complex network quotient space model, which is used for solving the problem of rapid path search in an undirected weighted network and finding out the 'optimal path' of any two nodes in the network.
The invention relates to a construction method of a complex network quotient space model, which is characterized in that a hierarchical network model is established according to different node granularity classifications of side weights in a non-directional weighted network; the method is characterized in that: starting from an initial network serving as a finest granularity quotient space, extracting different weights from the network to form an edge weight set, selecting the maximum weight from the edge weight set to perform equivalence relation classification, merging the edges equal to the maximum weight according to the equivalence relation to form an equivalence class, and obtaining a quotient space with thicker granularity; then, selecting weights from the second largest weight value to the smallest weight value in the edge weight set to carry out equivalence relation classification, merging the edges which are larger than or equal to the selected weight value in the obtained quotient space with thicker granularity to form an equivalence class according to the equivalence relation, and enabling the weight value of the edge of the network node in the same equivalence class to be larger than or equal to the weight value selected by the equivalence relation and smaller than the last selected weight value, thereby forming the quotient space with thicker granularity; this operation is continued until the coarsest granularity quotient space is classified: for the connected network, only one element is in the quotient space with the coarsest granularity, and for the non-connected network, the number of the elements in the quotient space with the coarsest granularity is equal to the number of the connected branches; and arranging the quotient spaces from fine to coarse according to granularity to form a hierarchical quotient space chain, and giving the hierarchical number of each node according to the position of each node in different quotient spaces on the hierarchical quotient space chain.
The specific operation steps can be described as follows:
firstly, giving an expression form of specific constituent elements according to a quotient space theory aiming at a specific complex network:
for a given undirected weighted network G (X, E), a node z belongs to X to form a node set X, all edges E form an edge set E, and the weight t (E) on the edge E belongs to 0, d]D is the maximum weight in each side of the network, and the set of edge weights with k different weights is denoted as { d 1 ＞d 2 ＞...＞d k }; according to the equivalence relation R (d) i ) I =1,.. K, the connection weight is greater than or equal to d i The node of edge e of (a) belongs to an equivalence class, and the corresponding quotient spaceIs formed by
Note node set X = X 0 The elements in X are represented by X j 0 Indicating that the weight on the edge in the network is greater than or equal to d 1 Is the edge set E 0 I.e. the edgeTo which is attached
Denotes, quotient space (X) 0 ，E 0 )；
Quotient space X 1 Middle element x j 1 Represents an equivalence relation R (d) 1 ) In the equivalence class, the weight on the edge in the network is greater than or equal to d 2 And is less than d 1 Is the edge set E 1 I.e. by
Set of edges in node set X
Marked as e jt 1 (ii) a Quotient space (X) 1 ，E 1 )；
Quotient space X i Middle element x j i Represents the equivalence relation R (d) i ) In the equivalence class, the weight on the edge in the network is greater than or equal to d i+1 And is less than d i Is the edge set E i I.e. by
Set of edges in node set XIs marked with e jt i (ii) a The quotient space is obtained as: (X) i ，E i )，i＝0，1，...，k；
For each quotient space X i The ordering of elements in (1), is noted
Arranging each quotient space from fine to coarse according to granularity to form a hierarchical quotient space chain X 0 ＞X 1 ＞...＞X k ；
The elements on the node set X are represented by hierarchical numbers: let z ∈ X, expressed as k + 1-dimensional integer as follows: z = (z) 0 ，z 1 ，...，z k ) Let p denote i ：X→X i Is a natural projection
z is at X i In the t-th element, then z i ＝t；
Quotient space (X) i ，E i ) Each element x of m i Introducing a matrix P of its quotient space m i Let x m i Is composed of X i-1 The s elements form an s × s dimensional matrix
Wherein: phi denotes in quotient space (X) i-1 ，E i-1 ) At x in the topology map t i-1 And x j i-1 There is no edge in between, e ((x) t i-1 ，x j i-1 ) Contain one or more values, representing x t i-1 And x j i-1 One or more paths exist between; by
And
forming an antisymmetric matrix P m i (ii) a Thus quotient space X i Can be represented by { P j i J =1, · m } denotes;
according to the equivalence relation R (d) 1 ) Operating said undirected weighted network G (X, E) to obtain a plurality of equivalence classes, to obtain a quotient space (X) 1 ，E 1 )，
Its corresponding matrix is P 1 1 ，...，P n1 1 (ii) a According to the equivalence relation R (d) 2 ) Quotient space (X) 1 ，E 1 ) Operating to obtain several equivalence classes, quotient space (X) 2 ，E 2 )，
Its corresponding matrix is P 1 2 ，...，P n2 2 (ii) a According to the equivalence relation R (d) 3 ) Quotient space (X) 2 ，E 2 ) Operating to obtain several equivalence classes, quotient space (X) 3 ，E 3 )，
Its corresponding matrix is P 1 3 ，...，P n3 3 (ii) a 8230, and by analogy, 8230; according to the equivalence relation R (d) i ) Quotient space (X) i-1 ，E i-1 ) Operating to obtain several equivalence classes, quotient space (X) i ，E i )，It corresponds toIs P 1 i ，...，P ni i I is more than or equal to 1 and less than or equal to k, k is the number of elements in the edge weight set, and the operation is carried out until the quotient space (X) j ，E j ) In (C) X j 1 ≦ j ≦ k having only one element or to quotient space (X) k ，E k ) (ii) a For each quotient space X i The elements in the (1) are sorted, and each quotient space is arranged from fine to coarse according to granularity to form a hierarchical quotient space chain X 0 ＞X 1 ＞...＞X j Finally, the elements on the node set X are represented by a hierarchical structure, and the hierarchical numbers z = (z) of all the elements on the node set X are obtained 0 ，z 1 ，...，z k ) Z ∈ X, wherein when j < k, z = (z) 0 ，z 1 ，...，z j ) Z belongs to X, so far, the quotient space granularity classification of the network G (X, E) is completed; at this time, if j < k, the quotient space (X) obtained by the network classification j ，E j ) Only one element in the list; if j = k, the last quotient space (X) obtained by the network classification k ，E k ) Only one element or more than one element in all quotient spaces;
in the above symbols, "∈" indicates "belonging", "+" indicates "arbitrary", "@" indicates "existing", "\58408;" indicates "equivalent".
The undirected weighted network in the invention is a complex network comprising a traffic network, an electric power network or a computer network.
For a road traffic network, a node z in the model can be corresponding to specific places such as cities, towns and the like and each intersection, a side e in the model represents that the cities, the towns and the roads are arranged among the intersections, a weight t (e) of the network side e is set according to the road sections, the roads and the vehicle speed conditions of the roads, and the driving speed per hour and the traffic light setting interval can be set according to the road section types and the road surface widths of the roads: and for the road sections with good road sections, wide road surfaces, large allowable speed per hour and large traffic light intervals, the weight of the corresponding network side is large, otherwise, the corresponding weight is small.
