CN103476051A - Method for evaluating importance of nodes in communication network - Google Patents

Method for evaluating importance of nodes in communication network Download PDF

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CN103476051A
CN103476051A CN2013104133879A CN201310413387A CN103476051A CN 103476051 A CN103476051 A CN 103476051A CN 2013104133879 A CN2013104133879 A CN 2013104133879A CN 201310413387 A CN201310413387 A CN 201310413387A CN 103476051 A CN103476051 A CN 103476051A
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戚银城
姚杰
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North China Electric Power University
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Abstract

本发明涉及一种通信网节点重要性评价方法,本发明属于网络中节点分析技术领域。一种通信网节点重要性评价方法,该方法包括以下步骤:步骤1:根据实际的通信网建立有权网络数学模型;步骤2:分别计算加权网络的节点度数k、节点介数b、特征向量指标Ce和紧密度指标Cc基本指标,并进行规格化;步骤3:对F1、F2、F3和F4,进行线性组合加权得到综合评价的最终得分F;步骤4:根据综合评价的最终得分F值的大小对n个节点进行排序,取排序靠前的节点,作为实际的通信网中的重要节点,从而确定实际的通信网中节点的重要性。本发明首先利用带宽对实际的通信网络进行加权,然后通过综合评价实现节点重要性的排序。

Figure 201310413387

The invention relates to a communication network node importance evaluation method, which belongs to the technical field of node analysis in the network. A communication network node importance evaluation method, the method comprising the following steps: step 1: establish a weighted network mathematical model according to the actual communication network; step 2: calculate the node degree k, node betweenness b, and eigenvector of the weighted network respectively The index C e and the compactness index C c are the basic indexes and normalized; Step 3: Perform linear combination weighting on F 1 , F 2 , F 3 and F 4 to obtain the final score F of the comprehensive evaluation; Step 4: According to the comprehensive The size of the final score F value of the evaluation sorts the n nodes, and takes the top-ranked nodes as important nodes in the actual communication network, so as to determine the importance of the nodes in the actual communication network. The invention first uses the bandwidth to weight the actual communication network, and then realizes the ordering of node importance through comprehensive evaluation.

Figure 201310413387

Description

一种通信网节点重要性评价方法A Method for Evaluating the Importance of Communication Network Nodes

技术领域technical field

本发明涉及一种通信网节点重要性评价方法,本发明属于网络中节点分析技术领域。The invention relates to a communication network node importance evaluation method, which belongs to the technical field of node analysis in the network.

背景技术Background technique

随着通信和信息技术的迅猛发展,通信网的覆盖面逐步扩大,承载的业务逐渐增多,在现代大型网络系统中的作用越来越重要。与此同时,通信网的节点数将不断增长,其复杂程度不断增加,所以对通信网的节点重要性研究还是很有必要的。近年来复杂网络节点重要性的评估方法主要有:With the rapid development of communication and information technology, the coverage of communication network is gradually expanding, and the business carried by it is gradually increasing, and its role in modern large-scale network systems is becoming more and more important. At the same time, the number of nodes in the communication network will continue to increase, and its complexity will continue to increase, so it is still necessary to study the importance of nodes in the communication network. In recent years, the evaluation methods for the importance of complex network nodes mainly include:

(1)基于该节点与其他节点之间的连接:最简单地是把节点的度作为节点重要性的衡量标准,认为与节点相连的边越多则该节点越重要,显然这种评估方法具有片面性,有些重要的核心节点并不一定具有较大的连接度,比如只有两条边相连的桥节点。(1) Based on the connection between the node and other nodes: the simplest way is to use the degree of the node as the measure of the importance of the node. It is considered that the more edges connected to the node, the more important the node is. Obviously, this evaluation method has One-sidedness, some important core nodes do not necessarily have a large degree of connectivity, such as bridge nodes connected by only two edges.

(2)基于节点(集)删除的方法:核心思想是“重要性等价于该节点(集)被删除后对网络的破坏性”。对网络中重要节点的发掘,是通过节点(集)删除前后网络连通性、性能的变化来反映的。很多文献都用到了节点的删除方法,即假设节点失效,通过比较删除节点前后网络性能的变化来评估节点重要度。节点删除法存在的问题是如果多个节点的删除都使得网络不连通,那么这些节点的重要度将是一致的,从而使得评估结果不精确。(2) The method based on node (set) deletion: the core idea is "the importance is equivalent to the destructiveness to the network after the node (set) is deleted". The excavation of important nodes in the network is reflected by the changes in network connectivity and performance before and after the node (set) is deleted. Many literatures have used the node deletion method, that is, assuming that the node fails, the importance of the node is evaluated by comparing the changes in network performance before and after the node is deleted. The problem with the node deletion method is that if the deletion of multiple nodes makes the network disconnected, the importance of these nodes will be consistent, which will make the evaluation result inaccurate.

