Discrete Dynamics in Nature and Society

Volume 2015, Article ID 172720, 5 pages

http://dx.doi.org/10.1155/2015/172720

## Influence of Dynamical Change of Edges on Clustering Coefficients

Department of Mathematics, Taiyuan Institute of Technology, Taiyuan, Shanxi 030008, China

Received 24 August 2014; Accepted 19 September 2014

Academic Editor: Li Li

Copyright © 2015 Yuhong Ruan and Anwei Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Clustering coefficient is a very important measurement in complex networks, and it describes the average ratio between the actual existent edges and probable existent edges in the neighbor of one vertex in a complex network. Besides, in a complex networks, the dynamic change of edges can trigger directly the evolution of network and further affect the clustering coefficients. As a result, in this paper, we investigate the effects of the dynamic change of edge on the clustering coefficients. It is illustrated that the increase and decrease of the clustering coefficient can be effectively controlled by adding or deleting several edges of the network in the evolution of complex networks.

#### 1. Introduction

Clustering coefficient is one of the most important quantities in complex networks which can depict the average number of the ratio of the actual existence sides of the neighbors of the point and the sides that may exist in the neighbors of the point in the complex networks. At present, there are two different but the most basic definitions of clustering coefficients. Firstly, Watts and Strogatz proposed the concept of the clustering coefficients in their creative small-world network model which is denoted by [1]. Secondly, shortly after Watts and Strogatz proposed the clustering coefficients, Newman et al. defined a concept similar to the former “transitivity,” which is denoted by [2].

Although the clustering coefficients is widely used in the study of complex networks, scholars do not have clustering coefficients discontinuous [3–19]. Shi et al. made a good overview on the research in this area and proposed a general method of calculating the clustering coefficients [20]. Soffer and Vazquez proposed a definition of clustering coefficients independent of degree [21].

In the complex networks, dynamic change of the edges directly led to the dynamic evolution of the network and thus affect the variation of the clustering coefficients. And one of the core features of complex networks is a huge number of nodes and edges. This feature directly affects the complexity of calculating the clustering coefficient. In this paper, how dynamical changes of the edges affect the clustering coefficient is deeply presented which can reveal the impact of changes of the edge in the quality and quantity on the clustering coefficients. In addition, the obtained results show that, in the evolution of complex networks, we can make the clustering coefficients increase or decrease by deleting or adding the certain edges of networks.

#### 2. Basic Concepts

For a given complex network, denotes the adjacency matrix of the network, denotes the number of nodes in the network, denotes the degree of node , denotes the set of neighbors of node , denotes the number of edges in the neighbors of node , and denotes the clustering coefficient of node ; namely,with global clustering coefficient:

As a special case, if , then .

For sake of description, we define a set by the adjacency matrix of the network. The set is used to describe a set of triangles with a shared edge . And we use the to denote the number of these triangles. In fact, when the edge exists, is equal to the number of nodes in . We show an example in Figure 1.