arXiv subjects

Prasad Raghavendra

Publications and source records attributed to Prasad Raghavendra.

At least 19 recordsLinked to original sources

arXiv2026

Improved Certificates for Independence Number in Semirandom Hypergraphs

We study the problem of efficiently certifying upper bounds on independence number of $\ell$-uniform hypergraphs in semirandom models. This is a notoriously hard problem, with efficient algorithms failing to approximate the independence number within an $n^{1-ε}$ factor in worst-case. A folklore reduction to graph case yields a weak $O(\sqrt{n/p})$ bound, and spectral certificates[GKM22] achieve $O(\sqrt{n}.polylog(n)/p^{2/\ell})$. In this work, we prove sharper bounds that eliminate logarithmic factors in $n$ and nearly attain the optimal threshold of $O(\sqrt{n}/p^{1/\ell})$. We also show matching low-degree polynomial lower bounds. Our certificates are designed using the proofs-to-algorithms paradigm via degree-$2\ell$ Sum-of-Squares(SoS) relaxation. The technically challenging case is odd-arity hypergraphs, where we employ a tensor-based analysis reducing the problem to bounding operator norm of random chaos matrices. Previous bounds[AMP21,RT23] have a logarithmic dependence, which we remove using recent matrix concentration inequalities[BBvH23,BLNvH25]; we believe this maybe useful in other hypergraph problems. Since we deploy our certificates in SoS framework, the bounds continue to hold for monotone adversaries. Additionally, we construct a 'quiet' planted distribution supported on independent sets of size $k=o(\sqrt{n}/p^{1/\ell})$ that is low-degree indistinguishable from random hypergraphs. Prior to this work, the problem of constructing a quiet planted distribution in sparse regimes was open even for graphs[JPR+22,Pot22]. This is in contrast to recovering a planted independent set, where the threshold is $k \gtrsim \sqrt{n}/p^{1/2(\ell-1)}$ (matching lower bounds in a concurrent work[FS26]). As application, our certificates combine with an SoS relaxation of an r-coloring system to recover a planted r-colorable subhypergraph under strong adversaries of [LPR25].

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arXiv2025

The communication complexity of distributed estimation

We study an extension of the standard two-party communication model in which Alice and Bob hold probability distributions $p$ and $q$ over domains $X$ and $Y$, respectively. Their goal is to estimate \[ \mathbb{E}_{x \sim p,\, y \sim q}[f(x, y)] \] to within additive error $\varepsilon$ for a bounded function $f$, known to both parties. We refer to this as the distributed estimation problem. Special cases of this problem arise in a variety of areas including sketching, databases and learning. Our goal is to understand how the required communication scales with the communication complexity of $f$ and the error parameter $\varepsilon$. The random sampling approach -- estimating the mean by averaging $f$ over $O(1/\varepsilon^2)$ random samples -- requires $O(R(f)/\varepsilon^2)$ total communication, where $R(f)$ is the randomized communication complexity of $f$. We design a new debiasing protocol which improves the dependence on $1/\varepsilon$ to be linear instead of quadratic. Additionally we show better upper bounds for several special classes of functions, including the Equality and Greater-than functions. We introduce lower bound techniques based on spectral methods and discrepancy, and show the optimality of many of our protocols: the debiasing protocol is tight for general functions, and that our protocols for the equality and greater-than functions are also optimal. Furthermore, we show that among full-rank Boolean functions, Equality is essentially the easiest.

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arXiv2025

On optimal distinguishers for Planted Clique

In a distinguishing problem, the input is a sample drawn from one of two distributions and the algorithm is tasked with identifying the source distribution. The performance of a distinguishing algorithm is measured by its advantage, i.e., its incremental probability of success over a random guess. A classic example of a distinguishing problem is the Planted Clique problem, where the input is a graph sampled from either $G(n,1/2)$ -- the standard Erdős-Rényi model, or $G(n,1/2,k)$ -- the Erdős-Rényi model with a clique planted on a random subset of $k$ vertices. The Planted Clique Hypothesis asserts that efficient algorithms cannot achieve advantage better than some absolute constant, say $1/4$, whenever $k=n^{1/2-Ω(1)}$. In this work, we aim to precisely understand the optimal distinguishing advantage achievable by efficient algorithms on Planted Clique. We show the following results under the Planted Clique hypothesis: 1. Optimality of low-degree polynomials: No efficient algorithm can beat the advantage the optimal low-degree polynomial. Concretely, this means that the advantage of any efficient algorithm is at most $(1+o(1))\cdot k^2/(\sqrtπn)$, which is optimal in light of a simple edge-counting algorithm achieving this bound. 2. Harder planted distributions: There is an efficiently sampleable distribution $\mathcal{P}^*$ supported on graphs containing $k$-cliques such that no efficient algorithm can distinguish $\mathcal{P}^*$ from $G(n,1/2)$ with advantage $n^{-d}$ for an arbitrarily large constant $d$. In other words, there exist alternate planted distributions that are much harder than $G(n,1/2,k)$. Along the way, we prove a constructive hard-core lemma for a broad class of distributions with respect to low-degree polynomials. This result is applicable much more widely beyond Planted Clique and might be of independent interest.

