In signal processing (SP) often we need to compute sample-wise distance between two signals, e.g. computing MSE. SP on graphs concerns the case when the domain of the signal has some intrinsic structure represented as weighted undirected graphs. The weighs could represent, for example, the correlation between the signal values in each dimension. I was wondering if there is a generalize notion of sample-wise euclidean distance for SP on graphs which takes into account the structure of the graph?
p.s. “The Emerging Field of Signal Processing on Graphs” discusses the distances as open issues.