Basis selection for Galerkin projections of graph transfer operators

Determine a principled method for selecting basis functions in Galerkin projections of graph transfer operators that approximate random-walk dynamics on graphs so that the dominant spectral properties and, consequently, the underlying cluster structure are preserved in the reduced representation.

Background

In the context of projecting random-walk dynamics on graphs to a lower-dimensional subspace via Galerkin approximation, the authors note that this projection can reduce the computational complexity of subsequent spectral analyses.

However, preserving the essential spectral features—particularly the dominant spectrum that encodes cluster structure—is nontrivial and currently lacks a principled, general approach for choosing appropriate basis functions. The cited work [KT24] provides related examples, but a general solution remains to be identified.