The non-backtracking random walk and its usage for node and edge clustering

TL;DR

This paper explores the spectral properties of the non-backtracking matrix and Laplacian, developing techniques for node and edge clustering in sparse graphs, especially within stochastic block models.

Contribution

It introduces new relationships between eigenvalues of non-backtracking matrices and Laplacians, along with inflation-deflation methods for clustering in sparse stochastic block models.

Findings

Eigenvalues of the non-backtracking matrix relate linearly to those of the Laplacian.

Inflation-deflation techniques improve clustering accuracy in sparse graphs.

Detection of bottlenecks via the symmetrized non-backtracking Laplacian.

Abstract

Relation between the real eigenvalues of the non-backtracking matrix and those of the non-backtracking Laplacian is considered with respect to node clustering. For this purpose we use the real eigenvalues of the transition probability matrix (when the random walk goes through the oriented edges with the rule of ``not going back in the next step'') which have a linear relation to those of the non-backtracking Laplacian of Jost and Mulas. ``Inflation--deflation'' techniques are also developed for clustering the nodes of the original graph when it comes from the sparse stochastic block model of Bordenave and Decelle. Via the symmetrized normalized non-backtracking Laplacian, ``bottlenecks'' in the non-backtracking graph are detected, where the random walk goes through rarely in any direction.