  In the design of decentralized networked systems, it is useful to know
whether a given network topology can sustain stable dynamics. We consider a
basic version of this problem here: given a vector space of sparse real
matrices, does it contain a stable (Hurwitz) matrix? Said differently, is a
feedback channel (corresponding to a non-zero entry) necessary for
stabilization or can it be done without. We provide in this paper a set of
necessary and a set of sufficient conditions for the existence of stable
matrices in a vector space of sparse matrices. We further prove some properties
of the set of sparse matrix spaces that contain Hurwitz matrices. The
conditions we exhibit are most easily stated in the language of graph theory,
which we thus adopt in this paper.
