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.

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

Although the clustering coefficients is widely used in the study of complex networks, scholars do not have clustering coefficients discontinuous [

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.

For a given complex network,

As a special case, if

For sake of description, we define a set

The dashed line represents the triangles with a shared edge

In this paper, we do not consider deleting the associated edge of suspension node. One edge

(1) After the edge

(2)

Similarly,

As mentioned above, when

Consider the following:

Consider the following:

Consider the following:

If edge

In this case, we just consider

The changing amount of

with

Similarly,

After adding an edge

From the above analysis, one can see that, by adding or deleting an edge in the complex networks, the clustering coefficients of the complex networks may change. If we continue in-depth to analysis the above conclusions under the premise without destroying network connectivity, the following conclusions can be clearly gained.

Deleting (adding) the edge

If the edge

Based on the above analysis, if

Similarly, based on formula (

If

When

When

If

If

It is obtained directly from formula (

If

In this paper, we present a systematic study on the effects of dynamic change of edges on clustering coefficients. It was found that the increase and decrease of the clustering coefficient can be effectively characterized by adding or deleting some edges of the network in the evolution of complex networks.

For the adaptive network, the susceptible may keep away from the infective for the reason that the susceptible individuals have the ability to recognize the infective group and avoid connecting with them [

The authors declare that there is no conflict of interests regarding the publication of this paper.