  We propose a new method for performing multiscale analysis of functions
defined on the vertices of a finite connected weighted graph. Our approach
relies on a random spanning forest to downsample the set of vertices, and on
approximate solutions of Markov intertwining relation to provide a subgraph
structure and a filter bank leading to a wavelet basis of the set of functions.
Our construction involves two parameters q and q'. The first one controls the
mean number of kept vertices in the downsampling, while the second one is a
tuning parameter between space localization and frequency localization. We
provide an explicit reconstruction formula, bounds on the reconstruction
operator norm and on the error in the intertwining relation, and a Jackson-like
inequality. These bounds lead to recommend a way to choose the parameters q and
q'. We illustrate the method by numerical experiments.