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arXiv2025

Locally Stationary Distributions: A Framework for Analyzing Slow-Mixing Markov Chains

Many natural Markov chains fail to mix to their stationary distribution in polynomially many steps. Often, this slow mixing is inevitable since it is computationally intractable to sample from their stationary measure. Nevertheless, Markov chains can be shown to always converge quickly to measures that are locally stationary, i.e., measures that don't change over a small number of steps. These locally stationary measures are analogous to local minima in continuous optimization, while stationary measures correspond to global minima. While locally stationary measures can be statistically far from stationary measures, do they enjoy provable theoretical guarantees that have algorithmic implications? We study this question in this work and demonstrate three algorithmic applications of locally stationary measures: 1. We show that Glauber dynamics on the hardcore model can be used to find independent sets of size $Ω\left(\frac{\log d}{d} \cdot n\right)$ in triangle-free graphs of degree at most $d$. 2. Let $W$ be a symmetric real matrix with bounded spectral diameter and $v$ be a unit vector. Given the matrix $M = λvv^\top + W$ with a planted rank-one spike along vector $v$, for sufficiently large constant $λ$, Glauber dynamics on the Ising model defined by $M$ samples vectors $x \in \{\pm 1\}^n$ that have constant correlation with the vector $v$. 3. Let $M = A_{\mathbf{G}} - \frac{d}{n}\mathbf{1}\mathbf{1}^\top$ be a centered version of the adjacency matrix where the graph $\mathbf{G}$ is drawn from a sparse 2-community stochastic block model. We show that for sufficiently large constant signal-to-noise ratio, Glauber dynamics on the Ising model defined by $M$ samples vectors $x \in \{\pm 1\}^n$ that have constant correlation with the hidden community vector $\mathbfσ$.

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arXiv2024

Certifying Euclidean Sections and Finding Planted Sparse Vectors Beyond the $\sqrt{n}$ Dimension Threshold

We consider the task of certifying that a random $d$-dimensional subspace $X$ in $\mathbb{R}^n$ is well-spread - every vector $x \in X$ satisfies $c\sqrt{n} \|x\|_2 \leq \|x\|_1 \leq \sqrt{n}\|x\|_2$. In a seminal work, Barak et. al. showed a polynomial-time certification algorithm when $d \leq O(\sqrt{n})$. On the other hand, when $d \gg \sqrt{n}$, the certification task is information-theoretically possible but there is evidence that it is computationally hard [MW21,Cd22], a phenomenon known as the information-computation gap. In this paper, we give subexponential-time certification algorithms in the $d \gg \sqrt{n}$ regime. Our algorithm runs in time $\exp(\widetilde{O}(n^{\varepsilon}))$ when $d \leq \widetilde{O}(n^{(1+\varepsilon)/2})$, establishing a smooth trade-off between runtime and the dimension. Our techniques naturally extend to the related planted problem, where the task is to recover a sparse vector planted in a random subspace. Our algorithm achieves the same runtime and dimension trade-off for this task.