For the power network, nodes z in the model can be corresponding to a distribution transformer, a booster and a step-down transformer, an edge e in the model represents that a line exists among the distribution transformer, the booster and the step-down transformer, and a weight t (e) of the edge e is set according to a transmission voltage, a total length of the line, transmission power and the like of the transmission line: the corresponding weight value of high transmission voltage, short total line length and large transmission power is large, otherwise, the weight value is small.
For a computer network, a node z in a model can be corresponding to a user or a router, an edge e in the model indicates that a network line exists between the user and the router, and a weight t (e) of the edge e is set according to the path length, the time delay, the bandwidth, the load and the communication cost of the network line: the network circuit has small load, low communication cost, large bandwidth, less time delay, large weight of short path and small weight of the opposite.
The construction of the complex network quotient space model can be realized by a computer, and the operation process is as follows:
using two-dimensional arrays W [ N ]][N]The undirected authorized network G (X, E) is stored, files can be stored in a large-scale complex network, and the files can be read line by line when the undirected authorized network G (X, E) is accessed; using the array D [ k ]]Set of edge weights { d } 1 ＞d 2 ＞...＞d k }; using array P2]To represent a quotient space (X) i ，E i ) Each element x of m i Matrix P corresponding to quotient space m i Of (1) containsMeaning, the objects in the array are linked lists; store quotient space (X) with linked list X i ，E i ) The middle node set is in X-layer status and uses one-dimensional array point _ num [ N ]]To store z hierarchy number z = (z) of each node 0 ，z 1 ，...，z k ) The sequence, N is the number of G (X, E) nodes of the undirected weighted network, and L is the number of particle grading layers;
firstly, initializing an array P [ ] to be empty, initializing a linked list X to be a linked list formed by elements in a node set X, initializing an array point _ num [ ] to be a label of the element in the node set X, and assigning a value of L to be 0;
then from two-dimensional array W [ N ]][N]Reading the network, taking it greater than or equal to DL]Is less than D [ L-1 ]]When L =0, only D [ L ] is selected as the group]A logarithmic value of greater than or equal to D [ L ]]Is recorded in array P [ ]]Middle P [ L +1 ]]According to P [ L +1]Can obtain quotient space X 1 To update the linked list X and then update the array point _ num [ 2]]L is 1 on the original basis;
if the linked list X only has one point, the program is ended; otherwise, when L is not equal to k, turning to the step 2, wherein L is equal to k, and ending the program;
and storing the finally obtained complex network quotient space model into the hierarchical quotient space chain and each node hierarchical number when the complex network quotient space model is stored in a computer.
The complex network quotient space model obtained by the construction method is a hierarchical network of undirected weighted network based on different node granularity classifications of the weights on the sides of the network; the method is characterized in that: according to the equivalence relation R (d) i ) And obtaining the hierarchical serial number of each node by using the hierarchical quotient space chain of granularity classification.
The complex network quotient space model obtained by the construction method is a hierarchical construction of the quotient space model by classifying all actual complex objects of the undirected weighted network capable of being abstracted into points and edges according to the equivalence relations of different weights on the edges of the network to form a quotient set and form a corresponding quotient space. The initial network is a quotient space with the finest granularity, the complex network is constructed into a hierarchical quotient space chain with the granularity arranged in sequence from thin to thick by utilizing equivalence relation classification, and the hierarchical number of each node is given according to the position of each node in different quotient spaces on the hierarchical quotient space chain. When classifying into the coarsest quotient space, there are two possible situations: one, only one element is in the coarsest quotient space formed by the network, and the classification can be terminated when the condition occurs; two, the network forms more than one element in the coarsest quotient space in granularity, which indicates that the network has a plurality of connected branches, and some nodes have no path.
The invention utilizes the equivalence relation R (d) in the quotient space theory i ) The granularity layering thought of the introduced quotient space model classifies weighted edges with the same level into the same equivalence class, so that the quotient space (X) i-1 ，E i-1 ) Different elements of the quotient set in (A) are in a coarser granularity quotient space (X) i ，E i ) The quotient set in the method belongs to the same element, embodies the structural characteristics of the distribution condition of the weight t (e) of the edge e of the complex network, and also provides a layering method for layering representation of the complex network. According to the equivalence relation R (d) i ) Progressively forming a hierarchical quotient space chain X 0 ＞X 1 ＞...＞X k Is favorable for forming worlds with different granularities and fully reflects the weight t (e) distribution condition of the edge e of the complex network. Obtaining each node z hierarchy number z = (z) 0 ，z 1 ，..，z k ) The process of (a) is to recognize that node z in a complex network is in position in the complex network: path conditions of other nodes to this node. In view of the overall view, the utility model,the construction of the complex network quotient space model reflects the characteristics of the complex network quantitatively, and the model provides convenience for searching the 'best path' among nodes in the complex network and the bottleneck problem of complex network storage on computer application of a specific large-scale complex network; quotient space (X) in the invention i ，E i ) Each element x of (2) m i Introducing a matrix P corresponding to the quotient space m i Representation, for the sake of hierarchical representation, whereby only a store of each quotient space (X) is required i ，E i ) Each element x of m i Matrix P corresponding to quotient space m i Instead of storing the hierarchical quotient space chain, the space expenditure can be greatly reduced.
According to the hierarchical numbering (z) of any two nodes z, z' in the complex network quotient space model constructed for the specific network 0 ，z 1 ，...，z k ) And (z' 0 ，z′ 1 ，...，z′ k ) The path distribution condition of the 'optimal path' between two nodes can be found visually by the different numbers of the corresponding numbers in the hierarchical quotient space chain, and the 'optimal path' between any two nodes in the network can be found out according to a topological graph formed between different elements under the same quotient space in the hierarchical quotient space chain.
Therefore, the 'best path' search based on the complex network quotient space model can find the hierarchical quotient space chain and the hierarchical number of each node of the complex network quotient space model constructed according to the invention for the starting point and the end point to be searched in the complex network, search the communication paths of two points from the quotient space with the coarsest granularity, search from the quotient space with the coarser granularity, refine the quotient space step by step until the quotient space with the finest granularity is used for finding the 'best path' of any two points in the complex network, and simultaneously search out a plurality of 'best paths' in the network. The method comprises the steps of searching paths on the basis of an established quotient space model of the complex network to search main roads and roads with good road conditions in a road traffic network to the greatest extent, designing a rapid, practical and driver-oriented humanized path search algorithm, and solving the best driving route in cities and among cities; selecting a high-voltage and low-loss transmission route for electric power transportation in an electric power system; high-traffic network paths are found for network users in a computer network, and the blind and directionless complexity of tentative path searching is solved.