(3)节点收缩方法:通过收缩与该节点相连的边,认为收缩后得到的网络凝聚程度越高则该节点越重要。网络凝聚程度指标则主要考虑节点的连接度以及经过节点的最短路径,来评估节点对网络的贡献大小。节点收缩法是网络中衡量和评估节点重要性的一种行之有效的方法。其优点主要是:不需要对节点进行移除,在应用上有着较广泛的基础。与此同时,节点收缩法还存在一定的缺点,主要有:没有办法对对称节点进行评价,对于一般节点,也不易控制其收缩范围。(3) Node shrinkage method: By shrinking the edges connected to the node, it is considered that the higher the degree of network cohesion obtained after shrinking, the more important the node is. The network cohesion index mainly considers the connection degree of nodes and the shortest path passing through nodes to evaluate the contribution of nodes to the network. Node contraction method is an effective method to measure and evaluate the importance of nodes in the network. Its main advantages are: no need to remove nodes, and it has a broad base in application. At the same time, there are still some shortcomings in the node shrinkage method, mainly including: there is no way to evaluate the symmetrical nodes, and it is not easy to control the shrinkage range of the general nodes.

复杂网络的基本网络拓扑参数包括节点的度,介数,特征向量,紧密度等。节点的度数是指连接该节点的边数,反映的是一个节点对于网络中其它节点的直接影响力。The basic network topology parameters of complex networks include node degree, betweenness, eigenvector, compactness, etc. The degree of a node refers to the number of edges connecting the node, which reflects the direct influence of a node on other nodes in the network.

介数刻画了信息流经给定节点的可能性,任一节点的介数均会随着经过该节点的信息流的增加而增大,利用介数可以确定信息负载繁重的网络节点。Brandes介数中心性算法是由Ulrik Brandes提出求解介数的算法,核心思想是任取一个节点为源节点,通过宽度优先搜索查找其它节点到该节点的最短路径,然后计算这些最短路径所对应的介数值。累加以图中任意节点为源节点的介数值,就得到图中所有节点和边的最终介数值。The betweenness describes the possibility of information flowing through a given node. The betweenness of any node will increase with the increase of the information flow passing through the node. The betweenness can be used to determine the network nodes with heavy information load. The Brandes betweenness centrality algorithm is an algorithm proposed by Ulrik Brandes to solve the betweenness. The core idea is to take a node as the source node, find the shortest path from other nodes to the node through breadth-first search, and then calculate the corresponding intermediate value. By accumulating the betweenness value of any node in the graph as the source node, the final betweenness value of all nodes and edges in the graph is obtained.

特征向量可以用来分析那种通过具有高度值的相邻节点所获得的间接影响力,不仅能直接反映网络的中心地位,还适合描述节点的长期影响力。综合考虑各个参数的意义,对各个参数进行线性组合,可以克服单一指标描述中心节点的缺陷,更能反应节点在网络中的中心位置。The eigenvector can be used to analyze the indirect influence obtained by the adjacent nodes with height values, which not only directly reflects the centrality of the network, but also is suitable for describing the long-term influence of nodes. Considering the significance of each parameter comprehensively and linearly combining each parameter can overcome the defect of a single index describing the central node, and can better reflect the central position of the node in the network.

紧密度为该节点到达所有其它节点的距离之和的倒数,用于刻画网络中的节点通过网络到达网络中其它节点的难易程度,反映的是节点通过网络对其他节点施加影响的能力,更能反映网络的全局结构。节点间的距离可以由Floyd算法求出,其主要思想是:从代表任意2个顶点vi到vj的距离的带权邻接矩阵开始,每次插入一个顶点vk,然后将vi到vj间的已知最短路径与插入顶点vk作为中间顶点(一条路径中除始点和终点外的其他顶点)时可能产生的vi到vj路径距离比较,取较小值以得到新的距离矩阵。The closeness is the reciprocal of the sum of the distances from the node to all other nodes. It is used to describe the difficulty of a node in the network to reach other nodes in the network through the network. It reflects the ability of a node to exert influence on other nodes through the network. It can reflect the global structure of the network. The distance between nodes can be obtained by the Floyd algorithm. The main idea is: start from the weighted adjacency matrix representing the distance between any two vertices v i and v j , insert a vertex v k each time, and then connect v i to v Compare the known shortest path between j and the path distance from v i to v j that may be generated when inserting vertex v k as an intermediate vertex (other vertices in a path except the start point and end point), and take the smaller value to obtain a new distance matrix.

发明内容Contents of the invention

本发明的目的在于克服现有技术的不足,在网络特征参数综合的基础上通过通信网中的带宽进行加权,充分利用通信网的固有性质,解决通信网中节点的重要性排序问题,提供了一种通信网节点重要性评价方法。The purpose of the present invention is to overcome the deficiencies of the prior art, to carry out weighting through the bandwidth in the communication network on the basis of the synthesis of network characteristic parameters, to make full use of the inherent properties of the communication network, to solve the problem of ordering the importance of nodes in the communication network, and to provide A communication network node importance evaluation method.

一种通信网节点重要性评价方法,该方法包括以下步骤:A communication network node importance evaluation method, the method comprises the following steps:

步骤1:根据实际的通信网建立有权网络数学模型;Step 1: Establish an authorized network mathematical model according to the actual communication network;

实际的通信网中,节点个数为n,边的条数为m,则该实际的通信网的有权网络数学模型用图GG及连接矩阵H=[hij]描述如下:In the actual communication network, the number of nodes is n, and the number of edges is m, then the weighted network mathematical model of the actual communication network is described by graph G G and connection matrix H=[h ij ] as follows:

GG=(N,L)   (1)G G =(N,L) (1)

式中:N为通信网中节点的集合,N={n1,n2,n3......nn};In the formula: N is the set of nodes in the communication network, N={n 1 ,n 2 ,n 3 ......n n };

