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This paper develops an adaptive version of Mallat's scattering transform forsignals on graphs. The main results are norm bounds for the layers of thetransform, obtained from a version of a Beurling-Deny inequality that permitsto remove the nonlinear steps in the scattering transform. Under statisticalassumptions on the input signal, the norm bounds can be refined. The conceptspresented here are illustrated with an application to traffic counts whichexhibit characteristic daily and weekly periodicities. Anomalous trafficpatterns which deviate from these expected periodicities produce a response inthe scattering transform.

A scattering transform for graphs based on heat semigroups, with an  application for the detection of anomalies in positive time series with  underlying periodicities