Analyses of the Clustering Coefficient and the Pearson Degree Correlation Coefficient of Chung's Duplication Model

Recent advances in gene expression profiling and proteomics techniques have spawn considerable interest in duplication models for modelling the evolution and growth of biological networks. In this paper, we consider the duplication model studied by Chung et al. It seems (to the best of our knowledge) that both the clustering coefficient and the Pearson degree correlation coefficient of this model have not been studied analytically. For such a model, we study the degree of a randomly selected vertex and derive first-order differential equations for its mean, second moment, and third moment. We also study the degrees of the two vertices that appear at both ends of a randomly selected edge and derive an approximation for the expected product of the degrees of these two vertices. Using these results, we obtain closed-form approximations for the clustering coefficient and the Pearson degree correlation coefficient of the duplication model. For the clustering coefficient, numerical results calculated from our approximation and the corresponding simulation results agree extremely well for the whole evolution process. For the Pearson degree correlation coefficient, there is some discrepancy at early times between the simulation results and the numerical results. However, as time goes on, the discrepancy diminishes. We present an asymptotic approximation by keeping only the dominant terms in the clustering coefficient and the degree correlation coefficient. Numerical study indicates that relative approximation error can decrease slowly with time, when the selection probability of the model is near some special values.