Relevance of Network Characteristics to Controllability Degree

This article studies the controllability degree via analyzing the condition number of Gramian matrix. Our aim is to explore how the network characteristics affect the controllability degree. Specifically, we prove that a large time parameter would worsen the controllability degree. The time parameter could be understood as the network coupling strength. For directed path networks, we derive how edge weights and time parameter jointly determine the best controllability degree. Furthermore, we prove that either adding a new edge or enhancing an existing edge weight appropriately would worsen the controllability degree. Moreover, through the numerical simulation of external inputs deployment, we find a significant statistical relationship between the controllability index and the controllability degree. In this article, the Gramian matrix reveals the importance of network characteristics that cannot be captured by classic Kalman rank condition or Popov–Belevitch–Hautus (PBH) test.