L为一组有权边的集合,L={l1,l2,l3......lm};L is a set of weighted edges, L={l 1 ,l 2 ,l 3 ......l m };

连接矩阵H中元素hij定义如下:The element h ij in the connection matrix H is defined as follows:

Figure BDA0000380650470000031
Figure BDA0000380650470000031

边权邻接矩阵WG如下:The edge weight adjacency matrix W G is as follows:

Figure BDA0000380650470000032
Figure BDA0000380650470000032

其中,边权邻接矩阵WG的矩阵元素WGij为:Among them, the matrix element W Gij of the edge weight adjacency matrix W G is:

Figure BDA0000380650470000033
Figure BDA0000380650470000033

式中,Bij为节点i与节点j间线路的权值;In the formula, Bi j is the weight of the line between node i and node j;

则加权后的连接矩阵HQ可以表示为:Then the weighted connection matrix HQ can be expressed as:

HQ=H*WG   (5)H Q =H*W G (5)

式中,*表示矩阵中对应元素相乘;In the formula, * indicates that the corresponding elements in the matrix are multiplied;

步骤2:分别计算加权网络的节点度数ki、节点介数bi、特征向量指标Ce(i)和紧密度指标Cc(i)基本指标,并进行规格化,得到规格化度指标F1、规格化介数指标F2、规格化特征向量指标F3、规格化紧密度指标F4Step 2: Calculate the basic indexes of node degree k i , node betweenness b i , eigenvector index C e (i) and compactness index C c (i) of the weighted network respectively, and perform normalization to obtain the normalization degree index F 1. Normalized betweenness index F 2 , normalized eigenvector index F 3 , and normalized compactness index F 4 :

1)规格化度指标F1 1) Normalization index F 1

第i个节点的度数ki是连接该节点的边的个数,即The degree k i of the i-th node is the number of edges connecting the node, that is

kk ii == ΣΣ jj == 11 nno Hh QQ [[ ii ,, jj ]] -- -- -- (( 66 ))

对ki进行规格化,可得规格化度指标F1如下:By normalizing ki , the normalization degree index F 1 can be obtained as follows:

F1=ki/(n-1)   (7)F 1 =k i /(n-1) (7)

2)规格化介数指标F2 2) Normalized betweenness index F 2

第i个节点的介数bi刻画了网络中的节点对于信息流动的影响力;设网络具有n个节点,则节点i的介数bi定义为:The betweenness b i of the i-th node describes the influence of nodes in the network on information flow; if the network has n nodes, then the betweenness b i of node i is defined as:

bb ii == ΣΣ sthe s ≠≠ tt ≠≠ ii δδ stst (( ii )) -- -- -- (( 88 ))

δδ stst (( ii )) == gg stst (( ii )) // gg stst -- -- -- (( 99 ))

式中,δst(i)表示通过该节点(边)的最短路径条数占所有最短路径的比例,gst表示节点s和节点t之间的最短路径数;gst(i)表示节点s和节点t之间经过节点i的最短路径数目,介数bi可利用Brandes介数中心性算法得到;In the formula, δ st (i) represents the ratio of the number of shortest paths passing through the node (edge) to all shortest paths; g st represents the number of shortest paths between node s and node t; g st (i) represents the The number of shortest paths passing through node i between node t and node t, the betweenness b i can be obtained by using the Brandes betweenness centrality algorithm;

对bi进行规格化,得到规格化介数指标F2如下:Normalize bi to get the normalized betweenness index F 2 as follows:

F2=2bi/(n-1)(n-2)   (10)F 2 =2b i /(n-1)(n-2) (10)

3)规格化特征向量指标F3 3) Normalized eigenvector index F 3

设λ为矩阵HQ的主特征值,e=(e1,e2,…,en)为λ对应的特征向量,第i个节点的特征向量指标Ce(i)定义为:Let λ be the main eigenvalue of the matrix H Q , e=(e 1 ,e 2 ,…,e n ) be the eigenvector corresponding to λ, and the eigenvector index C e (i) of the i-th node is defined as:

CC ee (( ii )) == 11 λλ ΣΣ jj == 11 nno hh ijij ee jj ,, ii == 1,21,2 ,, .. .. .. ,, nno -- -- -- (( 1111 ))

其中λ与e满足:Where λ and e satisfy:

HQ·e=λ·e   (12)H Q e=λ e (12)

对Ce(i)进行规格化,可得规格化特征向量指标F3如下:Normalize C e (i), the normalized feature vector index F 3 can be obtained as follows:

F3=Ce(i)/max(Ce)   (13)F 3 =C e (i)/max(C e ) (13)

4)规格化紧密度指标F4 4) Normalized tightness index F 4

第i个节点的紧密度指标Cc(i)定义为该节点到达所有其它节点的距离之和的倒数,即:The closeness index C c (i) of the i-th node is defined as the reciprocal of the sum of distances from the node to all other nodes, namely:

CC cc (( ii )) == 11 // ΣΣ jj == 11 nno dd ijij -- -- -- (( 1414 ))

其中,dij为连接任意两个节点i和j的最短路径长度,可由Floyd算法求出;Among them, d ij is the shortest path length connecting any two nodes i and j, which can be obtained by Floyd algorithm;

对Cc(i)进行规格化,可得规格化紧密度指标F4如下:By normalizing C c (i), the normalized compactness index F 4 can be obtained as follows:

