Spectral coarse grained controllability of complex networks

With the accumulation of interaction data from various systems, a fundamental question in network science is how to reduce the sizes while keeping certain properties of complex networks. Combined the spectral coarse graining theory and the structural controllability of complex networks, we explore the structural controllability of undirected complex networks during coarse graining processes. We evidence that the spectral coarse grained controllability (SCGC) properties for the Erdös-Rényi (ER) random networks, the scale-free (SF) random networks and the small-world (SW) random networks are distinct from each other. The SW networks are very robust, while the SF networks are sensitive during the coarse graining processes. As an emergent properties for the dense ER networks, during the coarse graining processes, there exists a threshold value of the coarse grained sizes, which separates the controllability of the reduced networks into robust and sensitive to coarse graining. Investigations on some real-world complex networks indicate that the SCGC properties are varied among different categories and different kinds of networks, some highly organized social or biological networks are more difficult to be controlled, while many man-made power networks and infrastructure networks can keep the controllability properties during the coarse graining processes. Furthermore, we speculate that the SCGC properties of complex networks may depend on their degree distributions. The associated investigations have potential implications in the control of large-scale complex networks, as well as in the understanding of the organization of complex networks.