Large algebraic connectivity fluctuations in spatial network ensembles imply a predictive advantage from node location information

TL;DR

This paper investigates the fluctuations in algebraic connectivity in Random Geometric Graphs, revealing significant variability influenced by node distribution and boundary conditions, and provides a predictive analytical expression.

Contribution

It provides the first characterization of algebraic connectivity distribution in RGGs, quantifies fluctuations, and derives a closed-form expected value for periodic boundary conditions.

Findings

Algebraic connectivity fluctuates up to 30% around the mean in large RGGs.

Fluctuations depend on dimensionality, boundary conditions, and node distribution.

A closed-form expression for expected algebraic connectivity in periodic RGGs is derived.

Abstract

A Random Geometric Graph (RGG) ensemble is defined by the disordered distribution of its node locations. We investigate how this randomness drives sample-to-sample fluctuations in the dynamical properties of these graphs. We study the distributional properties of the algebraic connectivity which is informative of diffusion and synchronization timescales in graphs. We use numerical simulations to provide the first characterisation of the algebraic connectivity distribution for RGG ensembles. We find that the algebraic connectivity can show fluctuations relative to its mean on the order of $30 \%$, even for relatively large RGG ensembles ($N=10^5$). We explore the factors driving these fluctuations for RGG ensembles with different choices of dimensionality, boundary conditions and node distributions. Within a given ensemble, the algebraic connectivity can covary with the minimum degree and can also be affected by the presence of density inhomogeneities in the nodal distribution. We also derive a closed-form expression for the expected algebraic connectivity for RGGs with periodic boundary conditions for general dimension.