Canonical Operators on Graphs

There is widespread current interest in distributed control of networked systems, e.g. [4], [5], [6], [12], [13], [15]. Much of the work to date centered on linear control laws, and has taken advantage the last twenty years of development in spectral graph theory. In particular the graph Laplacian, in various incarnations, has seen use as a stabilizing feedback. The property of the Laplacian used in these works has been essentially the fact that it is the generator of a reversible continuous time ergodic Markov chain: it has one zero eigenvalue and all others are strictly positive. The study we wish to propose is broader. We wish to ask which linear feedback laws are possible for actors which must communicate on a (possibly directed) network. The coarse grain answer to this question is: those laws which respect the network structure. The present work, in initiating this study, precisely defines and characterizes in some detail classes of canonical (di)graph operators constructed from the incidence relations. These ideas are implicit or glossed over in a number of earlier publications; we felt that there will be those readers who, like us, benefit from the careful codification of properties. Our methods have a pronounced geometric and functorial flavor. There is a literature which has also taken this perspective; see e.g. [8], [11], [19]. We then turn to characterizing the graph Laplacian as constructed