Measuring Mixing Patterns In Complex Networks By Spearman Rank Correlation Coefficient

Complex network is an interdiscipline that described the complex relationship in our real world. A large number of empirical studies indicate that many real-world sys-tems can be abstracted as complex networks. They recover the rich diversity of systems, and are found to have lots of common structural features.A random model of networks with specific degree sequences, is often considered as a zero model to test the structural properties of complex networks. In order to cap-ture the inner structures and behaviors of real-world networked systems, along with the degree distribution, the mixing patterns in networks are also essential. It plays an important role in many fields, such as mean distance, robustness, stability, percola-tion thresholds, epidemic spreading and synchronization of oscillators. By studying the statistical parameters to the two ends of edges of a network, such as the correlation co-efficient and the mean, we can then determine the joint degree probability distribution, which is decisive for the structure, function and dynamics of the network.To measure the mixing patterns in complex networks, Newman introduced a method based on the Pearson correlation coefficient. However, further studies indicate that the Pearson coefficient has a serious drawback. Its value critically depends on the size and degree distribution of networks. In particular, it would converge to zero for large scale-free networks. This drawback strongly impedes the quantitative comparison of different networks. This point cannot be ignored, as the size of nowaday networks is getting larger and larger, for instance, scientific collaboration networks, the World-Wide Web.In addition, when given the degree sequence and the correlation coefficient, the traditional way to get such a network is the rewiring procedure, but it is inefficiency. Therefore, searching for methods which can generate networks with priori correlation coefficients is very meaningful to the research of complex networks.In this thesis, to solve the above drawback of the Pearson-coefficient-based mea-sure, we propose a novel method based on Spearman rank correlation coefficient to measure the mixing patterns in complex networks. In mathematics, Spearman coeffi-cient is usually used to quantify how well two columns of data monotonically depend on each other. It is rank-based, nonparametric, independent of the network size and degree distribution. Hence it could overcome the aforementioned drawback of Pearson coefficient and work better.Moreover, we discover that the normalized rank orders of stubs are statistically in a linear correlation, and the correlation coefficient is just the Spearman coefficient. We argue that this kind of correlation universally exists in complex networks, and we verify it on both empirical and artificial networks. It can be used as an essential factor to determine the point degree probability distribution of networks.Based on the linear relationship above and Marrows’ results on ranking models, we give a simple exponent and Gauss approximate form to describe the jointed de-gree probability distribution of networks. On the basis of the jointed degree probability distribution, we could directly generate a network with any prescribed Spearman coef-ficient. We test the exponent and the Gauss model with simulations, and find them in good agreement with the theory. Moreover, comparing with the classical rewiring pro-cedure, networks created by the exponent and Gauss model also have good randomness and the computation complexity is reduced.