F4=Cc(i)i(n-1)   (15)F 4 =C c (i)i(n-1) (15)

步骤3:对加权网络的n个节点的规格化度指标F1、规格化介数指标F2、规格化特征向量指标F3和规格化紧密度指标F4,进行线性组合加权得到每个节点综合评价的最终得分F如下:Step 3: For the normalization degree index F 1 , the normalization betweenness index F 2 , the normalized eigenvector index F 3 , and the normalized closeness index F 4 of n nodes in the weighted network, perform linear combination weighting to obtain each node The final score F of the comprehensive evaluation is as follows:

Ff == ΣΣ kk == 11 44 αα kk Ff kk -- -- -- (( 1616 ))

其中,αk是权重系数, Σ k = 1 4 α k = 1 ; k = 1,2,3,4 ; Among them, α k is the weight coefficient, Σ k = 1 4 α k = 1 ; k = 1,2,3,4 ;

步骤4:根据n个节点综合评价的最终得分F值的大小对n个节点进行排序,取排序靠前的节点,作为实际的通信网中的重要节点,从而确定电力通信网中节点的重要性。Step 4: Sort the n nodes according to the final score F value of the comprehensive evaluation of n nodes, and take the top-ranked nodes as important nodes in the actual communication network, so as to determine the importance of nodes in the power communication network .

本发明的有益效果:本发明首先利用带宽对实际的通信网络进行加权,再对网络拓扑的度、介数、特征向量和紧密度等基本参数进行综合,实现节点重要性的排序。该发明对网络各个方面都进行考虑,特别适合于通信网络,对相关问题的方案设计有一定的借鉴意义,同时对于网络的维护也有重要意义。Beneficial effects of the present invention: the present invention first uses the bandwidth to weight the actual communication network, and then synthesizes basic parameters such as degree, betweenness, eigenvector and compactness of the network topology to realize the ranking of node importance. The invention considers all aspects of the network, is especially suitable for the communication network, has certain reference significance for the scheme design of related problems, and is also of great significance for the maintenance of the network.

附图说明Description of drawings

图1是本发明方法的流程框图;Fig. 1 is a block flow diagram of the inventive method;

图2是本发明的实例的网络拓扑结构示意图;Fig. 2 is the network topology schematic diagram of the example of the present invention;

图3是本发明的加权后的网络拓扑结构示意图。Fig. 3 is a schematic diagram of the weighted network topology of the present invention.

具体实施方式Detailed ways

下面结合附图对本发明的实施方式作进一步说明:Embodiments of the present invention will be further described below in conjunction with accompanying drawings:

如图1所示,本发明方法的流程框图,一种通信网节点重要性评价方法,其特征在于,该方法包括以下步骤:As shown in Figure 1, the flowchart of the method of the present invention, a communication network node importance evaluation method, is characterized in that, the method comprises the following steps:

步骤1:根据实际的通信网建立有权网络数学模型;Step 1: Establish an authorized network mathematical model according to the actual communication network;

实际的通信网的有权网络中,节点个数为n,边的条数为m,则该实际的通信网的有权网络的数学模型用图GG及连接矩阵H=[hij]描述如下:In the weighted network of the actual communication network, the number of nodes is n, and the number of edges is m, then the mathematical model of the weighted network of the actual communication network is described by graph G G and connection matrix H=[h ij ] as follows:

GG=(N,L)   (1)G G =(N,L) (1)

式中,N为通信网中节点的集合,N={n1,n2,n3......nn};In the formula, N is the set of nodes in the communication network, N={n 1 ,n 2 ,n 3 ......n n };

L为一组有权边的集合,L={l1,l2,l3......lm};L is a set of weighted edges, L={l 1 ,l 2 ,l 3 ......l m };

连接矩阵H中元素hij定义如下:The element h ij in the connection matrix H is defined as follows:

Figure BDA0000380650470000071
Figure BDA0000380650470000071

见图2,具有连接的网络拓扑图的连接矩阵表示出来如下:See Figure 2, the connection matrix of the connected network topology graph is shown as follows:

Hh == 00 11 00 00 00 00 00 00 00 00 00 00 00 00 11 00 11 00 00 00 00 00 00 00 00 00 00 00 00 11 00 11 11 00 00 00 00 00 00 00 00 00 00 00 11 00 11 00 11 11 11 00 00 00 00 00 00 00 11 11 00 11 11 00 00 11 00 00 00 00 00 00 00 00 11 00 11 00 00 11 11 00 00 00 00 00 00 11 11 11 00 11 00 00 11 00 00 00 00 00 00 11 00 00 11 00 11 00 11 00 00 00 00 00 00 11 00 00 00 11 00 11 11 11 00 00 00 00 00 00 11 11 00 00 11 00 11 11 00 00 00 00 00 00 00 11 11 11 11 11 00 00 00 00 00 00 00 00 00 00 00 00 11 11 00 00 11 00 00 00 00 00 00 00 00 00 00 00 00 11 00 11 00 00 00 00 00 00 00 00 00 00 00 00 11 00

边权邻接矩阵WG如下:The edge weight adjacency matrix W G is as follows:

Figure BDA0000380650470000073
Figure BDA0000380650470000073

其中,边权邻接矩阵WG的矩阵元素WGij为:Among them, the matrix element W Gij of the edge weight adjacency matrix W G is:

式中,Bij为节点i与节点j间线路的权值。In the formula, B ij is the weight of the line between node i and node j.

