  Many real world graphs, such as the graphs of molecules, exhibit structure at
multiple different scales, but most existing kernels between graphs are either
purely local or purely global in character. In contrast, by building a
hierarchy of nested subgraphs, the Multiscale Laplacian Graph kernels (MLG
kernels) that we define in this paper can account for structure at a range of
different scales. At the heart of the MLG construction is another new graph
kernel, called the Feature Space Laplacian Graph kernel (FLG kernel), which has
the property that it can lift a base kernel defined on the vertices of two
graphs to a kernel between the graphs. The MLG kernel applies such FLG kernels
to subgraphs recursively. To make the MLG kernel computationally feasible, we
also introduce a randomized projection procedure, similar to the Nystr\"om
method, but for RKHS operators.
