Kleinberg's Grid Reloaded

TL;DR

This paper reevaluates Kleinberg's grid model for small-world networks using simulations, demonstrating its robustness in practice, providing tighter performance bounds, and aligning results with real-world data.

Contribution

It introduces a new simulation algorithm with dynamic rejection sampling, offering detailed analysis and improved bounds on Kleinberg's model performance.

Findings

Kleinberg's augmented grid is more robust in practice than asymptotic predictions.

The new algorithm accelerates simulations, enabling detailed numerical evaluation.

Results align with real-life small-world network experiments.

Abstract

One of the key features of small-worlds is the ability to route messages with few hops only using local knowledge of the topology. In 2000, Kleinberg proposed a model based on an augmented grid that asymptotically exhibits such property. In this paper, we propose to revisit the original model from a simulation-based perspective. Our approach is fueled by a new algorithm that uses dynamic rejection sampling to draw augmenting links. The speed gain offered by the algorithm enables a detailed numerical evaluation. We show for example that in practice, the augmented scheme proposed by Kleinberg is more robust than predicted by the asymptotic behavior, even for very large finite grids. We also propose tighter bounds on the performance of Kleinberg's routing algorithm. At last, we show that fed with realistic parameters, the model gives results in line with real-life experiments.