Dilations and degeneracy in network controllability

Network controllability asserts a perspective that the structure—the location of edges that connect nodes—of the network contains important information about fundamental characteristics of our ability to change the behavior that evolves on these networks. It can be used, for example, to determine the parts of the system that when influenced by outside controlling signals, can ultimately steer the behavior of the entire network. One of the challenges in utilizing the ideas from network controllability on real systems is that there is typically more than one potential solution (often many) suggested by the topology of the graph that perform equally well. Picking a single candidate from this degenerate solution set over others should be properly motivated, however, to-date our understanding of how these different options are related has been limited. In this work, we operationalize the existing notion of a dilation into a framework that provides clarity on the source of this control degeneracy and further elucidates many of the existing results surrounding degeneracy in the literature.