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 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 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 < 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 α - 1 n · 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 α approaches 2. Further, we show that when 0 ≤ α < 2 or α ≥ 3 the expected number of steps is O ( log 2 n ) , while for α = 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.