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Optimal Graph Stretching for Distributed Averaging

Florine W. Dekker, Zekeriya Erkin, Mauro Conti

TL;DR

This work analyzes how increasing a graph's girth via edge removals affects the convergence of distributed averaging. It formalizes the optimal graph stretching problem, introduces three edge-removal strategies, and combines stretching with leaf minimisation and convergence-oriented heuristics. The key finding is that girth elevation significantly slows convergence, with Most-Cycles removal minimizing the performance hit, while a subsequent repair phase can recover some speed; leaf minimisation has little direct effect. The results motivate the development of high-girth-specific heuristics and distributed approaches for geometry-aware topology optimization in networked consensus tasks.

Abstract

The performance of distributed averaging depends heavily on the underlying topology. In various fields, including compressed sensing, multi-party computation, and abstract graph theory, graphs may be expected to be free of short cycles, i.e. to have high girth. Though extensive analyses and heuristics exist for optimising the performance of distributed averaging in general networks, these studies do not consider girth. As such, it is not clear what happens to convergence time when a graph is stretched to a higher girth. In this work, we introduce the optimal graph stretching problem, wherein we are interested in finding the set of edges for a particular graph that ensures optimal convergence time under constraint of a minimal girth. We compare various methods for choosing which edges to remove, and use various convergence heuristics to speed up the searching process. We generate many graphs with varying parameters, stretch and optimise them, and measure the duration of distributed averaging. We find that stretching by itself significantly increases convergence time. This decrease can be counteracted with a subsequent repair phase, guided by a convergence time heuristic. Existing heuristics are capable, but may be suboptimal.

Optimal Graph Stretching for Distributed Averaging

TL;DR

This work analyzes how increasing a graph's girth via edge removals affects the convergence of distributed averaging. It formalizes the optimal graph stretching problem, introduces three edge-removal strategies, and combines stretching with leaf minimisation and convergence-oriented heuristics. The key finding is that girth elevation significantly slows convergence, with Most-Cycles removal minimizing the performance hit, while a subsequent repair phase can recover some speed; leaf minimisation has little direct effect. The results motivate the development of high-girth-specific heuristics and distributed approaches for geometry-aware topology optimization in networked consensus tasks.

Abstract

The performance of distributed averaging depends heavily on the underlying topology. In various fields, including compressed sensing, multi-party computation, and abstract graph theory, graphs may be expected to be free of short cycles, i.e. to have high girth. Though extensive analyses and heuristics exist for optimising the performance of distributed averaging in general networks, these studies do not consider girth. As such, it is not clear what happens to convergence time when a graph is stretched to a higher girth. In this work, we introduce the optimal graph stretching problem, wherein we are interested in finding the set of edges for a particular graph that ensures optimal convergence time under constraint of a minimal girth. We compare various methods for choosing which edges to remove, and use various convergence heuristics to speed up the searching process. We generate many graphs with varying parameters, stretch and optimise them, and measure the duration of distributed averaging. We find that stretching by itself significantly increases convergence time. This decrease can be counteracted with a subsequent repair phase, guided by a convergence time heuristic. Existing heuristics are capable, but may be suboptimal.

Paper Structure

This paper contains 35 sections, 2 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph theory
  4. Basics
  5. Spectral theory
  6. Spectral graph theory
  7. Distributed averaging
  8. Related work
  9. Convergence
  10. Spectral theory
  11. Degree relations
  12. Comparisons
  13. Girth
  14. Moore bound
  15. High-girth graph constructions
  16. ...and 20 more sections

Figures (5)

  • Figure 1: Analysis of edges after stretching to a desired girth
  • Figure 2: Convergence heuristics after stretching to a desired girth
  • Figure 4: Number of leaves remaining after leaf minimisation
  • Figure 6: Convergence time after leaf minimisation
  • Figure 8: Convergence time after heurisical optimisation, with columns indicating the graph type, and rows indicating the heuristic that was optimised

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4