如图3所示的通信网中节点7与节点11之间边的带宽为2.5GHz,其它的边带宽为1GHz,对节点7与节点11之间的边赋予2.5的权值,其它边的权值为1,边权连接矩阵如下:In the communication network shown in Figure 3, the bandwidth of the edge between node 7 and node 11 is 2.5GHz, and the bandwidth of other edges is 1GHz. The edge between node 7 and node 11 is given a weight of 2.5, and the weight of other edges The value is 1, and the edge weight connection matrix is as follows:

WW GG == 00 11 00 00 00 00 00 00 00 00 00 00 00 00 11 00 11 00 00 00 00 00 00 00 00 00 00 00 00 11 00 11 11 00 00 00 00 00 00 00 00 00 00 00 11 00 11 00 11 11 11 00 00 00 00 00 00 00 11 11 00 11 11 00 00 11 00 00 00 00 00 00 00 00 11 00 11 00 00 11 11 00 00 00 00 00 00 11 11 11 00 11 00 00 2.52.5 00 00 00 00 00 00 11 00 00 11 00 11 00 11 00 00 00 00 00 00 11 00 00 00 11 00 11 11 11 00 00 00 00 00 00 11 11 00 00 11 00 11 11 00 00 00 00 00 00 00 11 2.52.5 11 11 11 00 00 00 00 00 00 00 00 00 00 00 00 11 11 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 11 00 11 00 00 00 00 00 00 00 00 00 00 00 00 11 00

则加权后的连接矩阵HQ可以表示为:Then the weighted connection matrix HQ can be expressed as:

HQ=H*WG   (5)H Q =H*W G (5)

式中,*表示矩阵中对应元素相乘,边权邻接矩阵HQ体现了加权后的连接矩阵的变化。In the formula, * indicates that the corresponding elements in the matrix are multiplied, and the edge weight adjacency matrix H Q reflects the change of the weighted connection matrix.

则加权后的连接矩阵为:Then the weighted connection matrix is:

Hh QQ == Hh ** WW GG == 00 11 00 00 00 00 00 00 00 00 00 00 00 00 11 00 11 00 00 00 00 00 00 00 00 00 00 00 00 11 00 11 11 00 00 00 00 00 00 00 00 00 00 00 11 00 11 00 11 11 11 00 00 00 00 00 00 00 11 11 00 11 11 00 00 11 00 00 00 00 00 00 00 00 11 00 11 00 00 11 11 00 00 00 00 00 00 11 11 11 00 11 00 00 2.52.5 00 00 00 00 00 00 11 00 00 11 00 11 00 11 00 00 00 00 00 00 11 00 00 00 11 00 11 11 11 00 00 00 00 00 00 11 11 00 00 11 00 11 11 00 00 00 00 00 00 00 11 2.52.5 11 11 11 00 00 00 00 00 00 00 00 00 00 00 00 11 11 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 11 00 11 00 00 00 00 00 00 00 00 00 00 00 00 11 00

步骤2:分别计算加权网络的节点度数k、节点介数b、特征向量指标Ce和紧密度指标Cc等基本指标,并进行规格化:Step 2: Calculate the basic indicators such as node degree k, node betweenness b, eigenvector index C e and compactness index C c of the weighted network respectively, and perform normalization:

1)规格化度指标F1 1) Normalization index F 1

第i个节点的度数ki是连接该节点的边的个数,The degree k i of the i-th node is the number of edges connecting the node,

kk ii == ΣΣ jj == 11 nno Hh QQ [[ ii ,, jj ]] -- -- -- (( 66 ))

对ki进行规格化,可得规格化度指标F1如下:By normalizing ki , the normalization degree index F 1 can be obtained as follows:

F1=ki/(n-1)   (7)F 1 =k i /(n-1) (7)

2)规格化介数指标F2 2) Normalized betweenness index F 2

第i个节点的介数bi刻画了网络中的节点对于信息流动的影响力。设网络具有n个节点,则节点i的介数bi定义为:The betweenness b i of the i-th node describes the influence of nodes in the network on information flow. Assuming that the network has n nodes, the betweenness bi of node i is defined as:

bb ii == ΣΣ sthe s ≠≠ tt ≠≠ ii δδ stst (( ii )) -- -- -- (( 88 ))

δδ stst (( ii )) == gg stst (( ii )) // gg stst -- -- -- (( 99 ))

式中,δst(i)表示通过该节点(边)的最短路径条数占所有最短路径的比例,gst表示节点s和节点t之间的最短路径数;gst(i)表示节点s和节点t之间经过节点i的最短路径数目。介数bi可利用Brandes介数中心性算法得到,具体步骤为:In the formula, δ st (i) represents the ratio of the number of shortest paths passing through the node (edge) to all shortest paths; g st represents the number of shortest paths between node s and node t; g st (i) represents the The number of shortest paths between node i and node t passing through node i. The betweenness b i can be obtained by using the Brandes betweenness centrality algorithm, and the specific steps are:

定义基于源节点s的任意节点i的介数值:Define the betweenness value for any node i based on the source node s:

δδ sthe s ·&Center Dot; (( ii )) == ΣΣ tt ∈∈ NN gg stst (( ii )) -- -- -- (( 1010 ))

其中N为图G的节点集合,则:Where N is the node set of graph G, then:

bb ii == ΣΣ sthe s ≠≠ tt ≠≠ ii δδ stst (( tt )) == ΣΣ sthe s ∈∈ NN ΣΣ tt ∈∈ NN δδ stst (( ii )) == ΣΣ sthe s ∈∈ NN δδ sthe s ·· (( ii )) -- -- -- (( 1111 ))

而δ(i)则可以通过以s为根节点,对图G的一次宽度优先遍历求得。And δ (i) can be obtained by a breadth-first traversal of the graph G with s as the root node.