The hierarchical quotient space chain of the complex network quotient space model and each node z are hierarchically numbered z = (z is the hierarchical quotient space chain of the complex network quotient space model and z is the hierarchical number of the complex network quotient space model 0 ，z 1 ，...，z k ) Storing in memory, finding out hierarchical number for the starting point and end point to be searched in complex network, and counting from the last number z of hierarchical number k Starting the comparison to search for the "best path" of two points up to z 0 That is, the "best path" of two points is searched from the coarsest quotient space, and then the quotient space is gradually refined from the coarser quotient space to the finest quotient space (X) 0 ，E 0 ) Therefore, the 'best path' of any two points can be quickly searched, and a plurality of 'best paths' in a complex network can be simultaneously searched.
Description of the drawings:
FIG. 1 is a complex network quotient space model classification topological diagram of a 10-node undirected weighted network diagram.
The specific implementation mode is as follows:
for an undirected weighted network, the network model can be applied to networks such as a road traffic network, a power grid and the Internet. Examples of specific uses of the complex cyber-spatial model of the present invention in different types of undirected networks are given below, respectively.
Example 1: complex network quotient space model for constructing road traffic network
According to the second technical standard of highway engineering in China, highways are divided into five grades, and urban roads are divided into four types, namely express roads, main roads, secondary roads and branches. Specific urban road segments generally include various types of highways, streets, roads, roadways, streets, and the like. In cities, countryside and towns, highway sections comprise highways, national roads, provincial roads, countryside roads and the like, and the roads comprise 4 lanes, 3 lanes, 2 lanes, single lanes and the like. In the road traffic network diagram to be established, the network G (X, E) and the node z may correspond to specific locations such as cities, towns and the like and each intersection, and the weight t (E) on the network edge E is determined according to the road section, the road and the vehicle speed condition of the road. When a road traffic network map is constructed for a certain town set region with communicated roads, the nodes z of the road traffic network map can correspond to specific places of cities and all intersections, and the weight t (e) on the network edge e can be set according to the road section type and the road surface width of the roads, the allowed driving speed per hour and the traffic light setting interval: and for the road sections with good road sections, wide road surfaces, large allowable speed per hour and large traffic light intervals, the weight value of the corresponding network side is large, otherwise, the corresponding weight value is small.
If the selected city locations and all the intersections in the road network between cities in a certain region are marked as 1,2,3,4,5,6,7,8,9 and 10 according to a certain sequence; the traffic road conditions between cities are represented by a weighted network edge e: the weight of the network edge between two cities connected without highway is 0, the weight of other network edges is determined by the conditions of the road section, the road and the vehicle speed of the highway, for example, the weight of the network edge of a highway with 4 lanes and a vehicle speed of more than 80km/h is represented as 10, the weight of 5 is represented as national road, 3 is lane and a vehicle speed of 80km/h, the weight of 3 is represented as provincial road, 2 is lane and a vehicle speed of 60km/h, the weight of 1 is represented as city, town road, single lane and a vehicle speed of less than 60km/h, thereby establishing a undirected road traffic complex network, which has 10 points in total and is marked as a node set X {1,2,3,4,5,6,7,8,9, 10}, and the weight t (e) on the edge e is set as: {10,5,3,1}; the weight values between the nodes 1 and 2,3 and 4, 6 and 9 are 10, the weight values between the nodes 2 and 4,5 and 6,7 and 10, 8 and 10 are 5, the weight values between the nodes 2 and 5,3 and 7,5 and 8,6 and 7 are 3, the weight values between the nodes 1 and 3,4 and 6,4 and 7,7 and 9,8 and 9, 9 and 10 are 1, and no connection exists between other nodes.
The construction process of the complex network quotient space model is as follows: the weight between nodes 1 and 2,3 and 4, 6 and 9 is 10, so (X) 0 ，E 0 )
x 5 0 ，x 7 0 ，x 8 0 ，x 10 0 There are 10 elements in total.
R (10) equivalence classes: forming an equivalence class by setting the weight value on the edge of the network to be more than or equal to 10, wherein the nodes 1 and 2 are the equivalence class, the nodes 3 and 4 are the equivalence class, the nodes 6 and 9 are the equivalence class, and a quotient set is obtained
There are 7 elements, corresponding to the following matrix: the weight between nodes 1 and 2 is 10, have
The weight between nodes 3 and 4 is 10, have
The weight between nodes 6 and 9 is 10, have
To obtainQuotient space (X) 1 ，E 1 ) From X 1 The middle 7 elements, since the weight between nodes 2 and 4,5 and 6,7 and 10, 8 and 10 is 5, the quotient space (X) 1 ，E 1 ) In x 1 1 And x 2 1 Is connected to x 3 1 And x 4 1 Having an edge, x 5 1 、x 6 1 Are each independently of x 7 1 Connected, others have no edges.
R (5) equivalence: the weight value on the edge in the network is more than or equal to 5 and less than 10 to form an equivalence class, node x 1 1 And x 2 1 Is an equivalence class, x 3 1 And x 4 1 Is an equivalence class, x 5 1 、x 6 1 And x 7 1 Is an equivalence class, quotient setThere are three elements in total, and the corresponding matrix is as follows: the weight between nodes 2 and 4 is 5, havingThe weight between nodes 5 and 6 is 5, have
The weight between nodes 7 and 10, 8 and 10 is 5, has
Quotient space (X) 2 ，E 2 ) From X 2 The 3 middle elements, since the weight between nodes 2 and 5,3 and 7,5 and 8,6 and 7 is 3, the quotient space (X) 2 ，E 2 ) In x 1 2 、x 2 2 And x 3 2 And forming a complete communication graph.
R (3) equivalence: forming an equivalence class with the weight value of more than or equal to 3 and less than 5 on the edge in the network, x 1 2 、x 2 2 And x 3 2 Is an equivalence class having
Quotient space (X) 3 ，E 3 ) Only one isThe matrix of each element is as follows: the weight between nodes 2 and 5,3 and 7,5 and 8,6 and 7 is 3, havingQuotient space (X) 3 ，E 3 )，X 3 Only one element of which constitutes the topology map. Form a hierarchical quotient space chain X 0 ＞X 1 ＞X 2 ＞X 3 Finally, the elements on X are represented by a hierarchical structure, and the node 1 is positioned on X 0 ，X 1 ，X 2 ，X 3 Is the first element in (1), so there is 1= (1, 1), node 2 is at X 0 Is a second element in X 1 ，X 2 ，X 3 All of them are the first elements, so that there are 2= (2, 1), \8230;, node 10 is at X 0 Is a tenth element in X 1 Is a seventh element in X 2 Is a third element in X 3 Is the first element in (1), so that 10= (10,7,3,1) is present.
