Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The non-backtracking transition probability matrix and its usage for node clustering

Marianna Bolla

TL;DR

The paper investigates how real eigenvalues of the non-backtracking transition matrix $\mathcal{T}$ relate to node clustering in sparse graphs, by connecting $\mathcal{T}$ to the non-backtracking matrix $\bm{B}$ and the non-backtracking Laplacian $\mathcal{L}$. It develops an inflation–deflation clustering framework that leverages the generalized eigenproblem $\bm{B}\mathbf{z}=\lambda\mathcal{D}_{row}\mathbf{z}$ and the PT-symmetric structure to relate edge-space eigenvectors to node-space representations, enabling effective spectral clustering in sparse stochastic block models. The authors derive SVD decompositions for $\bm{B}$ and $\bm{B}^T$, establish perturbation bounds (Bauer--Fike) that bound eigenvalue shifts under model perturbations, and show that real, structural eigenvalues of $\mathcal{T}$ remain separated from the bulk, aligning with block structure. Collectively, these results provide a principled, theory-backed approach to clustering using non-backtracking spectra in sparse networks, with practical guidance for inflating edge-eigenvectors to node representations and applying $k$-means. The work deepens understanding of the spectral properties of non-backtracking operators and offers a robust pathway for clustering in sparse SBM regimes.

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,Mulas. ``Inflation--deflation'' techniques are also developed for clustering the nodes of the non-backtracking graph. With further processing, it leads to the clustering of the nodes of the original graph, which usually comes from a sparse stochastic block model of Bordenave,Decelle.

The non-backtracking transition probability matrix and its usage for node clustering

TL;DR

The paper investigates how real eigenvalues of the non-backtracking transition matrix relate to node clustering in sparse graphs, by connecting to the non-backtracking matrix and the non-backtracking Laplacian . It develops an inflation–deflation clustering framework that leverages the generalized eigenproblem and the PT-symmetric structure to relate edge-space eigenvectors to node-space representations, enabling effective spectral clustering in sparse stochastic block models. The authors derive SVD decompositions for and , establish perturbation bounds (Bauer--Fike) that bound eigenvalue shifts under model perturbations, and show that real, structural eigenvalues of remain separated from the bulk, aligning with block structure. Collectively, these results provide a principled, theory-backed approach to clustering using non-backtracking spectra in sparse networks, with practical guidance for inflating edge-eigenvectors to node representations and applying -means. The work deepens understanding of the spectral properties of non-backtracking operators and offers a robust pathway for clustering in sparse SBM regimes.

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,Mulas. ``Inflation--deflation'' techniques are also developed for clustering the nodes of the non-backtracking graph. With further processing, it leads to the clustering of the nodes of the original graph, which usually comes from a sparse stochastic block model of Bordenave,Decelle.
Paper Structure (4 sections, 7 theorems, 68 equations)

This paper contains 4 sections, 7 theorems, 68 equations.

Table of Contents

  1. Non-backtracking graphs
  2. The non-backtracking transition probability matrix
  3. Relation between the eigenvalues of $\bm{B}$ and $\cal T$
  4. Clustering

Key Result

Theorem 1

Assume that $\bm{B}$ is irreducible and diagonalizable. Then the eigenvalues of ${\cal T} ={\cal D}_{row}^{-1} \bm{B}$ are allocated within the closed circle of center $\mathbf{0}$ and radius 1 of the complex plane $\mathbb{C}$, and 1 is a single real eigenvalue. Furthermore, the right eigenvectors

Theorems & Definitions (14)

  • Definition 1: Right and left eigenvectors
  • Theorem 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 4 more