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arXiv2024

Robust recovery for stochastic block models, simplified and generalized

We study the problem of $\textit{robust community recovery}$: efficiently recovering communities in sparse stochastic block models in the presence of adversarial corruptions. In the absence of adversarial corruptions, there are efficient algorithms when the $\textit{signal-to-noise ratio}$ exceeds the $\textit{Kesten--Stigum (KS) threshold}$, widely believed to be the computational threshold for this problem. The question we study is: does the computational threshold for robust community recovery also lie at the KS threshold? We answer this question affirmatively, providing an algorithm for robust community recovery for arbitrary stochastic block models on any constant number of communities, generalizing the work of Ding, d'Orsi, Nasser & Steurer on an efficient algorithm above the KS threshold in the case of $2$-community block models. There are three main ingredients to our work: (i) The Bethe Hessian of the graph is defined as $H_G(t) \triangleq (D_G-I)t^2 - A_Gt + I$ where $D_G$ is the diagonal matrix of degrees and $A_G$ is the adjacency matrix. Empirical work suggested that the Bethe Hessian for the stochastic block model has outlier eigenvectors corresponding to the communities right above the Kesten-Stigum threshold. We formally confirm the existence of outlier eigenvalues for the Bethe Hessian, by explicitly constructing outlier eigenvectors from the community vectors. (ii) We develop an algorithm for a variant of robust PCA on sparse matrices. Specifically, an algorithm to partially recover top eigenspaces from adversarially corrupted sparse matrices under mild delocalization constraints. (iii) A rounding algorithm to turn vector assignments of vertices into a community assignment, inspired by the algorithm of Charikar \& Wirth \cite{CW04} for $2$XOR.

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arXiv2024

Omnipredictors for Regression and the Approximate Rank of Convex Functions

Consider the supervised learning setting where the goal is to learn to predict labels $\mathbf y$ given points $\mathbf x$ from a distribution. An \textit{omnipredictor} for a class $\mathcal L$ of loss functions and a class $\mathcal C$ of hypotheses is a predictor whose predictions incur less expected loss than the best hypothesis in $\mathcal C$ for every loss in $\mathcal L$. Since the work of [GKR+21] that introduced the notion, there has been a large body of work in the setting of binary labels where $\mathbf y \in \{0, 1\}$, but much less is known about the regression setting where $\mathbf y \in [0,1]$ can be continuous. Our main conceptual contribution is the notion of \textit{sufficient statistics} for loss minimization over a family of loss functions: these are a set of statistics about a distribution such that knowing them allows one to take actions that minimize the expected loss for any loss in the family. The notion of sufficient statistics relates directly to the approximate rank of the family of loss functions. Our key technical contribution is a bound of $O(1/\varepsilon^{2/3})$ on the $ε$-approximate rank of convex, Lipschitz functions on the interval $[0,1]$, which we show is tight up to a factor of $\mathrm{polylog} (1/ε)$. This yields improved runtimes for learning omnipredictors for the class of all convex, Lipschitz loss functions under weak learnability assumptions about the class $\mathcal C$. We also give efficient omnipredictors when the loss families have low-degree polynomial approximations, or arise from generalized linear models (GLMs). This translation from sufficient statistics to faster omnipredictors is made possible by lifting the technique of loss outcome indistinguishability introduced by [GKH+23] for Boolean labels to the regression setting.

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arXiv2022

Noise stability on the Boolean hypercube via a renormalized Brownian motion

We consider a variant of the classical notion of noise on the Boolean hypercube which gives rise to a new approach to inequalities regarding noise stability. We use this approach to give a new proof of the Majority is Stablest theorem by Mossel, O'Donnell, and Oleszkiewicz, improving the dependence of the bound on the maximal influence of the function from logarithmic to polynomial. We also show that a variant of the conjecture by Courtade and Kumar regarding the most informative Boolean function, where the classical noise is replaced by our notion, holds true. Our approach is based on a stochastic construction that we call the renormalized Brownian motion, which facilitates the use of inequalities in Gaussian space in the analysis of Boolean functions.

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arXiv2021

Matrix Discrepancy from Quantum Communication

We develop a novel connection between discrepancy minimization and (quantum) communication complexity. As an application, we resolve a substantial special case of the Matrix Spencer conjecture. In particular, we show that for every collection of symmetric $n \times n$ matrices $A_1,\ldots,A_n$ with $\|A_i\| \leq 1$ and $\|A_i\|_F \leq n^{1/4}$ there exist signs $x \in \{ \pm 1\}^n$ such that the maximum eigenvalue of $\sum_{i \leq n} x_i A_i$ is at most $O(\sqrt n)$. We give a polynomial-time algorithm based on partial coloring and semidefinite programming to find such $x$. Our techniques open a new avenue to use tools from communication complexity and information theory to study discrepancy. The proof of our main result combines a simple compression scheme for transcripts of repeated (quantum) communication protocols with quantum state purification, the Holevo bound from quantum information, and tools from sketching and dimensionality reduction. Our approach also offers a promising avenue to resolve the Matrix Spencer conjecture completely -- we show it is implied by a natural conjecture in quantum communication complexity.

