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 ∈ (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 = Clog d/d for a sufficiently large constant C is 1/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.

In the graph clustering problem with a planted solution, the input is a graph on \(n\) vertices partitioned into \(k\) clusters, and the task is to infer the clusters from graph structure. A standard assumption is that clusters induce well-connected subgraphs (i.e. \(\Omega(1)\)-expanders), and connections between clusters are sparse (i.e. clusters form \(\epsilon\)-sparse cuts). Such a graph defines the clustering uniquely up to \(\approx \epsilon\) misclassification rate, and efficient algorithms for achieving this rate are known. While this vanilla version of graph clustering is extremely well studied, in the practice of graph analysis vertices of the graph are typically equipped with labels, or features, that provide additional information on cluster ids of the vertices. For example, each vertex could be equipped with a cluster label that is corrupted independently with probability \(\delta\). Using either of the two sources of information separately leads to misclassification rate \(\min \{\epsilon, \delta\}\), but can one combine the two to achieve misclassification rate \(\approx \epsilon\delta\)?

Given a vertex in a \((k,\varphi,\epsilon)\)-clusterable graph, i.e. a graph whose vertex set can be partitioned into a disjoint union of \(\varphi\)-expanders of size \(\approx n/k\) with outer conductance bounded by \(\epsilon\), can one quickly tell which cluster it belongs to? This is a classical question going back to the expansion testing problem of Goldreich and Ron’11 (the case of \(k=2\)) that has received a lot of attention in the literature. For \(k=2\) a sample of \(\approx n^{1/2+O(\epsilon/\varphi^{2})}\) logarithmic length walks from a given vertex approximately determines its cluster membership by the birthday paradox: two vertices whose random walk samples are ’close’ are likely in the same cluster, and otherwise in different clusters.

Testing graph cluster structure has been a central object of study in property testing since the foundational work of Goldreich and Ron [STOC'96] on expansion testing, i.e. the problem of distinguishing between a single cluster (an expander) and a graph that is far from a single cluster. More generally, a $(k, ε)$-clusterable graph $G$ is a graph whose vertex set admits a partition into $k$ induced expanders, each with outer conductance bounded by $ε$. A recent line of work initiated by Czumaj, Peng and Sohler [STOC'15] has shown how to test whether a graph is close to $(k, ε)$-clusterable, and to locally determine which cluster a given vertex belongs to with misclassification rate $\approx ε$, but no sublinear time algorithms for learning the structure of inter-cluster connections are known. As a simple example, can one locally distinguish between the `cluster graph' forming a line and a clique? In this paper, we consider the problem of testing the hierarchical cluster structure of $(k, ε)$-clusterable graphs in sublinear time. Our measure of hierarchical clusterability is the well-established Dasgupta cost, and our main result is an algorithm that approximates Dasgupta cost of a $(k, ε)$-clusterable graph in sublinear time, using a small number of randomly chosen seed vertices for which cluster labels are known. Our main result is an $O(\sqrt{\log k})$ approximation to Dasgupta cost of $G$ in $\approx n^{1/2+O(ε)}$ time using $\approx n^{1/3}$ seeds, effectively giving a sublinear time simulation of the algorithm of Charikar and Chatziafratis [SODA'17] on clusterable graphs. To the best of our knowledge, ours is the first result on approximating the hierarchical clustering properties of such graphs in sublinear time.

We study the maximum weight matching problem in the random-order semi-streaming model and in the robust communication model. Unlike many other sublinear models, in these two frameworks, there is a large gap between the guarantees of the best known algorithms for the unweighted and weighted versions of the problem. In the random-order semi-streaming setting, the edges of an n-vertex graph arrive in a stream in a random order. The goal is to compute an approximate maximum weight matching with a single pass over the stream using O(npolylog n) space. Our main result is a (2/3-ε)-approximation algorithm for maximum weight matching in random-order streams, using space O(n log n log R), where R is the ratio between the heaviest and the lightest edge in the graph. Our result nearly matches the best known unweighted (2/3+ε₀)-approximation (where ε₀ ∼ 10^{-14} is a small constant) achieved by Assadi and Behnezhad [Assadi and Behnezhad, 2021], and significantly improves upon previous weighted results. Our techniques also extend to the related robust communication model, in which the edges of a graph are partitioned randomly between Alice and Bob. Alice sends a single message of size O(npolylog n) to Bob, who must compute an approximate maximum weight matching. We achieve a (5/6-ε)-approximation using O(n log n log R) words of communication, matching the results of Azarmehr and Behnezhad [Azarmehr and Behnezhad, 2023] for unweighted graphs.

In a graph bisection problem, we are given a graph $G$ with two equally-sized unlabeled communities, and the goal is to recover the vertices in these communities. A popular heuristic, known as spectral clustering, is to output an estimated community assignment based on the eigenvector corresponding to the second-smallest eigenvalue of the Laplacian of $G$. Spectral algorithms can be shown to provably recover the cluster structure for graphs generated from probabilistic models, such as the Stochastic Block Model (SBM). However, spectral clustering is known to be non-robust to model mis-specification. Techniques based on semidefinite programming have been shown to be more robust, but they incur significant computational overheads. In this work, we study the robustness of spectral algorithms against semirandom adversaries. Informally, a semirandom adversary is allowed to ``helpfully'' change the specification of the model in a way that is consistent with the ground-truth solution. Our semirandom adversaries in particular are allowed to add edges inside clusters or increase the probability that an edge appears inside a cluster. Semirandom adversaries are a useful tool to determine the extent to which an algorithm has overfit to statistical assumptions on the input. On the positive side, we identify a wide range of semirandom adversaries under which spectral bisection using the _unnormalized_ Laplacian is strongly consistent, i.e., it exactly recovers the planted partitioning. On the negative side, we show that in many of these settings, _normalized_ spectral bisection outputs a partitioning that makes a classification mistake on a constant fraction of the vertices. Finally, we demonstrate numerical experiments that complement our theoretical findings.