The Laplacian is a critical concept in network analysis and shape tasks. Given its limitation to a single scale of the data set, it was recently imbued with ideas from persistent homology, leading to the \(p\)-persistent \(q\)-combinatorial Laplacian of a simplicial complex. This construction, colloquially referred to as a persistent Laplacian can be shown to be intricately linked to multi-scale topological information, such as the persistent Betti numbers of a data set.
Motivated by the utility of persistent Laplacians, this paper presents a new way of calculating their spectral information, specifically, the roots of their characteristic polynomials, by homotopy continuation. Homotopy continuation addresses the problem of solving a system \(f\) of polynomial equations by starting with an easy-to-solve system \(g\), building a homotopy between \(f\) and \(g\), and, finally, tracking the roots of \(g\) to those of \(f\). The authors demonstrate the feasibility of this continuation method and provide empirical evidence of the utility of using spectral information from persistent Laplacians to characterise aromatic molecules, for instance.