arXiv subjects

Miklós Z. Rácz

Publications and source records attributed to Miklós Z. Rácz.

15 recordsLinked to original sources

arXiv2026

On the largest common subtree of uniform attachment trees

We study the largest common subtree of two independent unlabeled uniform attachment trees (also known as random recursive trees). Our main result shows that, when the two trees have $n$ vertices each, their largest common subtree has at least $n^{0.83}$ vertices with high probability. This is obtained by starting with the common subtree induced by the Ulam--Harris labels in the two trees and improving using local optimization steps. We also give some upper bounds and bounds for general random tree growth models. We leave as an intriguing open question to understand the magnitude of the size of the largest common subtree.

math.PR↗

arXiv2026

Correlated uniform attachment trees

We introduce and study a new model of correlated uniform attachment (UA) trees, where correlation is sprinkled throughout the time evolution of the process. In this model, two UA trees are grown in parallel, and at each time step a new node is added to each tree, with an edge between it and a uniformly chosen existing vertex in the respective tree. The two choices of attachment are correlated: with probability $α$, the edges attach to nodes with the same time label in both trees, and with probability $1-α$, the choices are made independently. We study fundamental detection and estimation questions for this model, given two \emph{unlabeled} trees. In our main result, we construct a consistent estimator of the correlation parameter $α$, as the size of the trees goes to infinity. The construction of our statistic relies on two key ideas. First, we use Jordan centrality to identify subsets of vertices of each tree whose intersection has a sufficient number of common early vertices. The second idea is that, across multiple time scales, it is possible to approximately determine the labels of vertices that have attached to these early vertices, using the sizes of fringe subtrees. Our analysis includes novel quantitative bounds on the fraction of early vertices that remain central, which are of independent interest in the network archaeology literature.

math.PR↗

arXiv2026

Optimal Hardness of Online Algorithms for Large Common Induced Subgraphs

We study the problem of efficiently finding large common induced subgraphs of two independent Erdős--Rényi random graphs $G_1, G_2 \sim \mathbb{G}(n,1/2)$. Recently, Chatterjee and Diaconis showed that the largest common induced subgraph of $G_1$ and $G_2$ has size $(4-o(1))\log_2 n$ with high probability. We first show that a simple greedy online algorithm finds a common induced subgraph of $G_1$ and $G_2$ of size $(2-o(1)) \log_2 n$ with high probability. Our main result shows that no online algorithm can find a common induced subgraph of $G_1$ and $G_2$ of size at least $(2+\varepsilon) \log_2 n$ with probability bounded away from $0$ as $n \to \infty$. Together, these results provide evidence that this problem exhibits a computation-to-optimization gap. To prove the impossibility result, we show that the solution space of the problem exhibits a version of the (multi) overlap gap property (OGP), and utilize an interpolation argument recently developed by Gamarnik, Kizildağ, and Warnke that connects OGP and online algorithms.

cs.DS↗

arXiv2026

The statistical threshold for planted matchings and spanning trees

In this paper, we study the problem of detecting the presence of a planted perfect matching or spanning tree in an Erdős--Rényi random graph. More precisely, we study the hypothesis testing problem where the statistician observes a graph on $n$ vertices. Under the null hypothesis, the graph is a realization of an Erdős--Rényi random graph $G(n,q)$, while under the alternative hypothesis, the graph is the union of an Erdős--Rényi random graph and a random perfect matching (or random spanning tree). In order to avoid trivial detection by counting edges, we adjust the alternative hypothesis so that the expected number of edges under both distributions coincides. We prove that in both problems, when $q\gg n^{-1/2}$, no test can perform better than random guessing, while for $q\ll n^{-1/2}$, there exist computationally efficient tests that guess correctly with high probability.

