Analysis and perturbation of degree correlation in complex networks

Degree correlation is an important topological property common to many real-world networks such as such as the protein-protein interactions and the metabolic networks. In the letter, the statistical measures for characterizing the degree correlation in networks are investigated analytically. We give an exact proof of the consistency for the statistical measures, reveal the general linear relation in the degree correlation, which provide a simple and interesting perspective on the analysis of the degree correlation in complex networks. By using the general linear analysis, we investigate the perturbation of the degree correlation in complex networks caused by the simple structural variation such as the ―rich club‖. The results show that the assortativity of homogeneous networks such as the Erdös-Rényi graphs is easily to be affected strongly by the simple structural changes, while it has only slight variation for heterogeneous networks with broad degree distribution such as the scale-free networks. Clearly, the homogeneous networks are more sensitive for the perturbation than the heterogeneous networks. PACS: 89.75.Hc Networks and genealogical trees PACS: 89.75.Fb Structures and organization in complex systems Introduction. Complex networks provide a useful tool for investigating the topological structure and statistical properties of complex systems with networked structures [1]. These networked systems have been found to possess many common topological properties. For example, many real-world networks such as the protein-protein interactions, the metabolic networks and the Internet exhibit the existence of the nontrivial correlation between degrees of nodes connected by edges [2-5]. Empirical studies show that almost all social networks display the property that highor low-degree nodes tend to connect to other nodes with similar degrees, which is referred to as ―assortative mixing‖. In biological and technological networks, high-degree nodes often preferably connect to low-degree nodes, which is referred to as ―dissassortative mixing‖. The degree correlation has important influence on the topological properties of networks and may impact related problems on networks such as stability [6], the robustness of networks against attacks [7], the network controllability [8], the traffic dynamics on networks [9, 10], the network synchronization [11-13], the spreading of information or infections and other dynamic processes [7, 13-24]. In order to characterize and understand such preference of connections in complex networks, many statistical measures and network models have been introduced and investigated [2, 3, 25-36]. For example, the average nearest neighbors’ degree of nodes (ANND) [3] and the degree correlation coefficient (DCC) [2] are the classical measures for characterizing the degree correlation in networks. ANND could give the intuitional information of the connection preference in degree, and DCC is helpful for the quantitative comparison of the different networks. As an integral characterization of the degree correlation, DCC may miss some details of the degree correlation [37]. The generalized degree correlation coefficient proposed by Valdez et al [15] is a useful extension of DCC, which can help compare the difference of networks with the same DCC and exhibit the correlations at different levels. Generally, it may be more appropriate that the combinations of the measures are used for the analysis of the correlations in networks. In the letter, we will study analytically the classical statistical measures for characterizing the degree correlation in complex networks, and give some interesting results, including the exact proof of the consistency of the measures and the general linear relation in the degree correlation, which provides a simple and interesting perspective on the analysis of the degree correlation in networks. In terms of the general linear relation in the degree correlation, we analyze the perturbation of the degree correlation caused by the addition of few nodes and the ―rich club‖. Analysis of degree correlation in complex networks. The degree correlation between two nodes connected by edges can