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arXiv2021

On statistical inference when fixed points of belief propagation are unstable

Many statistical inference problems correspond to recovering the values of a set of hidden variables from sparse observations on them. For instance, in a planted constraint satisfaction problem such as planted 3-SAT, the clauses are sparse observations from which the hidden assignment is to be recovered. In the problem of community detection in a stochastic block model, the community labels are hidden variables that are to be recovered from the edges of the graph. Inspired by ideas from statistical physics, the presence of a stable fixed point for belief propogation has been widely conjectured to characterize the computational tractability of these problems. For community detection in stochastic block models, many of these predictions have been rigorously confirmed. In this work, we consider a general model of statistical inference problems that includes both community detection in stochastic block models, and all planted constraint satisfaction problems as special cases. We carry out the cavity method calculations from statistical physics to compute the regime of parameters where detection and recovery should be algorithmically tractable. At precisely the predicted tractable regime, we give: (i) a general polynomial-time algorithm for the problem of detection: distinguishing an input with a planted signal from one without; (ii) a general polynomial-time algorithm for the problem of recovery: outputting a vector that correlates with the hidden assignment significantly better than a random guess would.

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arXiv2020

Local Statistics, Semidefinite Programming, and Community Detection

We propose a new hierarchy of semidefinite programming relaxations for inference problems. As test cases, we consider the problem of community detection in block models. The vertices are partitioned into $k$ communities, and a graph is sampled conditional on a prescribed number of inter- and intra-community edges. The problem of detection, where we are to decide with high probability whether a graph was drawn from this model or the uniform distribution on regular graphs, is conjectured to undergo a computational phase transition at a point called the Kesten-Stigum (KS) threshold. In this work, we consider two models of random graphs namely the well-studied (irregular) stochastic block model and a distribution over random regular graphs we'll call the Degree Regular Block Model. For both these models, we show that sufficiently high constant levels of our hierarchy can perform detection arbitrarily close to the KS threshold and that our algorithm is robust to up to a linear number of adversarial edge perturbations. Furthermore, in the case of Degree Regular Block Model (DRBM), we show that below the Kesten-Stigum threshold no constant level can do so. In the case of the (irregular) Stochastic Block Model, it is known that efficient algorithms exist all the way down to this threshold, although none are robust to a linear number of adversarial perturbations of the graph when the average degree is small. More importantly, there is little complexity-theoretic evidence that detection is hard below the threshold. In the DRBM with more than two groups, it has not to our knowledge been proven that any algorithm succeeds down to the KS threshold, let alone that one can do so robustly, and there is a similar dearth of evidence for hardness below this point.

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arXiv2020

List Decodable Subspace Recovery

Learning from data in the presence of outliers is a fundamental problem in statistics. In this work, we study robust statistics in the presence of overwhelming outliers for the fundamental problem of subspace recovery. Given a dataset where an $α$ fraction (less than half) of the data is distributed uniformly in an unknown $k$ dimensional subspace in $d$ dimensions, and with no additional assumptions on the remaining data, the goal is to recover a succinct list of $O(\frac{1}α)$ subspaces one of which is nontrivially correlated with the planted subspace. We provide the first polynomial time algorithm for the 'list decodable subspace recovery' problem, and subsume it under a more general framework of list decoding over distributions that are "certifiably resilient" capturing state of the art results for list decodable mean estimation and regression.

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arXiv2019

Algorithms for Heavy-Tailed Statistics: Regression, Covariance Estimation, and Beyond

