  Graph spectral analysis can yield meaningful embeddings of graphs by
providing insight into distributed features not directly accessible in nodal
domain. Recent efforts in graph signal processing have proposed new
decompositions-e.g., based on wavelets and Slepians-that can be applied to
filter signals defined on the graph. In this work, we take inspiration from
these constructions to define a new guided spectral embedding that combines
maximizing energy concentration with minimizing modified embedded distance for
a given importance weighting of the nodes. We show these optimization goals are
intrinsically opposite, leading to a well-defined and stable spectral
decomposition. The importance weighting allows to put the focus on particular
nodes and tune the trade-off between global and local effects. Following the
derivation of our new optimization criterion and its linear approximation, we
exemplify the methodology on the C. elegans structural connectome. The results
of our analyses confirm known observations on the nematode's neural network in
terms of functionality and importance of cells. Compared to Laplacian
embedding, the guided approach, focused on a certain class of cells (sensory,
inter- and motoneurons), provides more biological insights, such as the
distinction between somatic positions of cells, and their involvement in low or
high order processing functions.
