The Degree of Nonholonomy in Distributed Computations

A network of locally interacting agents can be thought of as performing a distributed computation. But not all computations can be faithfully distributed. This paper discusses which global linear transformations can be computed in finite time using local weighting rules, i.e., rules which rely solely on information from adjacent nodes in a network. Additionally, it is shown that the degree of nonholonomy of the computation can be related to the underlying information exchange graph. The main result states that the degree of nonholonomy of the system dynamics is equal to D - 1 where D is the diameter of the information exchange graph. An optimal control problem is solved for finding the local interaction rules, and a simulation is performed to elucidate how optimal solutions can be obtained.