Random clique evolving network model and their communicability

Abstract

We introduce a notion of random clique evolving network, this network start from a complete subgraph of a -clique, where a is the size of the clique. In every time step T , m nodes are chosen from this network randomly, and forming a new complete subgraph of a -clique. In this way, this network grows in time steps. The numerical investigation shows that the cumulative degree distribution of this network takes an exponential function, which is the property of the homogeneous networks, and the clustering coefficient of this network is larger than that of the ER network. However, the characteristic path length of this network is the similar to that of the ER network, so this network shows the behaviors of small-world networks. Subsequent study shows that this network exhibits hierarchical modular structure for the clustering spectrum vs. k takes power-law. These results are in good agreement with the empirical results on many real-world complex networks, such as urban bus translation network or urban subway network, our model can explain the evolutionary procedure of these spatial networks. Whats more, we present a numerical investigation on the communicability of our model by the Estrada index EE ( G ), the Estrada index EE ( G ) of this network increases with decreasing the rata m / a at the same size N and the same average degree k >. The communicability of the urban public translation networks is very important, our results have a certain guiding significance for the construction of urban bus translation network and urban subway network.