Efficient and near-optimal algorithms for sampling small connected subgraphs

Abstract

We consider the following problem: given a graph G = (V, E) and an integer k, sample a connected induced k-node subgraph of G (also called k-graphlet) uniformly at random. The best algorithms known achieve ε-uniformity and are based on random walks or color coding. The random walk approach is elegant, but has a worst-case running time of ∆ Θ(k) log n ε where n = |V | and ∆ is the maximum degree of G. Color coding is more efficient, but requires a preprocessing phase with running time and space 2 Θ(k) O(m log 1 ε ) where m = |E|. Our main result is an algorithm for ε-uniform sampling with the following guarantees. The preprocessing runs in time k O(k) O 1 ε n log n and space O(n). This implies a sublinear preprocessing when G is dense enough. After, the algorithm yields independent ε-uniform k-graphlets in k O(k) O ( 1 ε ) 10 log 1 ε expected time per sample. The preprocessing phase computes in one pass an approximate ordering of G that makes rejection sampling efficient in the sampling phase, and the ε-uniformity is based on estimating cuts and coupling arguments. In fact, the algorithm derives from a much simpler algorithm which has O(m log ∆) preprocessing time and returns perfectly uniform k-graphlets from G in k O(k) O(log ∆) expected time per sample. To the best of our knowledge this is the first efficient algorithm for perfectly uniform graphlet sampling. In addition, we give an almost-tight bound for the random walk technique. More precisely, we show that the most commonly used random walk has mixing time k O(k) O(t(G)( ∆ δ ) k−1 log n) where t(G) is the mixing time of G and δ is its minimum degree. This improves on recent results and is tight up to a factor k O(k) δ log n.

Learning low-rank latent mesoscale structures in networks

“…This is often the case for image-processing applications [7,33,35], as it is straightforward to sample a k × k patch uniformly at random from an image. However, the similar problem of uniformly randomly sampling a k-node connected subnetwork from a network is not straightforward [3,16,22,58]. For our purpose of developing dictionary learning for networks, we use motif sampling, which was introduced recently in [29].…”