对bi进行规格化,得到规格化介数指标F2如下:Normalize bi to get the normalized betweenness index F 2 as follows:

F2=2bi/(n-1)(n-2)   (12)F 2 =2b i /(n-1)(n-2) (12)

3)规格化特征向量指标F3 3) Normalized eigenvector index F 3

设λ为矩阵HQ的主特征值,e=(e1,e2,…,en)为λ对应的特征向量,第i个节点的特征向量指标Ce(i)定义为:Let λ be the main eigenvalue of the matrix H Q , e=(e 1 ,e 2 ,…,e n ) be the eigenvector corresponding to λ, and the eigenvector index C e (i) of the i-th node is defined as:

CC ee (( ii )) == 11 λλ ΣΣ jj == 11 nno hh ijij ee jj ,, ii == 1,21,2 ,, .. .. .. ,, nno -- -- -- (( 1313 ))

其中λ与e满足:Where λ and e satisfy:

HQ·e=λ·e   (14)H Q e=λ e (14)

λ与e的求解步骤如下:The solution steps of λ and e are as follows:

(a)先求出矩阵的特征值:|HQ-λE|=0;(a) Find the eigenvalues of the matrix first: |H Q -λE|=0;

(b)对每个特征值λ求出(HQ-λE)X=0的基础解系e1,e2,…,en(b) Find the basic solution system e 1 , e 2 ,...,e n of (H Q -λE)X=0 for each eigenvalue λ;

(c)HQ的属于特征值λ的特征向量就是e1,e2,…,en的非零线性组合。(c) The eigenvector of H Q belonging to the eigenvalue λ is the non-zero linear combination of e 1 , e 2 ,…, e n .

对Ce(i)进行规格化,可得规格化特征向量指标F3如下:Normalize C e (i), the normalized feature vector index F 3 can be obtained as follows:

F3=Ce(i)/max(Ce)   (15)F 3 =C e (i)/max(C e ) (15)

4)规格化紧密度指标F4 4) Normalized tightness index F 4

第i个节点的紧密度指标Cc(i)定义为该节点到达所有其它节点的距离之和的倒数,即:The closeness index C c (i) of the i-th node is defined as the reciprocal of the sum of distances from the node to all other nodes, namely:

CC cc (( ii )) == 11 // ΣΣ jj == 11 nno dd ijij -- -- -- (( 1616 ))

其中,dij为节点i和j的最短路径长度,可由Floyd算法求出。具体计算过程如下:Among them, d ij is the shortest path length between nodes i and j, which can be obtained by Floyd algorithm. The specific calculation process is as follows:

Floyd算法所用的邻接矩阵D为:The adjacency matrix D used by the Floyd algorithm is:

Figure BDA0000380650470000103
Figure BDA0000380650470000103

定义一个距阵P用来记录所插入点的信息,P[i,j]表示从Vi到Vj需要经过的点,初始化P[i,j]=j。Define a matrix P to record the information of the inserted point, P[i,j] represents the point that needs to pass from V i to V j , and initialize P[i,j]=j.

把每个顶点分别插入图中,再分别比较插入顶点k后的距离与原距离的大小:Insert each vertex into the graph separately, and then compare the distance between the inserted vertex k and the original distance:

D[i,j]=min(D[i,j],D[i,k]+D[k,j])   (18)D[i,j]=min(D[i,j],D[i,k]+D[k,j]) (18)

如果D[i,j]的值变小,则P[i,j]=k。则最终在D中包含有两点之间最短路径长度的信息,而在P中则包含了最短通路径的信息。If the value of D[i,j] becomes smaller, then P[i,j]=k. Finally, D contains the information of the shortest path length between two points, and P contains the information of the shortest path.

对Cc(i)进行规格化,可得规格化紧密度指标F4如下:By normalizing C c (i), the normalized compactness index F 4 can be obtained as follows:

F4=Cc(i)i(n-1)   (19)F 4 =C c (i)i(n-1) (19)

分别计算未加权和加权后各个节点的规格化度、介数、特征向量以及紧密度指标,如表1,表2所示。Calculate the normalization degree, betweenness, eigenvector and compactness index of each node after unweighted and weighted respectively, as shown in Table 1 and Table 2.