Available node hierarchical numbering: 1= (1, 1), 2= (2, 1), 3= (3, 2, 1), 4= (4, 2, 1), 5= (5, 3,2, 1), 6= (6, 4,2, 1), 7= (7, 5,3, 1), 8= (8, 6,3, 1), 9= (9, 3,2, 1), 10= (10, 7,3, 1).
According to the hierarchical numbering of any two nodes in the complex network quotient space model construction method, the path distribution condition of the 'optimal path' of the two nodes can be found visually, for example, the hierarchical numbering between the nodes 1 and 2 is only different at the first position, the path weight is larger, the optimal path distribution condition is better found, and actually, a highway, 4 lanes and a path with the vehicle speed of 80km/h are arranged between the nodes 1 and 2, so that some actual conditions can be found; the difference of the layering numbers between the nodes 1 and 10 is large, the path weight distribution of the nodes not only has 10,5 but also has 3, the path is difficult to find, and actually, the 'best path' from the node 1 to the node 10 is 1- > 2- > 4- > 3- > 7- > 10. The 'best path' of any two nodes in the network can be found out according to the hierarchical quotient space chain.
FIG. 1 shows a schematic view of aA complex network quotient space model classification topological graph of an undirected weighted network graph of 10 nodes is provided. As shown in fig. 1: a undirected weighted network with a total of 10 points denoted as { x } 1 0 ，x 2 0 ，x 3 0 ，x 4 0 ，x 5 0 ，x 6 0 ，x 7 0 ，x 8 0 ，x 9 0 ，x 10 0 }, node x 1 0 And x 2 0 、x 3 0 And x 4 0 、x 6 0 And x 9 0 The weight between is 10, node x 2 0 And x 4 0 、x 5 0 And x 6 0 、x 7 0 And x 10 0 、x 8 0 And x 10 0 Weight value between 5, node x 2 0 And x 5 0 、x 3 0 And x 7 0 、x 5 0 And x 8 0 、x 6 0 And x 7 0 Is 3, node x 1 0 And x 3 0 、 x 4 0 And x 6 0 、x 4 0 And x 7 0 、x 7 0 And x 9 0 、x 8 0 And x 9 0 、x 9 0 And x 10 0 The weight value between the nodes is 1, other nodes are not connected, and the node x 1 0 And x 2 0 、x 3 0 And x 4 0 、x 6 0 And x 9 0 The small ellipses with a thin line in between, three in total, indicate that they are equivalence classes in R (10), node x 1 0 、x 2 0 、x 3 0 And x 4 0 ，x 5 0 、x 6 0 And x 9 0 Between which there are two large ellipses with thick black circlesAnd x 7 0 、 x 8 0 And x 10 0 There are large ellipses circled with thick black dashed lines between them, indicating that they are equivalence classes in R (5), and thus it is seen that it is a topological graph formed by hierarchical quotient space chains, from which the "best path" of any two nodes can be easily found: the positions of a point 1 and a point 10 are firstly found out in three large ellipses of a thick black coil, the left large ellipse of the thick black coil is found to have the point 1, the right large ellipse of the thick black dotted coil has the point 10, the side connecting the two large ellipses is the point 3 and the point 7, the side formed by connecting the point 3 and the point 7 is in the path of the point 1 to the point 10, then the path of the point 1 to the point 3 and the path of the point 7 to the point 10 are found, in the process of finding the point 1 to the point 3, a small ellipse containing two thin lines in the left large ellipse of the thick black coil classifies the point 1 and the point 3 into different equivalence classes, the small ellipse of the two thin lines is connected by the side between the point 2 and the point 4, then the side formed by connecting the point 2 and the point 4 is in the required path, no smaller equivalence relation between the point 1 and the point 2, the point 4 and the point 3, the point 7 and the point 10 is classified, the weight value of the side is 10, so the optimal path from the point 1 to the point 10 can be searched, and the's is 1 to the's > 2- > 4- > 3- > 10.
The method can quickly search the 'best path' of any two nodes in the network according to the complex network quotient space model, and the steps for searching the path from the point 1 to the point 10 in the complex network quotient space model for the network are as follows:
the starting point is represented by a hierarchy: x = (1,1,1,1), y = (10,7,3,1) because x 3 ＝y 3 The maximum path capacity d from 1 to 10 3 ＝3；
As a start end sequence: ((1, 1), (10, 7,3, 1)) due to x 3 ＝y 3 =1, so in matrix P 1 3 Calculating a path;
finding x, y at P 1 3 In the quotient space (X) 3 ，E 3 ) In finding out (X) 2 ，E 2 ) X in (1) 3 ＝1，y 3 =3 at X 3 IsThe position in a particular element being unique thereto1,3 position of an element of (1), so that P is taken 1 3 [1][3]: ((3, 2, 1), (7, 5, 3)), now inserted between x, y to give:
((1，1，1，1)，(3，2，1)，(7，5，3)，(10，7，3，1))[3，7]；
the 2 nd coordinates of ((1, 1), (3, 2, 1)) and ((7, 5, 3), (10, 7, 3)) are the same, and are therefore respectively at P 1 2 And P 3 2 Finding respective communication paths from P 1 2 [1][2]，P 3 2 [1][3]Obtaining: ((2, 1), (4, 2)) and ((7, 5), (10, 7)), inserting the above sequence to obtain
((1，1，1，1)，(2，1)，(4，2)，(3，2，1)，(7，5，3)，(7，5)，(10，7)，(10，7，3，1))[2，4]，[3，7]，[7，10]。
The 1 st coordinates of (1, 1), (2, 1)), ((4, 2), (3, 2, 1)), ((7, 5, 3), (7, 5)), ((10, 7), (10, 7,3, 1)) are the same and are therefore each present at P 1 1 [1][2]，P 2 1 [1][1]，P 7 1 Obtaining respective communication paths:
(1, 2), (4, 3), (7, 7) and (10, 10) by inserting the above sequence
((1，1，1，1)，(1，2)，(2，1)，(4，2)，(4，3)，(3，2，1)，(7，5，3)，(7，7)，(7，5)，(10，7)，(10，10)，(10，7，3，1))
\58339The [1,2], [2.4], [4,3], [3.7], [7,7] [7, 10], [10, 10]; and (3) finally obtaining: a path (1, 2,4,3,7, 10) from point 1 to point 10;
the construction of the quotient space model of the above-described 10-point network and the search for the "best path" may also be accomplished by computer operations, which include the steps of:
using two-dimensional arrays W [ N ]][N]To store undirected weighted network, using the array D [ k ]]Storing the weight value classification value, int D [ ]]= {10,5,3,1}; using array P2]To represent a matrix P of quotient space m i Means that the objects in the array are linked listsStore quotient space X with chaining table X i Node hierarchy status, using one-dimensional array point _ num [ N ]]To store the number sequence after each node is layered, where N is the number of nodes, and in this embodiment, N =10 is taken. The following is operated with the computer:
1. initializing an array P [ ] to be empty, a linked list X to be {1} - > {2} - > {3} - > {4} - > {5} - > {6} - > {7} - > {8} - > {9} - > {10}, an array point _ num [ ] being { {1, } {2, } {3, } {4, } {5, } {6, } {7, } {8, } {9, } {10, }, L =0;
