Effect of degree correlations on controllability of undirected networks
Authors: Xu Ming  , Xu Chuan-Yun  , Cao Ke-Fei  ,   Center for Nonlinear Complex Systems, School of Physics and Astronomy, Yunnan University, Kunming 650091, China;,   School of Mathematical Sciences, Kaili University, Kaili 556011, China;,   School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China
Published: December 2020

The controllability analysis of complex networks is of great importance for modern network science and engineering. Existing research shows that the controllability of a complex network is affected not only by the degree distribution of the network,but also by the degree correlation.Although the effect of degree correlations on the network controllability is well studied for directed networks,it is not yet very clear for the case of undirected networks.To explore the impact of degree correlations on the controllability of undirected networks and their corresponding generalized (bidirectional and directed) networks,in this paper,we use the simulated annealing algorithm to change the network degree correlation coefficients by link rewiring.First,the undirected Erdős-Rényi random network and the modified scale-free network are taken as example models to be investigated.Numerical simulations show that the controllability measure (density of driver nodes) of undirected networks decreases monotonically with the increase of the degree correlation coefficient under a constant degree distribution.Specifically,when the degree correlation coefficient changes from -1 to 0,the controllability measure decreases rapidly;while the decrease in the controllability measure is not obvious when the degree correlation coefficient changes from 0 to 1.Next,the bidirectional networks and some directed networks are considered;in these networks,the in-degree of each node is equal to its out-degree,thus link rewiring results in the simultaneous changes of various degree correlations (i.e.,in-in,in-out,out-in,and out-out degree correlations).Further investigations show that these bidirectional and directed networks also follow the above rule,which is verified by the two real networks.The increase of the degree correlation coefficient in undirected networks also implies the increases of various degree correlation coefficients in the corresponding directed networks.Although the effect of a single degree correlation on the controllability of directed networks is clear,the comprehensive effect of the simultaneous changes in various degree correlations on the network controllability cannot be additively and therefore directly estimated by the relevant results in the corresponding directed networks;namely,the effect of the degree correlation on the controllability in an undirected network has its special rule.Some explanations are given for this phenomenon.Moreover,for a large sparse network without self-loops,no matter how assortative or disassortative it is,its structural controllability and exact controllability are verified to be almost the same.These studies will deepen the understanding of the relationship between the network controllability and the network structure.