表1未加权网络各节点的规格化度、介数、特征向量、紧密度指标Table 1 The normalization degree, betweenness, eigenvector and compactness index of each node in the unweighted network

Figure BDA0000380650470000111
Figure BDA0000380650470000111

表2加权网络各节点的规格化度、介数、特征向量、紧密度指标Table 2 The normalization degree, betweenness, eigenvector and compactness index of each node in the weighted network

Figure BDA0000380650470000121
Figure BDA0000380650470000121

对加权网络的n个节点规格化度指标F1、规格化介数指标F2、规格化特征向量指标F3和规格化紧密度指标F4,进行线性组合加权得到每个节点综合评价的最终得分F如下:For the normalization degree index F 1 , normalization betweenness index F 2 , normalization eigenvector index F 3 , and normalization compactness index F 4 of n nodes in the weighted network, linear combination weighting is performed to obtain the final comprehensive evaluation of each node The score F is as follows:

Ff == ΣΣ kk == 11 44 αα kk Ff kk -- -- -- (( 2020 ))

其中,αk是权重系数, Σ k = 1 4 α k = 1 . k = 1,2,3,4 ; Among them, α k is the weight coefficient, Σ k = 1 4 α k = 1 . k = 1,2,3,4 ;

节点的规格化度、介数、特征向量以及紧密度的权重系数分别取0.2,0.25,0.35,0.2;The weight coefficients of normalization degree, betweenness, eigenvector and compactness of nodes are respectively 0.2, 0.25, 0.35, 0.2;

步骤4:根据n个节点综合评价的最终得分F值的大小对n个节点进行排序,取排序靠前的节点,作为实际的通信网中的重要节点,从而确定实际的通信网中节点的重要性。Step 4: Sort the n nodes according to the final score F value of the comprehensive evaluation of n nodes, and take the top-ranked nodes as the important nodes in the actual communication network, so as to determine the importance of the nodes in the actual communication network sex.

得到的节点重要性排序,如表3所示。The ranking of the obtained node importance is shown in Table 3.

表3节点重要性计算结果对比Table 3 Comparison of node importance calculation results

Figure BDA0000380650470000124
Figure BDA0000380650470000124

Figure BDA0000380650470000131
Figure BDA0000380650470000131

如图2所示的网络拓扑图中,对于未加权之前,利用综合法和收缩法首先计算出节点重要性,由表3可知,节点4、节点5、节点9、节点10是并列最重要的;对于加权后的网络,如图3,加权节点7-11,由综合法可以得到节点7和节点11的重要性明显提高,而对于收缩法仍是节点4、5、9、10重要,这样可以充分证明本方法对于加权网络中核心节点的选择具有一定的优势。尤其对于电力通信网来说,电力通信网中心节点比较重要,但其中心节点的度往往不高,根据不加权的方法很难找到中心节点,而由于中心节点的固有性质,加上带宽加权,通过综合法进行计算,可以很有效地找到中心节点。In the network topology diagram shown in Figure 2, before unweighting, the importance of nodes is first calculated by using the comprehensive method and the contraction method. From Table 3, it can be seen that node 4, node 5, node 9, and node 10 are the most important in parallel ; For the weighted network, as shown in Figure 3, the weighted nodes 7-11, the importance of nodes 7 and 11 can be obtained by the comprehensive method, and the importance of nodes 4, 5, 9, and 10 is still important for the contraction method, so It can be fully proved that this method has certain advantages for the selection of core nodes in the weighted network. Especially for the power communication network, the central node of the power communication network is more important, but the degree of the central node is often not high, it is difficult to find the central node according to the unweighted method, and due to the inherent nature of the central node, plus bandwidth weighting, The central node can be found efficiently by the comprehensive method for calculation.

Claims (1)

