The combinatorial graph Laplacian has been a fundamental object in the analysis of and optimization on graphs. Its spectral structure has been widely used in graph optimization problems (e.g, clustering), and it is also connected to network circuit theory. There is also a topological view of the graph Laplacian, which permits the extension of the graph Laplacian to the more general q-th combinatorial Laplacian \Delta_q^K for a simplicial complex K. In this way, the standard graph Laplacian is simply the 0-th case (when q=0) of this family of operators. Taking this topological view, Wang et al. introduced the so-called persistent Laplacian \Delta_q^{K,L simplicial complexes K \hookrightarrow L . In this paper, we present a thorough study of properties and algorithms for persistent Laplacians. We first prove that the nullity of  \Delta_q^{K,L} gives rise to the q-th persistent Betti number from K to L . We then present a first algorithm to compute the matrix representation of \Delta_q^{K,L}, which helps reveal insights into the meaning of persistent Laplacian. We next show a new relation between the persistent Laplacian and the so-called Schur complement of a matrix . This has several interesting implications. For example, in the graph case, this uncovers a relation with the notion of effective resistance , as well as a persistent version of the Cheeger inequality. This also gives a second , very simple algorithm to compute the q-th persistent Laplacian . This in turn leads to a new algorithm to compute the q-th persistent Betti number for a pair of spaces which can be significantly more efficient than existing algorithmsunder some conditions . Finally, we also study persistent Laplacians for a filtration of simplicial complexes, and present interesting stability results for their eigenvalues  .
We present a thorough study of the theoretical properties and devise efficient algorithms for thepersistent Laplacian, an extension of the standard combinatorial Laplacian to the setting of pairs (or, in more generality, sequences) of simplicial complexes K \hookrightarrow L , which was recently introduced by Wang, Nguyen, and Wei. In particular, in analogy with the non-persistent case, we first prove that the nullity of the q-th persistent Laplacian \Delta_q^{K,L} equals the q-th persistent Betti number of the inclusion (K \hookrightarrow L) . We then present an initial algorithm for finding a matrix representation of \Delta_q^{K,L}, which itself helps interpret the persistent Laplacian. We exhibit a novel relationship between the persistent Laplacian and the notion of Schur complement of a matrix which has several important implications. In the graph case, it both uncovers a link with the notion of effective resistance and leads to a persistent version of the Cheeger inequality. This relationship also yields an additional , very simple algorithm for finding (a matrix representation of) the q-th persistent Laplacian which in turn leads to a novel and fundamentally different algorithm for computing the q-th persistent Betti number for a pair (K,L) which can be significantly more efficient than standard algorithms . Finally, we study persistent Laplacians for simplicial filtrations and present novel stability results for their eigenvalues . Our work brings methods from spectral graph theory, circuit theory, and persistent homology together with a topological view of the combinatorial Laplacian on simplicial complexes .
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