2. from two-dimensional array W [ N ]][N]Reading the network, taking it greater than or equal to D [ 0]]A value of (= 10), a side having a logarithmic value larger than or equal to 10 is recorded in the array P [ (= 10) ]]In, P1]Among them { {1,2;1, 2- > {3,4;3,4} - > {6,9;6,9} according to P [1 ]]The quotient space X can be obtained i To update the linked list X {1,2} - > {3,4} - > {5} - > {6,9} - > {7} - > {8} - > {10}, and then update the one-dimensional array point _ num [ 2], ]]{ {1, } {2,1, } {3,2, } {4,2, } {5,3, } {6,4, } {7,5, } {8,6, } {9,4, } {10,7, }, L1 on an original basis;
3. from two-dimensional array W [ N ]][N]Reading the network, taking it greater than or equal to D1](= 5) less than D [ 0]]The edge with a logarithmic value of 5 or more and less than 10 is recorded in P, P2]Among them { {1,2;2, 4- > {3,4;5,6} - > {5,7;7,10 } > {6,7;8, 10} according to P [ 2]]Can obtain quotient space X 2 To update the linked list X {1,2,3,4} - > {5,6,9} - > {7,8, 10}, and then update the one-dimensional array point _ num [, [ 2] ] [, [ 4] ] - > (5, 6,9} - > ]]{ {1, } {2,1, } {3,2,1, } {4,2,1, } {5,3,2, } {6,4,2, } {7,5,3, } {8,6,3, } {9,4,2, } {10,7,3, }, L is the original oneAdding 1 on the basis;
4. from two-dimensional array W [ N ]][N]Reading the network, taking it out of the network, where D2 is greater than or equal to](= 3), less than D [1 ]]A side having a logarithmic value of 3 or more and less than 5 is recorded in the array P [ 2]]In the middle, P3]Among them { {1,2;2,5} - > {1,3;3,7} - > {2,3;5,8;6,7} according to P3]Can obtain quotient space X 3 To update the linked list X {1,2,3,4,5,6,9,7,8, 10}, and thenUpdating the one-dimensional array point _ num]{ {1, 1} {2,1} {3,2,1} {4,2, 1} {5,3,2,1} {6,4,2,1} {7,5,3,1} {8,6,3,1} {9,4,2,1} {10,7,3,1} }, since the linked list X has only one point, the procedure ends.
L =4, the array point _ num [ ]]Middle point _ num [1 ]]And point _ num [10 ]]Starting from the last number, the comparisons are all 1,L =3, the second to last comparison, 1,3, are different, and P [ L = 3]]Find {1,3; * ， * find {1,3;3,7} from point _ num [ 2], [ 2]]Find point _ num [ 3] in]And point _ num [ 7]]Since L =3, there is {3,2,1} {7,5,3}, and 3 and 7 are inserted between paths 1 and 10 to obtain 1,3, 7, and 10, L =2;
6. compare {1, 1} {3,2,1} with {7,5,3} {10,7,3,1}, at P [ L =2]Find {1,2; * ， * and {5,7; * ， * find {1,2;2,4 and {5,7;7, 10, so there are {2,1} {4,2} and {7,5} {10,7}, inserting 2,4 and 7, 10 in the path respectively, resulting in 1,2,4,3,7, 10, l =1;
7. comparison of {1, 1} {2,1}, {4,2} {3,2,1}, {7,5,3}, {7,5}, {10,7} {10,7,3,1}, in P [ L = 1}, in the following]Find {1,2; * ， * }、{4，3； * ， * }、 {7，7； * ， * }、{10，10； * ， * due to the fact that like {7,7; * ， * }{10，10； * ， * it is not necessary to find {1,2; * ， * }、{4，3； * ， * find {1,2;1,2}, {3,4;3,4}, {3,4;3,4 is not equal to {4,3; * ， * completely matched, so there are {4,3;4,3}. The possible 1 to 10 paths are 1,2,4,3,7, 10.
The 'best path' between any two nodes can be conveniently searched by applying a complex network quotient space model, and then the digital code of the 'best path' corresponds to the city and the name of each intersection, so that the 'best path' between any two cities can be obtained. If the requirements of the driving on high speed, national road and lane are not strict, only the speed is required, the side with the weight of 10 can be changed into 5.
By constructing a complex network quotient space model and searching paths on the basis of the model, the humanized road which is good in road traveling condition and faces to a driver is searched out in a road traffic network as quickly as possible, so that the optimal driving route in and among cities is solved.
Example 2: constructing a complex network quotient space model of an electric power transmission network
In the process of power transmission in a power system, according to the specific practical conditions that the voltage of remote power transmission in China is 110 kilovolt, 220 kilovolt and 330 kilovolt, 500 kilovolt ultrahigh-voltage power transmission lines are built in recent years, and 1150 kilovolt experimental ultrahigh-voltage power transmission lines are built abroad, a network G (X, E) is built, a node z is each distribution transformer, a booster and a step-down transformer, a weight t (E) on an edge E is determined according to the transmission voltage, the total length of the line, the transmission power and the like of a transmission line, and the corresponding weight of the transmission voltage, the total length of the line and the transmission power is large, otherwise, the weight is small.
Now, an electric power system in a certain area is selected, the names of all the distribution transformers, boosters and step-down transformers in the area are numbered as 1,2,3,4,5,6,7,8,9, 10, 11 and 12 according to a certain sequence, the connection states of all the distribution transformers, the boosters and the step-down transformers are represented by weighted edges e, the weight of no connection is 0, other weights are determined according to the transmission voltage, the total length of lines, the transmission power and the like of a transmission line, the edge with the weight of 9 represents that the transmission voltage is 500 kilovolts, the path is short, the transmission power is high, the weight of 7 represents that the transmission voltage is 330 kilovolts, the path is short, the transmission power is high, the edge with the weight of 4 represents that the transmission voltage is 220 kilovolts, the path is short, the transmission power is not high, and the edge with the weight of 1 represents that the transmission voltage is 110 kilovolts, the path is long and the transmission power is low. Establishing a quotient space model of the complex network, and regarding a non-directional weighted network G (X, E), totally 12 points are marked as X {1,2,3,4,5,6,7,8,9, 10, 11, 12}, and the weight t (E) on the edge E is set as: {9,7,4,1}. The weight between nodes 1 and 2,4 and 5,6 and 7, 9 and 10 is 9, the weight between nodes 2 and 3, 8 and 9,8 and 11 is 7, the weight between nodes 1 and 4,3 and 5, 5 and 10,6 and 10,7 and 8, 10 and 11, 11 and 12 is 4, the weight between nodes 2 and 4,3 and 7,4 and 6,4 and 10,6 and 9, 9 and 11, 8 and 12 is 1, and there is no connection between other nodes.
