Analysis and perturbation of degree correlation in complex networks 1, 2, 3 1, 2 Ju Xiang , Ke Hu4() , Yan Zhang5(), Tao Hu6 and Jian-Ming Li 1Neuroscience Research Center, Changsha Medical University, Changsha 410219, China; 2Department of Anatomy, Histology and Embryology, Changsha Medical University, Changsha 410219, China 3Department of Basic Medical Sciences, Changsha Medical University, Changsha 410219, China 4Department of Physics, Xiangtan University, Xiangtan 411105, Hunan, China 5Department of Computer Science, Changsha Medical University, Changsha 410219, China 6College of Science, QiLu University of Technology, Jinan 250353, Shandong, China Corresponding author. Email: [email protected], [email protected], [email protected] Abstract – 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 (DCC) [2] are the classical measures for characterizing the investigating the topological structure and statistical properties degree correlation in networks. ANND could give the of complex systems with networked structures [1]. These intuitional information of the connection preference in degree, networked systems have been found to possess many common and DCC is helpful for the quantitative comparison of the topological properties. For example, many real-world networks different networks. As an integral characterization of the degree such as the protein-protein interactions, the metabolic networks correlation, DCC may miss some details of the degree and the Internet exhibit the existence of the nontrivial correlation [37]. The generalized degree correlation coefficient correlation between degrees of nodes connected by edges [2-5]. proposed by Valdez et al [15] is a useful extension of DCC, Empirical studies show that almost all social networks display which can help compare the difference of networks with the the property that high- or low-degree nodes tend to connect to same DCC and exhibit the correlations at different levels. other nodes with similar degrees, which is referred to as Generally, it may be more appropriate that the combinations of ―assortative mixing‖. In biological and technological networks, the measures are used for the analysis of the correlations in high-degree nodes often preferably connect to low-degree networks. nodes, which is referred to as ―dissassortative mixing‖. The In the letter, we will study analytically the classical degree correlation has important influence on the topological statistical measures for characterizing the degree correlation in properties of networks and may impact related problems on complex networks, and give some interesting results, including networks such as stability [6], the robustness of networks the exact proof of the consistency of the measures and the against attacks [7], the network controllability [8], the traffic general linear relation in the degree correlation, which provides dynamics on networks [9, 10], the network synchronization a simple and interesting perspective on the analysis of the [11-13], the spreading of information or infections and other degree correlation in networks. In terms of the general linear dynamic processes [7, 13-24]. relation in the degree correlation, we analyze the perturbation In order to characterize and understand such preference of the degree correlation caused by the addition of few nodes of connections in complex networks, many statistical measures and the ―rich club‖. and network models have been introduced and investigated [2, 3, 25-36]. For example, the average nearest neighbors’ degree Analysis of degree correlation in complex networks. - The of nodes (ANND) [3] and the degree correlation coefficient degree correlation between two nodes connected by edges can 1 be naturally characterized by the degree-degree joint measures can be given by the following transformation (from probability e jk , the probability that one of the two ends of a the equation (2)), randomly selected edge in a network will has node of 1 r[] kq je q kq jq degree j and another will has node of degree k . This quantity 2 k jk k k j q k j k j is a symmetric matrix in undirected network ( eejk kj ), and have e 1 and eq , where 1 jk jk j jk k [kq k ( k ) kq q k ( j )] q kp jp is the probability that the end of a randomly 2 k nn k j nn k k j j q k k j chosen edge in a network has a node of degree k, while p is k 1 the degree distribution of the network—the probability that a q kq[ k ( k ) k ( j )] 2 k j nn nn randomly chosen node in the network has degree k (the degree q jk of a node is the number of other nodes to which it connects in 1 the network). If e takes the value of qq in a network, {qkqkk [() k ()] j qjqk [() j kk ()]} jk jk 2 k j nn nn j k nn nn the network is usually considered to have no correlation of q j,() k j k degrees. While most of real-world networks always exhibit an qk q j( k j )[ k nn ( k ) k nn ( j )] obvious deviation from the value, meaning the existence of j,() k j k , degree correlation in the networks. However, the degree-degree 2 qkj q() k j joint probability is easily affected by statistical fluctuations in j,( k jk ) finite networks, and it is rather difficult to identify the tendency (3) of degree correlation in the network by the quantity. Therefore, other statistical measures may be more convenient and where, efficient. 2 2 2 qk q k ()() kq k kq k q j k j Statistical measures for degree correlation. One of the widely jk k jk used measures for degree correlation is ANND [3], [kqk q j ( k j ) jq j q k ( j k )] , (4) j,() k j k k( k ) jp ( j | k ) , (1) nn j c 2 qkj q() k j j,() k j k where p( j | k ) e e e q is the conditional c jk j jk jk k probability that an edge belonging to a node with degree k will and the constraint condition due to the topological constraint of connect to a node with degree j. This measure considers the the network is used, average degree of the neighbors of a node as a function of its k 2 q k() k q k . (5) degree k, and it can provide a clear indication for the presence k nn k k or absence of degree correlation in networks. When it is kk (Note that it is related to the sum of the degrees of ends of all independent of k, meaning that networks have no obvious edges in network). According to the equation (3), r=0, when correlation of degree. In homogeneous and uncorrelated knn () k C , meaning the absence of the degree correlation. networks, knn () k k , while it will increase with the 2 kj , qkj q( k j ) 0 . In assortative networks, increase of the heterogeneity of networks: knn ()/ k k k 22 kk() is a monotonically increasing function with k, so ( k kp and k k p ) in general nn k k k k k()() k k j >0 (when kj ). As a result, r will be a uncorrelated networks. In general correlated networks, ANND nn nn positive value in assortative networks, and thus the assortative will increases with k for assortative mixing, meaning that nodes mixing of degree is also called as positive correlation. In preferentially attach to other nodes with similar degrees, while disassortative networks, a monotonically descreasing kk() it will decrease with k for dissassortative mixing, meaning that nn corresponds to a negative r-value, so the disassortative mixing high-degree nodes preferentially attach to other low-degree of degree is often called negative correlation. Clearly, the two nodes. The representation above provides a plain interpretation measures for describing the degree correlation are absolutely for the origin of degree correlations. However, this quantity can consistent. give a clear but only qualitative characterization for the degree correlation in networks. Linear relation in degree correlation. As we know, ANND as a A more coarse-grained and quantitative characterization function of k gives a curve that varies with k. It can be for degree correlation in networks can be given by DCC [2], characterized by suitable fitting functions. For example, 1 researchers showed a power-law dependence of ANND on r jk() e q q , (2) degree [3, 38]. Ma and Szeta extended the Aboav-Wearie law to 2 jk j k q jk the analysis of degree correlation in complex networks [25]. In where 2k 2 q() kq 2 . It can describe the level of qjk k k k the study of complex systems, the linear analysis is often more degree correlation in networks by a scalar quantity appreciated, due to the simplicity and clarity of it.