1.一种通信网节点重要性评价方法,其特征在于,该方法包括以下步骤:1. a communication network node importance evaluation method, is characterized in that, the method comprises the following steps: 步骤1:根据实际的通信网建立有权网络数学模型;Step 1: Establish an authorized network mathematical model according to the actual communication network; 实际的通信网中,节点个数为n,边的条数为m,则该实际的通信网的有权网络数学模型用图GG及连接矩阵H=[hij]描述如下:In the actual communication network, the number of nodes is n, and the number of edges is m, then the weighted network mathematical model of the actual communication network is described by graph G G and connection matrix H=[h ij ] as follows: GG=(N,L)   (1)G G =(N,L) (1) 式中:N为通信网中节点的集合,N={n1,n2,n3......nn};In the formula: N is the set of nodes in the communication network, N={n 1 ,n 2 ,n 3 ......n n }; L为一组有权边的集合,L={l1,l2,l3......lm};L is a set of weighted edges, L={l 1 ,l 2 ,l 3 ......l m }; 连接矩阵H中元素hij定义如下:The element h ij in the connection matrix H is defined as follows:
Figure FDA0000380650460000011
Figure FDA0000380650460000011
边权邻接矩阵WG如下:The edge weight adjacency matrix W G is as follows:
Figure FDA0000380650460000012
Figure FDA0000380650460000012
其中,边权邻接矩阵WG的矩阵元素WGij为:Among them, the matrix element W Gij of the edge weight adjacency matrix W G is:
Figure FDA0000380650460000013
Figure FDA0000380650460000013
式中,Bij为节点i与节点j间线路的权值;In the formula, B ij is the weight of the line between node i and node j; 则加权后的连接矩阵HQ可以表示为:Then the weighted connection matrix HQ can be expressed as: HQ=H*WG   (5)H Q =H*W G (5) 式中,*表示矩阵中对应元素相乘;In the formula, * indicates that the corresponding elements in the matrix are multiplied; 步骤2:分别计算加权网络的节点度数ki、节点介数bi、特征向量指标Ce(i)和紧密度指标Cc(i)基本指标,并进行规格化,得到规格化度指标F1、规格化介数指标F2、规格化特征向量指标F3、规格化紧密度指标F4Step 2: Calculate the basic indexes of node degree k i , node betweenness b i , eigenvector index C e (i) and compactness index C c (i) of the weighted network respectively, and perform normalization to obtain the normalization degree index F 1. Normalized betweenness index F 2 , normalized eigenvector index F 3 , and normalized compactness index F 4 : 1)规格化度指标F1 1) Normalization index F 1 第i个节点的度数ki是连接该节点的边的个数,即The degree k i of the i-th node is the number of edges connecting the node, that is kk ii == ΣΣ jj == 11 nno Hh QQ [[ ii ,, jj ]] -- -- -- (( 66 )) 对ki进行规格化,可得规格化度指标F1如下:By normalizing ki , the normalization degree index F 1 can be obtained as follows: F1=ki/(n-1)   (7)F 1 =k i /(n-1) (7) 2)规格化介数指标F2 2) Normalized betweenness index F 2 第i个节点的介数bi刻画了网络中的节点对于信息流动的影响力;设网络具有n个节点,则节点i的介数bi定义为:The betweenness b i of the i-th node describes the influence of nodes in the network on information flow; if the network has n nodes, then the betweenness b i of node i is defined as: bb ii == ΣΣ sthe s ≠≠ tt ≠≠ ii δδ stst (( ii )) -- -- -- (( 88 )) δδ stst (( ii )) == gg stst (( ii )) // gg stst -- -- -- (( 99 )) 式中,δst(i)表示通过该节点(边)的最短路径条数占所有最短路径的比例,gst表示节点s和节点t之间的最短路径数;gst(i)表示节点s和节点t之间经过节点i的最短路径数目,介数bi可利用Brandes介数中心性算法得到;In the formula, δ st (i) represents the ratio of the number of shortest paths passing through the node (edge) to all shortest paths; g st represents the number of shortest paths between node s and node t; g st (i) represents the The number of shortest paths passing through node i between node t and node t, the betweenness b i can be obtained by using the Brandes betweenness centrality algorithm; 对bi进行规格化,得到规格化介数指标F2如下:Normalize bi to get the normalized betweenness index F 2 as follows: F2=2bi/(n-1)(n-2)   (10)F 2 =2b i /(n-1)(n-2) (10) 3)规格化特征向量指标F3 3) Normalized eigenvector index F 3 设λ为矩阵HQ的主特征值,e=(e1,e2,…,en)为λ对应的特征向量,第i个节点的特征向量指标Ce(i)定义为:Let λ be the main eigenvalue of the matrix H Q , e=(e 1 ,e 2 ,…,e n ) be the eigenvector corresponding to λ, and the eigenvector index C e (i) of the i-th node is defined as: CC ee (( ii )) == 11 λλ ΣΣ jj == 11 nno hh ijij ee jj ,, ii == 1,21,2 ,, .. .. .. ,, nno -- -- -- (( 1111 )) 其中λ与e满足:Where λ and e satisfy: HQ·e=λ·e   (12)H Q e=λ e (12) 对Ce(i)进行规格化,可得规格化特征向量指标F3如下:Normalize C e (i), the normalized feature vector index F 3 can be obtained as follows: F3=Ce(i)/max(Ce)   (13)F 3 =C e (i)/max(C e ) (13) 4)规格化紧密度指标F4 4) Normalized tightness index F 4 第i个节点的紧密度指标Cc(i)定义为该节点到达所有其它节点的距离之和的倒数,即:The closeness index C c (i) of the i-th node is defined as the reciprocal of the sum of distances from the node to all other nodes, namely: CC cc (( ii )) == 11 // ΣΣ jj == 11 nno dd ijij -- -- -- (( 1414 )) 其中,dij为连接任意两个节点i和j的最短路径长度,可由Floyd算法求出;Among them, d ij is the shortest path length connecting any two nodes i and j, which can be obtained by Floyd algorithm; 对Cc(i)进行规格化,可得规格化紧密度指标F4如下:By normalizing C c (i), the normalized compactness index F 4 can be obtained as follows: F4=Cc(i)i(n-1)   (15)F 4 =C c (i)i(n-1) (15) 步骤3:对加权网络的n个节点的规格化度指标F1、规格化介数指标F2、规格化特征向量指标F3和规格化紧密度指标F4,进行线性组合加权得到每个节点综合评价的最终得分F如下:Step 3: For the normalization degree index F 1 , the normalization betweenness index F 2 , the normalized eigenvector index F 3 , and the normalized closeness index F 4 of n nodes in the weighted network, perform linear combination weighting to obtain each node The final score F of the comprehensive evaluation is as follows: Ff == ΣΣ kk == 11 44 αα kk Ff kk -- -- -- (( 1616 )) 其中,αk是权重系数, Σ k = 1 4 α k = 1 ; k = 1,2 , 3,4 ; Among them, α k is the weight coefficient, Σ k = 1 4 α k = 1 ; k = 1,2 , 3,4 ; 步骤4:根据n个节点综合评价的最终得分F值的大小对n个节点进行排序,取排序靠前的节点,作为实际的通信网中的重要节点,从而确定电力通信网中节点的重要性。Step 4: Sort the n nodes according to the final score F value of the comprehensive evaluation of n nodes, and take the top-ranked nodes as important nodes in the actual communication network, so as to determine the importance of nodes in the power communication network .
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