  Radial-basis-function networks are traditionally defined for sets of
vector-based observations. In this short paper, we reformulate such networks so
that they can be applied to adjacency-matrix representations of weighted,
directed graphs that represent the relationships between object pairs. We
re-state the sum-of-squares objective function so that it is purely dependent
on entries from the adjacency matrix. From this objective function, we derive a
gradient descent update for the network weights. We also derive a gradient
update that simulates the repositioning of the radial basis prototypes and
changes in the radial basis prototype parameters. An important property of our
radial basis function networks is that they are guaranteed to yield the same
responses as conventional radial-basis networks trained on a corresponding
vector realization of the relationships encoded by the adjacency-matrix. Such a
vector realization only needs to provably exist for this property to hold,
which occurs whenever the relationships correspond to distances from some
arbitrary metric applied to a latent set of vectors. We therefore completely
avoid needing to actually construct vectorial realizations via
multi-dimensional scaling, which ensures that the underlying relationships are
totally preserved.
