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Nonlinear PageRank Problem for Local Graph Partitioning

Costy Kodsi, Dimosthenis Pasadakis

TL;DR

The paper develops a nonlinear PageRank framework for local graph partitioning using the Moore–Penrose inverse of the incidence matrix. It formulates a nonlinear PageRank problem with a tunable $p$-norm, solved numerically via Levenberg–Marquardt on a full-rank Jacobian, to produce a vertex ranking from which a local cluster is extracted by conductance-based sweep. The authors show that the nonlinear formulation reduces to the linear PageRank as the graph grows and provide a perturbation analysis indicating potential gains in cluster quality. Extensive experiments on synthetic (LFR and Gaussian) and real-world graphs (image datasets and the ORBIS Roman network) demonstrate that NPR achieves lower conductance and higher $F$-scores than state-of-the-art baselines across diverse settings, highlighting its practical impact for efficient local clustering.

Abstract

A nonlinear generalisation of the PageRank problem involving the Moore-Penrose inverse of an incidence matrix is developed for local graph partitioning purposes. The Levenberg-Marquardt method with a full rank Jacobian variant provides a strategy for obtaining a numerical solution to the generalised problem. Sets of vertices are formed according to the ranking supplied by the solution, and a conductance criterion decides upon the set that best represents the cluster around a starting vertex. Experiments on both synthetic and real-world inspired graphs demonstrate the capability of the approach to not only produce low conductance sets, but to also recover local clusters with an accuracy that consistently surpasses state-of-the-art algorithms.

Nonlinear PageRank Problem for Local Graph Partitioning

TL;DR

The paper develops a nonlinear PageRank framework for local graph partitioning using the Moore–Penrose inverse of the incidence matrix. It formulates a nonlinear PageRank problem with a tunable -norm, solved numerically via Levenberg–Marquardt on a full-rank Jacobian, to produce a vertex ranking from which a local cluster is extracted by conductance-based sweep. The authors show that the nonlinear formulation reduces to the linear PageRank as the graph grows and provide a perturbation analysis indicating potential gains in cluster quality. Extensive experiments on synthetic (LFR and Gaussian) and real-world graphs (image datasets and the ORBIS Roman network) demonstrate that NPR achieves lower conductance and higher -scores than state-of-the-art baselines across diverse settings, highlighting its practical impact for efficient local clustering.

Abstract

A nonlinear generalisation of the PageRank problem involving the Moore-Penrose inverse of an incidence matrix is developed for local graph partitioning purposes. The Levenberg-Marquardt method with a full rank Jacobian variant provides a strategy for obtaining a numerical solution to the generalised problem. Sets of vertices are formed according to the ranking supplied by the solution, and a conductance criterion decides upon the set that best represents the cluster around a starting vertex. Experiments on both synthetic and real-world inspired graphs demonstrate the capability of the approach to not only produce low conductance sets, but to also recover local clusters with an accuracy that consistently surpasses state-of-the-art algorithms.
Paper Structure (20 sections, 15 theorems, 94 equations, 5 figures, 1 table)

This paper contains 20 sections, 15 theorems, 94 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notions of graph theory
  4. Conductance and sweep-cut
  5. Matrix representation of graphs
  6. Moore-Penrose inverse
  7. Notation
  8. PageRank problem
  9. Nonlinear PageRank problem
  10. Numerical methodology
  11. Levenberg-Marquardt method
  12. Rank deficiency and the Jacobian
  13. Experiments
  14. Set-up
  15. Synthetic graphs
  16. ...and 5 more sections

Key Result

Proposition 2.1

For a simple and connected graph with $n$ vertices, $\text{rank} \, (B) = n - 1$.

Figures (5)

  • Figure 1: (a) Conductance and the related (b) $Fscore$ of local clusters for an increasing number of inter-community edges in LFR graphs.
  • Figure 2: Levenberg-Marquardt method for the NPR problem defined on an LFR graph with $\gamma = 0.3$ at (a) $p = 1.95$, (b) $p = 1.8$ and (c) $p = 1.6$.
  • Figure 3: (a) Example of eight groupings. (b) Local cluster based on the NPR problem solution. The starting vertex is shown as a yellow diamond and the edges comprising the boundary are in red. (c) Conductance and the related (d) $Fscore$ of local clusters for an increasing number of (400 point) groupings.
  • Figure 4: Levenberg-Marquardt method for the NPR problem defined on a graph with eight groupings at (a) $p = 1.95$, (b) $p = 1.8$ and (c) $p = 1.6$.
  • Figure : Distance (km)

Theorems & Definitions (36)

  • remark 1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 26 more