Scaling of Congestion in Small World Networks

TL;DR

This paper analyzes how congestion scales in small world networks, showing that network structure and routing strategies significantly influence congestion levels, with implications for network design and optimization.

Contribution

It provides new theoretical bounds on congestion scaling in planar and non-planar small world networks, highlighting the impact of network properties and routing on congestion.

Findings

Congestion scales as O(N^2 / log N) in planar exponentially growing networks.

Without planarity, congestion can scale as low as O(N^{1+ε}).

Routing on shortest paths with bounded link weights results in negligible congestion variation.

Abstract

In this report we show that in a planar exponentially growing network consisting of $N$ nodes, congestion scales as $O(N^2/\log(N))$ independently of how flows may be routed. This is in contrast to the $O(N^{3/2})$ scaling of congestion in a flat polynomially growing network. We also show that without the planarity condition, congestion in a small world network could scale as low as $O(N^{1+\epsilon})$, for arbitrarily small $\epsilon$. These extreme results demonstrate that the small world property by itself cannot provide guidance on the level of congestion in a network and other characteristics are needed for better resolution. Finally, we investigate scaling of congestion under the geodesic flow, that is, when flows are routed on shortest paths based on a link metric. Here we prove that if the link weights are scaled by arbitrarily small or large multipliers then considerable changes in congestion may occur. However, if we constrain the link-weight multipliers to be bounded away from both zero and infinity, then variations in congestion due to such remetrization are negligible.