  We propose a novel random walk-based algorithm for unbiased estimation of
arbitrary functions of a weighted adjacency matrix, coined universal graph
random features (u-GRFs). This includes many of the most popular examples of
kernels defined on the nodes of a graph. Our algorithm enjoys subquadratic time
complexity with respect to the number of nodes, overcoming the notoriously
prohibitive cubic scaling of exact graph kernel evaluation. It can also be
trivially distributed across machines, permitting learning on much larger
networks. At the heart of the algorithm is a modulation function which
upweights or downweights the contribution from different random walks depending
on their lengths. We show that by parameterising it with a neural network we
can obtain u-GRFs that give higher-quality kernel estimates or perform
efficient, scalable kernel learning. We provide robust theoretical analysis and
support our findings with experiments including pointwise estimation of fixed
graph kernels, solving non-homogeneous graph ordinary differential equations,
node clustering and kernel regression on triangular meshes.
