Linear Software Models: Bipartite Isomorphism between Laplacian Eigenvectors and Modularity Matrix Eigenvectors

Abstract

We have recently shown that one can obtain the numbers and sizes of modules of a software system from the eigenvectors of its modularity matrix symmetrized and weighted by an affinity matrix. However such a weighting still demands a suitable definition of an affinity. This paper offers an alternative way to obtain the same results by means of the eigenvectors of a Laplacian matrix, directly obtained from the modularity matrix without the need of weighting. These two formalizations stand in a mutual isomorphism. We call it bipartite isomorphism since it is most straightforwardly shown by deriving the Laplacian from the modularity matrix and vice versa through the intermediate bipartite graph between two separate sets: the structors’ and the functionals’ sets. This isomorphism is also demonstrated through the equation defining the Laplacian in terms of the modularity matrix, or by the direct mapping of the respective matrices’ eigenvectors. Both matrices and the bipartite graph reflect one central idea: modules are connected components with high cohesion. The Laplacian matrix technique, of which the Fiedler vector is of central importance, is illustrated by case studies. An important claim of this paper is that, independently of the modularity matrix- and Laplacian matrix-specific properties, behind these two alternative matrices there is just one unified algebraic theory of software composition — the Linear Software Models — here concerning the application of the matrices’ eigenvectors to software modularity.