Structural controllability of general edge dynamics in complex network

Dynamic processes that occur on the edge of complex networks are relevant to a variety of real-world systems, where states are defined on individual edges, and nodes are active components with information processing capabilities. In traditional studies of edge controllability, all adjacent edge states are assumed to be coupled. In this paper, we release this all-to-all coupling restriction and propose a general edge dynamics model. We give a theoretical framework to study the structural controllability of the general edge dynamics and find that the set of driver nodes for edge controllability is unique and determined by the local information of nodes. Applying our framework to a large number of model and real networks, we find that there exist lower and upper bounds of edge controllability, which are determined by the coupling density, where the coupling density is the proportion of adjacent edge states that are coupled. Then we investigate the proportion of effective coupling in edge controllability and find that homogeneous and relatively sparse networks have a higher proportion, and that the proportion is mainly determined by degree distribution. Finally, we analyze the role of edges in edge controllability and find that it is largely encoded by the coupling density and degree distribution, and are influenced by in- and out-degree correlation.