The effect of power-law degrees on the navigability of small worlds: [extended abstract]

We analyze decentralized routing in small-world networks that combine a wide variation in node degrees with a notion of spatial embedding. Specifically, we consider a variation of Kleinberg's augmented-lattice model (STOC 2000), where the number of long-range contacts for each node is drawn from a power-law distribution. This model is motivated by the experimental observation that many "real-world" networks have power-law degrees. In such networks, the exponent α of the power law is typically between 2 and 3. We prove that, in our model, for this range of values, 2 < α < 3, the expected number of steps of greedy routing from any source to any target is O(logα-1 n) steps. This bound is tight in a strong sense. Indeed, we prove that the expected number of steps of greedy routing for a uniformly-random pair of source-target nodes is Ω(logα-1 n) steps. We also show that for α < 2 or α ≥ 3, greedy routing performs in Θ(log2 n) xexpected steps, and for α = 2, Θ(log1+ε n) expected steps are required, where 1/3 ≤ ε ≤ 1/2. To the best of our knowledge, these results are the first to formally quantify the effect of the power-law degree distribution on the navigability of small worlds. Moreover, they show that this effect is significant. In particular, as α approaches 2 from above, the expected number of steps of greedy routing in the augmented lattice with power-law degrees approaches the square-root of the expected number of steps of greedy routing in the augmented lattice with fixed degrees, although both networks have the same average degree.