The quotient space hierarchical representation process of the network comprises the following steps: (X) 0 ，E 0 ) 
x 3 0 ，x 8 0 ，x 11 0 ，x 12 0 There are 12 elements in total.
R (9) equivalence classes: de-bust set
There are 8 elements, and the corresponding matrix is as follows:
quotient space (X) 1 ，E 1 ) From X 1 8 elements in the structure, where x 1 1 And x 2 1 Are connected to x 5 1 Are each independently of x 6 1 、x 7 1 Connected, others have no edges.
R (7) equivalence classes: acquaintance set
There are 5 elements in total, and the corresponding matrix is as follows:
quotient space (X) 2 ，E 2 ) From X 2 The middle 5 elements constitute, x 1 2 And x 2 2 Are connected to x 2 2 And x 4 2 Are connected to x 3 2 And x 4 2 Is connected to x 4 2 And x 5 2 Connected, others have no edges.
R (4) equivalence classes:
only one element has the following matrix:
(ii) a Quotient space (X) 3 ，E 3 )，X 3 Only one element of which constitutes the topology.
Node hierarchical numbering: 1= (1, 1), 2= (2, 1), 3= (3, 2, 1), 4= (4, 3,2, 1), 5= (5, 3,2, 1), 6= (6, 4,3, 1), 7= (7, 4,3, 1), 8= (8, 5,4, 1), 9= (9, 6,4, 1), 10= (10, 6,4, 1), 11= (11, 7,4, 1), 12= (12, 8,5, 1). The model has been built and a "best path" from point 1 to point 11 is searched for according to the model as (1, 2,3,5, 10,9,8, 11).
The complex network quotient space model can be used for searching the optimal path between any two nodes, and then the digital codes of the optimal path are corresponding to the names of each distribution transformer, booster and step-down transformer, so that the optimal path between any two transmission points can be obtained.
By constructing the complex network business space model, the path search is facilitated on the basis of the complex network business space model, and a high-voltage and low-loss transmission route is selected for electric power transportation in an electric power system.
Example 3: constructing complex cyber-spatial models for routing in computer networks
In the computer network routing process, a network G (X, E) is established, a node z is a user or a router, and a weight t (E) on an edge E is determined by the path length, the time delay, the bandwidth, the load and the communication cost among nodes. Inside one area, node z is a user; outside the area, node z is a router. When searching a path, firstly judging whether a source user and a target user are in the same area, if so, searching by using a method that a node z is a user; otherwise, judging the home routers of the source user and the destination user, and then searching by using the method that the node z is the router. For the determination of the weight t (e) on the edge e, the load is small, the communication cost is low, the bandwidth is large, the time delay is small, the weight with short path is large, and otherwise, the weight is small.
A computer network system in a certain area is selected at present, each router is marked according to a certain sequence, the connection status of each matched router is represented by a weighted edge e, the weight without connection is 0, other weights are determined according to path length, time delay, bandwidth, load, communication cost and the like, the edge with the weight of 10 represents that the load is small, the communication cost is low, the bandwidth is large, the time delay is small and the path is short, the weight is 5 represents that the load is small, the communication cost is low, the bandwidth is large, the time delay is small and the path is short, the edge with the weight of 3 represents that the load is small, the communication cost is low, the bandwidth is not large, the time delay is prolonged and the path is long, and the edge with the weight of 1 represents that the load is large, the communication cost is large, the bandwidth is small, the time delay is long and the path is long. The method of the invention is applied to simulate 10 through computer simulation 4 Node and 10 4 Of computer network systems within nodesThe network and the undirected weighted network are respectively generated according to a random network model, a small-world network model and a scale-free network model, are stored in a text file, are respectively constructed by a quotient space granularity classification method, and are searched for a 50-point optimal path on the basis, in contrast, the Dijkstra algorithm and the Floyd algorithm cannot complete 10 points 4 Computer simulation experiment of path search of nodes, and for 10 2 And 10 3 Orders of magnitude search time for undirected weighted network paths is also long, and the searched paths are not "best paths". The results show that: it is convenient to search for the "best path" based on the complex network quotient space model.
The 'best path' between any two nodes can be conveniently searched by applying a complex network quotient space model, and then the digital code of the 'best path' corresponds to the name of each router, so that the 'best path' between any two routers can be obtained.
By means of the built complex network quotient space model, path searching is carried out on the basis of the model, a route formed by high-flow network paths in a computer network can be searched, and the throughput and the speed of the network are improved in a computer network system.
Claims (8)
1. A construction method of a complex network quotient space model is characterized in that a hierarchical network model is established according to different node granularity classifications of side weights in an undirected weighted network; the method is characterized in that: starting from an initial network serving as a finest granularity quotient space, extracting different weights in the network to form an edge weight set, selecting the maximum weight from the edge weight set to perform equivalence relation classification, merging the edges equal to the maximum weight according to the equivalence relation to form an equivalence class, and obtaining a quotient space with thicker granularity; then, selecting weights from the second largest weight value to the smallest weight value in the edge weight set to carry out equivalence relation classification, merging the edges which are larger than or equal to the selected weight value in the obtained quotient space with thicker granularity to form an equivalence class according to the equivalence relation, and enabling the weight value of the edge of the network node in the same equivalence class to be larger than or equal to the weight value selected by the equivalence relation and smaller than the last selected weight value, thereby forming the quotient space with thicker granularity; this operation is continued until the coarsest granularity quotient space is classified: for the connected network, only one element exists in the quotient space with the coarsest granularity, and for the non-connected network, the number of the elements in the quotient space with the coarsest granularity is equal to the number of the connected branches; and arranging the quotient spaces from fine to coarse according to granularity to form a hierarchical quotient space chain, and giving the hierarchical number of each node according to the position of each node in different quotient spaces on the hierarchical quotient space chain.
