  We consider the detection of activations over graphs under Gaussian noise,
where signals are piece-wise constant over the graph. Despite the wide
applicability of such a detection algorithm, there has been little success in
the development of computationally feasible methods with proveable theoretical
guarantees for general graph topologies. We cast this as a hypothesis testing
problem, and first provide a universal necessary condition for asymptotic
distinguishability of the null and alternative hypotheses. We then introduce
the spanning tree wavelet basis over graphs, a localized basis that reflects
the topology of the graph, and prove that for any spanning tree, this approach
can distinguish null from alternative in a low signal-to-noise regime. Lastly,
we improve on this result and show that using the uniform spanning tree in the
basis construction yields a randomized test with stronger theoretical
guarantees that in many cases matches our necessary conditions. Specifically,
we obtain near-optimal performance in edge transitive graphs, $k$-nearest
neighbor graphs, and $\epsilon$-graphs.
