Greedy routing in small-world networks with power-law degrees

Abstract

In this paper we study decentralized routing in small-world networks that combine a wide variation in node degrees with a notion of spatial embedding. Specifically, we consider a variant of J. Kleinberg’s grid-based small-world model in which (1) the number of long-range edges of each node is not fixed, but is drawn from a power-law probability distribution with exponent parameter $\alpha \ge 0$ and constant mean, and (2) the long-range edges are considered to be bidirectional for the purposes of routing. This model is motivated by empirical observations indicating that several real networks have degrees that follow a power-law distribution. The measured power-law exponent $\alpha$ for these networks is often in the range between 2 and 3. For the small-world model we consider, we show that when $2 < \alpha < 3$ the standard greedy routing algorithm, in which a node forwards the message to its neighbor that is closest to the target in the grid, finishes in an expected number of $O(\log ^{\alpha -1} n\cdot \log \log n)$ steps, for any source–target pair. This is asymptotically smaller than the $O(\log ^2 n)$ steps needed in Kleinberg’s original model with the same average degree, and approaches $O(\log n)$ as $\alpha$ approaches 2. Further, we show that when $0\le \alpha < 2$ or $\alpha \ge 3$ the expected number of steps is $O(\log ^2 n)$, while for $\alpha = 2$ it is $O(\log ^{4/3} n)$. We complement these results with lower bounds that match the upper bounds within at most a $\log \log n$ factor.