Deformed Laplacians and spectral ranking in directed networks

Abstract

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.

24 pages. In press in Applied and Computational Harmonic Analysis

References in corpus (3)

• The Lusternik-Schnirelmann theorem for graphs
• Estimation and Registration on Graphs
• Analysis of Crowdsourced Sampling Strategies for HodgeRank with Sparse Random Graphs

Cited by in corpus (3)

• Ranking and synchronization from pairwise measurements via SVD
• The Geometry of Synchronization Problems and Learning Group Actions
• The Limiting Eigenvalue Distribution of Iterated k-Regular Graph Cylinders