We study efficient algorithms for linear regression and covariance estimation in the absence of Gaussian assumptions on the underlying distributions of samples, making assumptions instead about only finitely-many moments. We focus on how many samples are needed to do estimation and regression with high accuracy and exponentially-good success probability. For covariance estimation, linear regression, and several other problems, estimators have recently been constructed with sample complexities and rates of error matching what is possible when the underlying distribution is Gaussian, but algorithms for these estimators require exponential time. We narrow the gap between the Gaussian and heavy-tailed settings for polynomial-time estimators with: 1. A polynomial-time estimator which takes $n$ samples from a random vector $X \in R^d$ with covariance $Σ$ and produces $\hatΣ$ such that in spectral norm $\|\hatΣ - Σ\|_2 \leq \tilde{O}(d^{3/4}/\sqrt{n})$ w.p. $1-2^{-d}$. The information-theoretically optimal error bound is $\tilde{O}(\sqrt{d/n})$; previous approaches to polynomial-time algorithms were stuck at $\tilde{O}(d/\sqrt{n})$. 2. A polynomial-time algorithm which takes $n$ samples $(X_i,Y_i)$ where $Y_i = \langle u,X_i \rangle + \varepsilon_i$ and produces $\hat{u}$ such that the loss $\|u - \hat{u}\|^2 \leq O(d/n)$ w.p. $1-2^{-d}$ for any $n \geq d^{3/2} \log(d)^{O(1)}$. This (information-theoretically optimal) error is achieved by inefficient algorithms for any $n \gg d$; previous polynomial-time algorithms suffer loss $Ω(d^2/n)$ and require $n \gg d^2$. Our algorithms use degree-$8$ sum-of-squares semidefinite programs. We offer preliminary evidence that improving these rates of error in polynomial time is not possible in the median of means framework our algorithms employ.

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arXiv2019

Extended Formulation Lower Bounds for Refuting Random CSPs

Random constraint satisfaction problems (CSPs) such as random $3$-SAT are conjectured to be computationally intractable. The average case hardness of random $3$-SAT and other CSPs has broad and far-reaching implications on problems in approximation, learning theory and cryptography. In this work, we show subexponential lower bounds on the size of linear programming relaxations for refuting random instances of constraint satisfaction problems. Formally, suppose $P : \{0,1\}^k \to \{0,1\}$ is a predicate that supports a $t-1$-wise uniform distribution on its satisfying assignments. Consider the distribution of random instances of CSP $P$ with $m = Δn$ constraints. We show that any linear programming extended formulation that can refute instances from this distribution with constant probability must have size at least $Ω\left(\exp\left(\left(\frac{n^{t-2}}{Δ^2}\right)^{\frac{1-ν}{k}}\right)\right)$ for all $ν> 0$. For example, this yields a lower bound of size $\exp(n^{1/3})$ for random $3$-SAT with a linear number of clauses. We use the technique of pseudocalibration to directly obtain extended formulation lower bounds from the planted distribution. This approach bypasses the need to construct Sherali-Adams integrality gaps in proving general LP lower bounds. As a corollary, one obtains a self-contained proof of subexponential Sherali-Adams LP lower bounds for these problems. We believe the result sheds light on the technique of pseudocalibration, a promising but conjectural approach to LP/SDP lower bounds.

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arXiv2019

Lifting Sum-of-Squares Lower Bounds: Degree-$2$ to Degree-$4$

The degree-$4$ Sum-of-Squares (SoS) SDP relaxation is a powerful algorithm that captures the best known polynomial time algorithms for a broad range of problems including MaxCut, Sparsest Cut, all MaxCSPs and tensor PCA. Despite being an explicit algorithm with relatively low computational complexity, the limits of degree-$4$ SoS SDP are not well understood. For example, existing integrality gaps do not rule out a $(2-\varepsilon)$-algorithm for Vertex Cover or a $(0.878+\varepsilon)$-algorithm for MaxCut via degree-$4$ SoS SDPs, each of which would refute the notorious Unique Games Conjecture. We exhibit an explicit mapping from solutions for degree-$2$ Sum-of-Squares SDP (Goemans-Williamson SDP) to solutions for the degree-$4$ Sum-of-Squares SDP relaxation on boolean variables. By virtue of this mapping, one can lift lower bounds for degree-$2$ SoS SDP relaxation to corresponding lower bounds for degree-$4$ SoS SDPs. We use this approach to obtain degree-$4$ SoS SDP lower bounds for MaxCut on random $d$-regular graphs, Sherington-Kirkpatrick model from statistical physics and PSD Grothendieck problem. Our constructions use the idea of pseudocalibration towards candidate SDP vectors, while it was previously only used to produce the candidate matrix which one would show is PSD using much technical work. In addition, we develop a different technique to bound the spectral norms of _graphical matrices_ that arise in the context of SoS SDPs. The technique is much simpler and yields better bounds in many cases than the _trace method_ -- which was the sole technique for this purpose.

