On the properties of Laplacian pseudoinverses

TL;DR

This paper investigates the mathematical properties of Laplacian pseudoinverses, revealing they are generally signed Laplacians with specific positivity properties, and explores their stability and structural characteristics.

Contribution

It demonstrates that Laplacian pseudoinverses are signed Laplacians with exponential positivity and shows this class is closed under pseudoinversion, extending to signed digraphs with weight balance.

Findings

Laplacian pseudoinverses are signed Laplacians with Perron-Frobenius properties.

The set of such signed Laplacians is closed under pseudoinversion.

Stability is guaranteed for weight-balanced signed digraphs.

Abstract

The pseudoinverse of a graph Laplacian is used in many applications and fields, such as for instance in the computation of the effective resistance in electrical networks, in the calculation of the hitting/commuting times for a Markov chain and in continuous-time distributed averaging problems. In this paper we show that the Laplacian pseudoinverse is in general not a Laplacian matrix but rather a signed Laplacian with the property of being an eventually exponentially positive matrix, i.e., of obeying a strong Perron-Frobenius property. We show further that the set of signed Laplacians with this structure (i.e., eventual exponential positivity) is closed with respect to matrix pseudoinversion. This is true even for signed digraphs, and provided that we restrict to Laplacians that are weight balanced also stability is guaranteed.