Optimal Degree Distributions for Uniform Small World Rings

TL;DR

This paper analyzes the optimal degree distributions for greedy routing in harmonic rings, showing that certain distributions minimize expected route length and are asymptotically optimal for small world networks.

Contribution

It generalizes previous bounds to broader degree distributions and proves that fixed degree distributions are optimal for greedy routing in harmonic rings.

Findings

Expected route length is $O(rac{ ext{log}^2 n}{ ext{mean degree}})$ for distributions with mean $ ext{mean degree}$ and max $O( ext{log} n)$.

Harmonic rings with these distributions are asymptotically optimal for greedy routing.

Fixed degree distributions with degrees $loor{ ext{mean degree}}$ or $ ext{ceiling}( ext{mean degree})$ minimize routing time.

Abstract

Motivated by Kleinberg's (2000) and subsequent work, we consider the performance of greedy routing on a directed ring of $n$ nodes augmented with long-range contacts. In this model, each node $u$ is given an additional $D_u$ edges, a degree chosen from a specified probability distribution. Each such edge from $u$ is linked to a random node at distance $r$ ahead in the ring with probability proportional to $1/r$, a "harmonic" distance distribution of contacts. Aspnes et al. (2002) have shown an $O(\log^2 n / \ell)$ bound on the expected length of greedy routes in the case when each node is assigned exactly $\ell$ contacts and, as a consequence of recent work by Dietzfelbinger and Woelfel (2009), this bound is known to be tight. In this paper, we generalize Aspnes' upper bound to show that any degree distribution with mean $\ell$ and maximum value $O(\log n)$ has greedy routes of expected length $O(\log^2n / \ell)$, implying that any harmonic ring in this family is asymptotically optimal. Furthermore, for a more general family of rings, we show that a fixed degree distribution is optimal. More precisely, if each random contact is chosen at distance $r$ with a probability that decreases with $r$, then among degree distributions with mean $\ell$, greedy routing time is smallest when every node is assigned $\floor{\ell}$ or $\ceiling{\ell}$ contacts.