2. The method for constructing the spatial model of the complex cyber-quotient as recited in claim 1, characterized in that:
for a given undirected weighted network G (X, E), a node z belongs to X to form a node set X, all edges E form an edge set E, and the weight t (E) on the edge E belongs to 0, d]D is the maximum weight in each side of the network, and the set of edge weights with k different weights is denoted as { d 1 ＞d 2 ＞...＞d k }; according to the equivalence relation R (d) 1 ) I = 1.. K, the connection weight is greater than or equal to d 1 The node of edge e of (a) belongs to an equivalence class, and the corresponding quotient space is
Note node set X = X 0 The elements in X are represented by X j 0 Indicating that the weight on the edge in the network is greater than or equal to d 1 Is the edge set E 0 I.e. the edge
To which is attachedDenotes, quotient space (X) 0 ，E 0 )；
Quotient space X 1 Middle element x j 1 Represents the equivalence relation R (d) 1 ) One equivalence class in, the weight on the edge in the network is largeIs equal to d 2 And is less than d 1 Is the edge set E 1 I.e. by
Set of edges in node set XMarked with e jt 1 (ii) a Dequotient space (X) 1 ，E 1 )；
Quotient space X i Middle element x j i Represents an equivalence relation R (d) i ) In the equivalence class, the weight on the edge in the network is greater than or equal to d i+1 And is less than d i Is the edge set E i I.e. by
Set of edges in node set XIs marked with e jt i (ii) a Get the quotient space as: (X) i ，E i )，i＝0，1，...，k；
For each quotient space X i The ordering of elements in (1), is noted
Arranging each quotient space from fine to coarse according to granularity to form a hierarchical quotient space chain X 0 ＞X 1 ＞...＞X k ；
The elements on the node set X are represented by hierarchical numbers: let z ∈ X, expressed as k + 1-dimensional integer as follows: z = (z) 0 ，z 1 ，...，z k ) Let p denote i ：X→X i Is a natural projection, so that
z is at X i In the t-th element, then z i ＝t；
Quotient space (X) i ，E i ) Each element x of m i Introducing a matrix P of its corresponding quotient space m i Let x m i Is composed of X i-1 The middle s elements are formed, then the s is multiplied by s dimensional matrix
Wherein: phi denotes in quotient space (X) i-1 ，E i-1 ) At x in the topology map t i-1 And x j i-1 There is no edge in between, e ((x) t i-1 ，x j i-1 ) Contain one or more values representing x t i-1 And x j i-1 One or more paths exist between; byAnd withForming an antisymmetric matrix P m i (ii) a Then quotient space X i Can be constructed from { P j i J =1,... M } represents;
according to the equivalence relation R (d) 1 ) Operating said undirected weighted network G (X, E) to obtain a plurality of equivalence classes, to obtain a quotient space (X) 1 ，E 1 )，
Its corresponding matrix is P 1 1 ，...，P n1 1 (ii) a According to the equivalence relation R (d) 2 ) Quotient space (X) 1 ，E 1 ) Operating to obtain several equivalence classes, quotient space (X) 2 ，E 2 )，
Its corresponding matrix is P 1 2 ，...，P n2 2 (ii) a According to the equivalence relation R (d) 2 ) Quotient space (X) 2 ，E 2 ) Operating to obtain several equivalence classes, quotient space (X) 3 ，E 3 )，
Its corresponding matrix is P 1 3 ，...，P n3 3 (ii) a 8230, and by analogy, 8230; according to the equivalence relation R (d) i ) Quotient space (X) i-1 ，E i-1 ) Operating to obtain several equivalence classes, quotient space (X) i ，E i )，
Its corresponding matrix is P 1 i ，...，P ni i I is more than or equal to 1 and less than or equal to k, k is the number of elements in the edge weight set and is operated to the quotient space (X) j ，E j ) In (C) X j 1 ≦ j ≦ k having only one element or to quotient space (X) k ，E k ) (ii) a For each quotient space X i The elements in the (1) are sorted, and each quotient space is arranged from fine to coarse according to granularity to form a hierarchical quotient space chain X 0 ＞X 1 ＞...＞X j Finally, the elements on the node set X are represented by a hierarchical structure, and the hierarchical numbers z = (z) of all the elements on the node set X are obtained 0 ，z 1 ，...，z k ) Z ∈ X, wherein when j < k, z = (z) 0 ，z 1 ，...，z j ) Z belongs to X, so far, the quotient space granularity classification of the network G (X, E) is completed; in this case, if j < k,the quotient space (X) obtained by the network classification j ，E j ) Has only one element; if j = k, the last quotient space (X) obtained by network classification k ，E k ) Only one element or more than one element in all quotient spaces;
in the above symbols, "∈" indicates "belonging", "+" indicates "arbitrary", "@" indicates "existing", "\58408;" indicates "equivalent".
3. The method for constructing a quotient space model of a complex network as claimed in claim 1, characterized in that the undirected weighted network is a complex network comprising a traffic network, an electric power network or a computer network.
4. The method as claimed in claim 1, wherein for a road traffic network, the node z in the model is corresponding to specific locations such as cities, towns and the like and each intersection, the edge e in the model represents the cities, the towns and the roads among the intersections, the weight t (e) of the edge e of the network is set according to the road sections, roads and vehicle speed conditions of the roads, and the driving speed and the traffic light setting interval are allowed to be set according to the road section types and the road surface widths of the roads: and for the road sections with good road sections, wide road surfaces, large allowable speed per hour and large traffic light intervals, the weight of the corresponding network side is large, otherwise, the corresponding weight is small.
5. The method for constructing a complex network quotient space model as claimed in claim 1, characterized in that for the power network, the nodes z in the model are corresponding to distribution transformers, voltage boosters and voltage reducers, the edge e in the model represents that there are lines among the distribution transformers, the voltage boosters and the voltage reducers, and the weight t (e) of the edge e is set according to the transmission voltage, the total length of the lines, the transmission power and the like of the transmission lines: the corresponding weight value of high transmission voltage, short total line length and large transmission power is large, otherwise, the weight value is small.
6. The method according to claim 1, wherein for a computer network, a node z in the model is corresponding to a user or a router, an edge e in the model indicates that there is a network line between the user and the router, and a weight t (e) of the edge e is set according to a path length, a delay, a bandwidth, a load, and a communication cost of the network line: the network circuit has small load, low communication cost, large bandwidth, less time delay and large weight of short path, otherwise the weight is small.
