Deformed Laplacians and spectral ranking in directed networks

A deformation of the combinatorial Laplacian is proposed, consisting in a generalization of several existing Laplacians. As particular cases of this construction, the dilation Laplacians are shown to be useful tools for ranking in directed networks of pairwise comparisons. The eigenvector with the smallest eigenvalue of the dilation Laplacians has the same sign on any connected graph, and provides directly a ranking score of its nodes. The ranking method, phrased in terms of a group synchronization problem, is applied to artificial and real data, and its performance is compared with other ranking strategies. A main feature of this approach is the presence of a deformation parameter enabling the emphasis of the top-$k$ objects in the ranking. Furthermore, inspired by these results, a family of random walks interpolating between the undirected random walk and the Pagerank random walk is also proposed.