The f-Adjusted Graph Laplacian: a Diagonal Modification with a Geometric Interpretation

Consider a neighborhood graph, for example a k-nearest neighbor graph, that is constructed on sample points drawn according to some density p. Our goal is to re-weight the graph’s edges such that all cuts and volumes behave as if the graph was built on a different sample drawn from an alternative density q. We introduce the f-adjusted graph and prove that it provides the correct cuts and volumes as the sample size tends to infinity. From an algebraic perspective, we show that its normalized Laplacian, denoted as the f-adjusted Laplacian, represents a natural family of diagonal perturbations of the original normalized Laplacian. Our technique allows to apply any cut and volume based algorithm to the f-adjusted graph, for example spectral clustering, in order to study the given graph as if it were built on an unaccessible sample from a different density. We point out applications in sample bias correction, data uniformization, and multi-scale analysis of graphs.