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

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 $$\alpha \ge 0$$α≥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 $$\alpha $$α for these networks is often in the range between 2 and 3. For the small-world model we consider, we show that when $$2 < \alpha < 3$$2<α<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 $$O(\log ^{\alpha -1} n\cdot \log \log n)$$O(logα-1n·loglogn) steps, for any source–target pair. This is asymptotically smaller than the $$O(\log ^2 n)$$O(log2n) steps needed in Kleinberg’s original model with the same average degree, and approaches $$O(\log n)$$O(logn) as $$\alpha $$α approaches 2. Further, we show that when $$0\le \alpha < 2$$0≤α<2 or $$\alpha \ge 3$$α≥3 the expected number of steps is $$O(\log ^2 n)$$O(log2n), while for $$\alpha = 2$$α=2 it is $$O(\log ^{4/3} n)$$O(log4/3n). We complement these results with lower bounds that match the upper bounds within at most a $$\log \log n$$loglogn factor.