math.ST↗

arXiv2024

Efficient Graph Matching for Correlated Stochastic Block Models

We study learning problems on correlated stochastic block models with two balanced communities. Our main result gives the first efficient algorithm for graph matching in this setting. In the most interesting regime where the average degree is logarithmic in the number of vertices, this algorithm correctly matches all but a vanishing fraction of vertices with high probability, whenever the edge correlation parameter $s$ satisfies $s^2 > α\approx 0.338$, where $α$ is Otter's tree-counting constant. Moreover, we extend this to an efficient algorithm for exact graph matching whenever this is information-theoretically possible, positively resolving an open problem of Rácz and Sridhar (NeurIPS 2021). Our algorithm generalizes the recent breakthrough work of Mao, Wu, Xu, and Yu (STOC 2023), which is based on centered subgraph counts of a large family of trees termed chandeliers. A major technical challenge that we overcome is dealing with the additional estimation errors that are necessarily present due to the fact that, in relevant parameter regimes, the latent community partition cannot be exactly recovered from a single graph. As an application of our results, we give an efficient algorithm for exact community recovery using multiple correlated graphs in parameter regimes where it is information-theoretically impossible to do so using just a single graph.

cs.DS↗

arXiv2024

Harnessing Multiple Correlated Networks for Exact Community Recovery

We study the problem of learning latent community structure from multiple correlated networks, focusing on edge-correlated stochastic block models with two balanced communities. Recent work of Gaudio, Rácz, and Sridhar (COLT 2022) determined the precise information-theoretic threshold for exact community recovery using two correlated graphs; in particular, this showcased the subtle interplay between community recovery and graph matching. Here we study the natural setting of more than two graphs. The main challenge lies in understanding how to aggregate information across several graphs when none of the pairwise latent vertex correspondences can be exactly recovered. Our main result derives the precise information-theoretic threshold for exact community recovery using any constant number of correlated graphs, answering a question of Gaudio, Rácz, and Sridhar (COLT 2022). In particular, for every $K \geq 3$ we uncover and characterize a region of the parameter space where exact community recovery is possible using $K$ correlated graphs, even though (1) this is information-theoretically impossible using any $K-1$ of them and (2) none of the latent matchings can be exactly recovered.

math.ST↗

arXiv2023

Average-case and smoothed analysis of graph isomorphism

We propose a simple and efficient local algorithm for graph isomorphism which succeeds for a large class of sparse graphs. This algorithm produces a low-depth canonical labeling, which is a labeling of the vertices of the graph that identifies its isomorphism class using vertices' local neighborhoods. Prior work by Czajka and Pandurangan showed that the degree profile of a vertex (i.e., the sorted list of the degrees of its neighbors) gives a canonical labeling with high probability when $n p_n = ω( \log^{4}(n) / \log \log n )$ (and $p_{n} \leq 1/2$); subsequently, Mossel and Ross showed that the same holds when $n p_n = ω( \log^{2}(n) )$. We first show that their analysis essentially cannot be improved: we prove that when $n p_n = o( \log^{2}(n) / (\log \log n)^{3} )$, with high probability there exist distinct vertices with isomorphic $2$-neighborhoods. Our first main result is a positive counterpart to this, showing that $3$-neighborhoods give a canonical labeling when $n p_n \geq (1+δ) \log n$ (and $p_n \leq 1/2$); this improves a recent result of Ding, Ma, Wu, and Xu, completing the picture above the connectivity threshold. Our second main result is a smoothed analysis of graph isomorphism, showing that for a large class of deterministic graphs, a small random perturbation ensures that $3$-neighborhoods give a canonical labeling with high probability. While the worst-case complexity of graph isomorphism is still unknown, this shows that graph isomorphism has polynomial smoothed complexity.

math.PR↗

arXiv2023

Matching Correlated Inhomogeneous Random Graphs using the $k$-core Estimator

We consider the task of estimating the latent vertex correspondence between two edge-correlated random graphs with generic, inhomogeneous structure. We study the so-called \emph{$k$-core estimator}, which outputs a vertex correspondence that induces a large, common subgraph of both graphs which has minimum degree at least $k$. We derive sufficient conditions under which the $k$-core estimator exactly or partially recovers the latent vertex correspondence. Finally, we specialize our general framework to derive new results on exact and partial recovery in correlated stochastic block models, correlated Chung-Lu graphs, and correlated random geometric graphs.

math.ST↗

arXiv2021

Tree trace reconstruction using subtraces

Tree trace reconstruction aims to learn the binary node labels of a tree, given independent samples of the tree passed through an appropriately defined deletion channel. In recent work, Davies, Rácz, and Rashtchian used combinatorial methods to show that $\exp(\mathcal{O}(k \log_{k} n))$ samples suffice to reconstruct a complete $k$-ary tree with $n$ nodes with high probability. We provide an alternative proof of this result, which allows us to generalize it to a broader class of tree topologies and deletion models. In our proofs, we introduce the notion of a subtrace, which enables us to connect with and generalize recent mean-based complex analytic algorithms for string trace reconstruction.

