Complex Network Evolution Model

In recent years, the study of complex network has attracted more and more attention. Empirical analysis on the evolution of complex network structure and the corresponding study of modeling are the foundation to fully understand all the functions and application of complex network. Evolving model can correctly capture the processes on the network and access how various processes influence the network topology, it also has an extremely important role to help us understand the functions and dynamic behavior of the network. At the early days of evolution model of complex network, most models only concerns the main macroscopic properties, such as the small-world effect and scale-free property, and so on. Just according to the macroscopic properties, it is difficult to give the credible assessment of the different evolution mechanism. Deeply researching the detailed statistical properties of networks, especially the properties of the local structure, and accurately evaluating the evolution mechanism according to these properties are the inevitable trend of network evolution model.With the development of study on network evolution model, the research focus is shifting to detailed local structures in recent years, such as the statistical analysis on motif, ring and close graph structure, and so on. Clique-degree is the typical measurement to depict the local environment of nodes, empirical analysis showed that many real networks exhibit power-law clique-degree distribution, this new statistical property provides a new measurement for network evolution model. But there is no simple mechanism to reproduce this distribution up to now, especially the exponent of the distribution decreasing with the increasing order of the clique. Finding the evolution mechanism, which partly characters the real networks, and building the correspondent model are the driving force of the research on network evolution models. In this paper, we considers the widely accepted characteristics currently, including the average shortest path, clustering coefficient, degree distribution and clique-degree distribution, and study the topology structure, evolution mechanism and model. We propose two network evolution models. The first evolution model is an improvement of the HK model with tunable clustering coefficient. The HK model reproduces the small-world effect and scale-free property at the same time, but it only considers the linear growth of evolution network, and neglects the important ingredient during the process of network evolution, which impacts the structure and degree distribution of networks to a great extent. In our improved model, the total connections exponentially increase with the network size, this accelerated growth is due to the connections established between the adding node and the old nodes. Numerical simulations indicate that the network generated by our model not only has the all statistical properties of original model, but also reproduces the power-law clique-degree distribution. Therefore, our improved model is more close to the realistic networks.The second one is an accelerated evolution model based on the common neighbor driven. It both considers the common neighbor driven mechanism and the accelerated growth, which is mainly due to the additional connections generated among the old nodes. This model provides a completely new common neighbor driven mechanism. That is to say, two unconnected old nodes in the network with the greater number of common neighbors are connected with the greater probability. Numerical simulations show that our model can reproduce the observed power-law clique-degree distribution, and the changes of power exponents coincide with the observed results. The mechanism proposed in our model accords with the recognition of the real networks and provides the extensive explanation. Our model provides the example to research the formation mechanism of local structures, and it indicates that common neighbor driven mechanism and accelerating growth are the essential factors leading to the emergence of local structures.