7. The method for constructing the spatial model of the complex cyber-quotient as claimed in claim 1, characterized in that the computer is operated by: using two-dimensional arrays W [ N ]][N]The undirected authorized network G (X, E) is stored, files can be stored in a large-scale complex network, and the files can be read line by line when the undirected authorized network G (X, E) is accessed; using the array D [ k ]]Set of edge weights { d } 1 ＞d 2 ＞...＞d k }; using array P [ 2]]To represent a quotient space (X) i ，E i ) Each element x of (2) m i Matrix P corresponding to quotient space m i The objects in the array are linked lists; store quotient space (X) with linked list X i ，E i ) The middle node set is in X-layer status and uses one-dimensional array point _ num [ N ]]To store z hierarchy number z = (z) of each node 0 ，z 1 ，...，z k ) The sequence, N is the number of nodes of the undirected weighted network G (X, E), and L is the number of particle grading times;
firstly, initializing an array P [ ] to be empty, initializing a linked list X to be a linked list formed by elements in a node set X, initializing an array point _ num [ ] to be a label of the element in the node set X, and assigning a value of L to be 0;
then from two-dimensional array W [ N ]][N]Reading the network, taking it greater than or equal to DL]Is less than D [ L-1 ]]When L =0, only D [ L ] is selected as the group]A logarithmic value of D [ L ] or more]Is recorded in array P [ ]]Middle P [ L +1 ]]According to P [ L +1 ]]Can obtain quotient space X 1 To update the linked list X and then update the array point _ num [ 2]]L is 1 on the original basis;
if the linked list X only has one point, the program is ended; otherwise, when L is not equal to k, turning to the step 2, wherein L is equal to k, and ending the program;
and storing the finally obtained complex network quotient space model into the hierarchical quotient space chain and each node hierarchical numbering when the complex network quotient space model is stored by a computer. Only if it is greater than or equal to D [ L ]]A logarithmic value of (1) is greater than or equal toD[L]Is recorded in array P [ ]]Middle P [ L +1 ]]According to P [ L +1]Can obtain quotient space X 1 To update the linked list X and then update the array point _ num]L is added with 1 on the original basis;
if the linked list X only has one point, the program is ended; otherwise, when L is not equal to k, turning to the step 2, wherein L is equal to k, and ending the program;
and storing the finally obtained complex network quotient space model into the hierarchical quotient space chain and each node hierarchical number when the complex network quotient space model is stored in a computer.
8. A complex network quotient space model constructed by the method of claim 1 is a hierarchical network of undirected weighted network based on different node granularity classifications of the weights on the edges of the network; the method is characterized in that: according to the equivalence relation R (d) i ) And obtaining the hierarchical serial number of each node by using the hierarchical quotient space chain of granularity classification.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CNA200710131393XA CN101114967A (en) | 2007-08-31 | 2007-08-31 | Method for constructing complex network quotient space model |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CNA200710131393XA CN101114967A (en) | 2007-08-31 | 2007-08-31 | Method for constructing complex network quotient space model |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CN101114967A true CN101114967A (en) | 2008-01-30 |
Family
ID=39023100
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CNA200710131393XA Pending CN101114967A (en) | 2007-08-31 | 2007-08-31 | Method for constructing complex network quotient space model |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN101114967A (en) |
Cited By (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN101330457B (en) * | 2008-07-24 | 2011-03-23 | 安徽大学 | Method for searching shortest route based on quotient space overlay model |
| CN101383748B (en) * | 2008-10-24 | 2011-04-13 | 北京航空航天大学 | Community division method in complex network |
| CN103281745A (en) * | 2013-06-03 | 2013-09-04 | 南昌大学 | Wireless sensing network routing method of quotient topology energy hierarchical game |
| CN106208040A (en) * | 2016-07-18 | 2016-12-07 | 国网河南省电力公司电力科学研究院 | The optimization method in power supply network path |
| CN106250814A (en) * | 2016-07-15 | 2016-12-21 | 中国民航大学 | A kind of finger venous image recognition methods based on hypersphere granulation quotient space model |
| CN107808217A (en) * | 2017-10-24 | 2018-03-16 | 山东大学 | A kind of Public Transport Transfer optimization method based on Big Dipper positioning with the volume of the flow of passengers |
-
2007
- 2007-08-31 CN CNA200710131393XA patent/CN101114967A/en active Pending
Cited By (8)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN101330457B (en) * | 2008-07-24 | 2011-03-23 | 安徽大学 | Method for searching shortest route based on quotient space overlay model |
| CN101383748B (en) * | 2008-10-24 | 2011-04-13 | 北京航空航天大学 | Community division method in complex network |
| CN103281745A (en) * | 2013-06-03 | 2013-09-04 | 南昌大学 | Wireless sensing network routing method of quotient topology energy hierarchical game |
| CN103281745B (en) * | 2013-06-03 | 2016-01-20 | 南昌大学 | A kind of wireless sense network route method of quotient topology energy hierarchical game |
| CN106250814A (en) * | 2016-07-15 | 2016-12-21 | 中国民航大学 | A kind of finger venous image recognition methods based on hypersphere granulation quotient space model |
| CN106250814B (en) * | 2016-07-15 | 2019-03-19 | 中国民航大学 | A kind of finger venous image recognition methods based on hypersphere granulation quotient space model |
| CN106208040A (en) * | 2016-07-18 | 2016-12-07 | 国网河南省电力公司电力科学研究院 | The optimization method in power supply network path |
| CN107808217A (en) * | 2017-10-24 | 2018-03-16 | 山东大学 | A kind of Public Transport Transfer optimization method based on Big Dipper positioning with the volume of the flow of passengers |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| Han et al. | Dynamic and multi-faceted spatio-temporal deep learning for traffic speed forecasting | |
| Cavalcante et al. | Roadside unit deployment for information dissemination in a VANET: An evolutionary approach | |
| CN101114968A (en) | Complex network quotient space model based path search method | |
| Sun et al. | Discovering time-dependent shortest path on traffic graph for drivers towards green driving | |
| Nishizeki et al. | Planar graphs: Theory and algorithms | |
| Tom et al. | Transit route network design using frequency coded genetic algorithm | |
| Bielli et al. | Genetic algorithms in bus network optimization | |
| CN110222907B (en) | Electric vehicle charging station planning method and terminal equipment | |
| CN108921366B (en) | Road network important road section screening method based on PageRank algorithm | |
| CN101114967A (en) | Method for constructing complex network quotient space model | |
| CN104391907B (en) | A kind of fast path method for searching of variable resolution degree | |
| JP2001504965A (en) | Using multiple cost levels for route discovery computation | |
| Rodríguez-Puente et al. | Algorithm for shortest path search in Geographic Information Systems by using reduced graphs | |
| Baum et al. | Speed-consumption tradeoff for electric vehicle route planning | |
| CN101330417A (en) | Quotient space overlay model for calculating network shortest path and building method thereof | |
| Battiston et al. | Multiplex and multilevel networks | |
| CN109951392B (en) | Intelligent routing method for medium and large networks based on deep learning | |
| CN110619426A (en) | Road network model construction method, device, equipment and storage medium | |
| Diveev et al. | Intelligent traffic flows control software for megapolis | |
| Cipriani et al. | A multimodal transit network design procedure for urban areas | |
| Wu et al. | Dynamic urban traffic flow behavior on scale-free networks | |
| Delgadillo et al. | Reducing the size of traveling salesman problems using vaccination by fuzzy selector | |
| CN101571404A (en) | Shortest path auction algorithm considering intersection turning | |
| Xu et al. | Discovery of important crossroads in road network using massive taxi trajectories | |
| Zhang et al. | Solving multi-class traffic assignment problem with genetic algorithm |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| C06 | Publication | ||
| PB01 | Publication | ||
| C10 | Entry into substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| C02 | Deemed withdrawal of patent application after publication (patent law 2001) | ||
| WD01 | Invention patent application deemed withdrawn after publication |