cs.DS↗

arXiv2020

Finding a planted clique by adaptive probing

We consider a variant of the planted clique problem where we are allowed unbounded computational time but can only investigate a small part of the graph by adaptive edge queries. We determine (up to logarithmic factors) the number of queries necessary both for detecting the presence of a planted clique and for finding the planted clique. Specifically, let $G \sim G(n,1/2,k)$ be a random graph on $n$ vertices with a planted clique of size $k$. We show that no algorithm that makes at most $q = o(n^2 / k^2 + n)$ adaptive queries to the adjacency matrix of $G$ is likely to find the planted clique. On the other hand, when $k \geq (2+ε) \log_2 n$ there exists a simple algorithm (with unbounded computational power) that finds the planted clique with high probability by making $q = O( (n^2 / k^2) \log^2 n + n \log n)$ adaptive queries. For detection, the additive $n$ term is not necessary: the number of queries needed to detect the presence of a planted clique is $n^2 / k^2$ (up to logarithmic factors).

math.CO↗

arXiv2018

Finding cliques using few probes

Consider algorithms with unbounded computation time that probe the entries of the adjacency matrix of an $n$ vertex graph, and need to output a clique. We show that if the input graph is drawn at random from $G_{n,\frac{1}{2}}$ (and hence is likely to have a clique of size roughly $2\log n$), then for every $δ< 2$ and constant $\ell$, there is an $α< 2$ (that may depend on $δ$ and $\ell$) such that no algorithm that makes $n^δ$ probes in $\ell$ rounds is likely (over the choice of the random graph) to output a clique of size larger than $α\log n$.

math.CO↗

arXiv2015

Multidimensional sticky Brownian motions as limits of exclusion processes

We study exclusion processes on the integer lattice in which particles change their velocities due to stickiness. Specifically, whenever two or more particles occupy adjacent sites, they stick together for an extended period of time, and the entire particle system is slowed down until the ``collision'' is resolved. We show that under diffusive scaling of space and time such processes converge to what one might refer to as a sticky reflected Brownian motion in the wedge. The latter behaves as a Brownian motion with constant drift vector and diffusion matrix in the interior of the wedge, and reflects at the boundary of the wedge after spending an instant of time there. In particular, this leads to a natural multidimensional generalization of sticky Brownian motion on the half-line, which is of interest in both queuing theory and stochastic portfolio theory. For instance, this can model a market, which experiences a slowdown due to a major event (such as a court trial between some of the largest firms in the market) deciding about the new market leader.

math.PR↗

arXiv2014

From trees to seeds: on the inference of the seed from large trees in the uniform attachment model

We study the influence of the seed in random trees grown according to the uniform attachment model, also known as uniform random recursive trees. We show that different seeds lead to different distributions of limiting trees from a total variation point of view. To do this, we construct statistics that measure, in a certain well-defined sense, global "balancedness" properties of such trees. Our paper follows recent results on the same question for the preferential attachment model.

math.PR↗

arXiv2014

Can one hear the shape of a population history?

Reconstructing past population size from present day genetic data is a major goal of population genetics. Recent empirical studies infer population size history using coalescent-based models applied to a small number of individuals. Here we provide tight bounds on the amount of exact coalescence time data needed to recover the population size history of a single, panmictic population at a certain level of accuracy. In practice, coalescence times are estimated from sequence data and so our lower bounds should be taken as rather conservative.

q-bio.PE↗

arXiv2014

On the influence of the seed graph in the preferential attachment model

We study the influence of the seed graph in the preferential attachment model, focusing on the case of trees. We first show that the seed has no effect from a weak local limit point of view. On the other hand, we conjecture that different seeds lead to different distributions of limiting trees from a total variation point of view. We take a first step in proving this conjecture by showing that seeds with different degree profiles lead to different limiting distributions for the (appropriately normalized) maximum degree, implying that such seeds lead to different (in total variation) limiting trees.

math.PR↗