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arXiv2019

High-dimensional estimation via sum-of-squares proofs

Estimation is the computational task of recovering a hidden parameter $x$ associated with a distribution $D_x$, given a measurement $y$ sampled from the distribution. High dimensional estimation problems arise naturally in statistics, machine learning, and complexity theory. Many high dimensional estimation problems can be formulated as systems of polynomial equations and inequalities, and thus give rise to natural probability distributions over polynomial systems. Sum-of-squares proofs provide a powerful framework to reason about polynomial systems, and further there exist efficient algorithms to search for low-degree sum-of-squares proofs. Understanding and characterizing the power of sum-of-squares proofs for estimation problems has been a subject of intense study in recent years. On one hand, there is a growing body of work utilizing sum-of-squares proofs for recovering solutions to polynomial systems when the system is feasible. On the other hand, a general technique referred to as pseudocalibration has been developed towards showing lower bounds on the degree of sum-of-squares proofs. Finally, the existence of sum-of-squares refutations of a polynomial system has been shown to be intimately connected to the existence of spectral algorithms. In this article we survey these developments.

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arXiv2019

List Decodable Learning via Sum of Squares

In the list-decodable learning setup, an overwhelming majority (say a $1-β$-fraction) of the input data consists of outliers and the goal of an algorithm is to output a small list $\mathcal{L}$ of hypotheses such that one of them agrees with inliers. We develop a framework for list-decodable learning via the Sum-of-Squares SDP hierarchy and demonstrate it on two basic statistical estimation problems {\it Linear regression:} Suppose we are given labelled examples $\{(X_i,y_i)\}_{i \in [N]}$ containing a subset $S$ of $βN$ {\it inliers} $\{X_i \}_{i \in S}$ that are drawn i.i.d. from standard Gaussian distribution $N(0,I)$ in $\mathbb{R}^d$, where the corresponding labels $y_i$ are well-approximated by a linear function $\ell$. We devise an algorithm that outputs a list $\mathcal{L}$ of linear functions such that there exists some $\hat{\ell} \in \mathcal{L}$ that is close to $\ell$. This yields the first algorithm for linear regression in a list-decodable setting. Our results hold for any distribution of examples whose concentration and anticoncentration can be certified by Sum-of-Squares proofs. {\it Mean Estimation:} Given data points $\{X_i\}_{i \in [N]}$ containing a subset $S$ of $βN$ {\it inliers} $\{X_i \}_{i \in S}$ that are drawn i.i.d. from a Gaussian distribution $N(μ,I)$ in $\mathbb{R}^d$, we devise an algorithm that generates a list $\mathcal{L}$ of means such that there exists $\hatμ \in \mathcal{L}$ close to $μ$. The recovery guarantees of the algorithm are analogous to the existing algorithms for the problem by Diakonikolas \etal and Kothari \etal. In an independent and concurrent work, Karmalkar \etal \cite{KlivansKS19} also obtain an algorithm for list-decodable linear regression using the Sum-of-Squares SDP hierarchy.

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arXiv2018

Average whenever you meet: Opportunistic protocols for community detection

Consider the following asynchronous, opportunistic communication model over a graph $G$: in each round, one edge is activated uniformly and independently at random and (only) its two endpoints can exchange messages and perform local computations. Under this model, we study the following random process: The first time a vertex is an endpoint of an active edge, it chooses a random number, say $\pm 1$ with probability $1/2$; then, in each round, the two endpoints of the currently active edge update their values to their average. We show that, if $G$ exhibits a two-community structure (for example, two expanders connected by a sparse cut), the values held by the nodes will collectively reflect the underlying community structure over a suitable phase of the above process, allowing efficient and effective recovery in important cases. In more detail, we first provide a first-moment analysis showing that, for a large class of almost-regular clustered graphs that includes the stochastic block model, the expected values held by all but a negligible fraction of the nodes eventually reflect the underlying cut signal. We prove this property emerges after a mixing period of length $\mathcal O(n\log n)$. We further provide a second-moment analysis for a more restricted class of regular clustered graphs that includes the regular stochastic block model. For this case, we are able to show that most nodes can efficiently and locally identify their community of reference over a suitable time window. This results in the first opportunistic protocols that approximately recover community structure using only polylogarithmic work per node. Even for the above class of regular graphs, our second moment analysis requires new concentration bounds on the product of certain random matrices that are technically challenging and possibly of independent interest.

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