On Sampling Edges Almost Uniformly

The Arboricity Captures the Complexity of Sampling Edges

“…In the context of social networks, preferential attachment graphs and additional generative models exhibit bounded arboricity [3,6,5], and this has also been empirically validated for many real-world graphs [18,16,24]. We describe a new algorithm for sampling edges almost uniformly whose runtime is O * (α d) where α is an upper bound on the arboricity of G. In the extremal case that α = Θ( √ m), the runtime of our algorithm is the same as that of [15] (up to poly-log factors). For smaller α, our algorithm is strictly faster.…”

“…For smaller α, our algorithm is strictly faster. In particular for α = O(1), the new algorithm is exponentially faster than that of [15]. We also prove matching lower bounds, showing that for all ranges of α, our algorithm is query-optimal, up to polylogarithmic factors and the dependence in 1 ε.…”

“…Furthermore, there are sublinear algorithms that are known to work when given access to uniform edges (e.g., [2]) and can be adapted to the case when the distribution over the edges is almost uniform (see Section 1.4 for details). An important observation is that in many cases it is crucial that the sampling distribution is pointwise-close to uniform rather than close with respect to the Total Variation Distance (henceforth TVD) -see the discussion in [15,Sec. 1.1].…”

“…Eden and Rosenbaum [15] recently showed that Θ * ( √ m d) queries are both sufficient and necessary for sampling edges almost uniformly, where d = 2m n denotes the average degree in the graph. (We use the notation O * to suppress factors that are polylogarithmic in n and polynomial in 1 ε.)…”

“…We also prove matching lower bounds, showing that for all ranges of α, our algorithm is query-optimal, up to polylogarithmic factors and the dependence in 1 ε. Furthermore, while not as simple as the algorithm of [15], our algorithm is still easy to implement and does not incur any large constants in the query complexity and running time, thus making in it suitable for practical applications.…”