The problem of recovering planted community structure in random graphs has received a lot of attention in the literature on the stochastic block model, where the input is a random graph in which edges crossing between different communities appear with smaller probability than edges induced by communities. The communities themselves form a collection of vertex-disjoint sparse cuts in the expected graph, and can be recovered, often exactly, from a sample as long as a separation condition on the intra- and inter-community edge probabilities is satisfied. In this paper, we ask whether the presence of a large number of overlapping sparsest cuts in the expected graph still allows recovery. For example, the $d$-dimensional hypercube graph admits $d$ distinct (balanced) sparsest cuts, one for every coordinate. Can these cuts be identified given a random sample of the edges of the hypercube where each edge is present independently with some probability $p\in (0, 1)$? We show that this is the case, in a very strong sense: the sparsest balanced cut in a sample of the hypercube at rate $p=C\log d/d$ for a sufficiently large constant $C$ is $1/\text{poly}(d)$-close to a coordinate cut with high probability. This is asymptotically optimal and allows approximate recovery of all $d$ cuts simultaneously. Furthermore, for an appropriate sample of hypercube-like graphs recovery can be made exact. The proof is essentially a strong hypercube cut sparsification bound that combines a theorem of Friedgut, Kalai and Naor on boolean functions whose Fourier transform concentrates on the first level of the Fourier spectrum with Karger's cut counting argument.

翻译：随机图中植入社区结构的恢复问题在随机块模型文献中受到广泛关注，其输入为随机图，其中跨越不同社区的边出现概率低于社区内部边。在期望图中，社区本身构成一系列顶点不相交的稀疏割，只要社区内与社区间边概率满足分离条件，通常可以从样本中精确恢复这些社区。本文探讨当期望图中存在大量重叠的最稀疏割时，是否仍能实现恢复。例如，$d$维超立方图具有$d$个不同的（平衡）最稀疏割，每个坐标对应一个割。给定超立方图边的随机样本（每条边以独立概率$p\in (0, 1)$存在），能否识别这些割？我们证明在极强意义下这是可行的：当采样率$p=C\log d/d$（$C$为足够大常数）时，超立方样本中最稀疏平衡割以高概率与某个坐标割的差异不超过$1/\text{poly}(d)$。该结果具有渐近最优性，且允许同时近似恢复所有$d$个割。进一步地，对于适当的类超立方图样本，可实现精确恢复。证明本质上是强超立方割稀疏化界，结合了Friedgut、Kalai和Naor关于傅里叶谱集中于第一层的布尔函数定理，以及Karger的割计数论证。