Testing correlation of unlabeled random graphs

Wu, Yihong; Xu, Jiaming; Yu, Sophie H.

doi:10.48550/arxiv.2008.10097

Cited by 5 publications

(21 citation statements)

References 42 publications

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“…As the first step, we leverage the previous truncated second moment computation in [WXY20] to conclude that the KL-divergence D(P Q) is negligible under the desired conditions. By expressing the mutual information as I(π; A, B) = n 2 D(P Q) − D(P Q) where D(P Q) = I(P ), this readily implies that I(π; A, B) = n 2 I(P )(1 + o(1)).…”

Section: Negative Results On Partial Recoverymentioning

confidence: 99%

“…This method, however, is often too loose as the second moment can be derailed by rare events. Thus a more robust version is by means of truncated second moment, which has been carried out in [WXY20] to bound TV (P A,B , Q A,B ) for studying the hypothesis testing problem in graph matching. Here we leverage the same result to bound the KL divergence.…”

Section: Proof Of Propositionmentioning

confidence: 99%

“…Recall that the null model is chosen to be Q A,B = P A ⊗ P B . As shown in [WXY20], for both the Gaussian and Erdős-Rényi graph model, it is possible to construct a high-probability event E satisfying the symmetry condition…”

Section: Proof Of Propositionmentioning

confidence: 99%

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Settling the Sharp Reconstruction Thresholds of Random Graph Matching

Wu¹,

Xu²,

Yu³

2021

Preprint

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This paper studies the problem of recovering the hidden vertex correspondence between two edge-correlated random graphs. We focus on the Gaussian model where the two graphs are complete graphs with correlated Gaussian weights and the Erdős-Rényi model where the two graphs are subsampled from a common parent Erdős-Rényi graph G(n, p). For dense graphs with p = n −o(1) , we prove that there exists a sharp threshold, above which one can correctly match all but a vanishing fraction of vertices and below which correctly matching any positive fraction is impossible, a phenomenon known as the "all-or-nothing" phase transition. Even more strikingly, in the Gaussian setting, above the threshold all vertices can be exactly matched with high probability. In contrast, for sparse Erdős-Rényi graphs with p = n −Θ(1) , we show that the all-or-nothing phenomenon no longer holds and we determine the thresholds up to a constant factor. Along the way, we also derive the sharp threshold for exact recovery, sharpening the existing results in Erdős-Rényi graphs [CK16, CK17].The proof of the negative results builds upon a tight characterization of the mutual information based on the truncated second-moment computation in [WXY20] and an "area theorem" that relates the mutual information to the integral of the reconstruction error. The positive results follows from a tight analysis of the maximum likelihood estimator that takes into account the cycle structure of the induced permutation on the edges.

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Section: Negative Results On Partial Recoverymentioning

confidence: 99%

Section: Proof Of Propositionmentioning

confidence: 99%

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Settling the Sharp Reconstruction Thresholds of Random Graph Matching

Wu¹,

Xu²,

Yu³

2021

Preprint

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“…In particular, Theorem 1.1 solves [43, Section 6, Open Problem 3]. It is worth emphasizing that (1.3) for α = 1 with np → ∞ was already proved in [43] (which even allows ǫ → 0 as long as ǫ ≫ n −1/3 ). While our method should also be able to give (1.3) for α = 1, we chose to exclude this case since the assumption α < 1 allows to avoid some technical complications.…”

Section: Introductionmentioning

confidence: 82%

“…Our work is closely related to and much inspired by a recent work [43], where a sharp threshold was established for α = 0 and upper and lower bounds on λ * up to a constant factor were established for α ∈ (0, 1]. In particular, Theorem 1.1 solves [43, Section 6, Open Problem 3].…”

Section: Introductionmentioning

confidence: 93%

Detection threshold for correlated Erdős-Rényi graphs via densest subgraphs

Ding¹,

Du²

2022

Preprint

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The problem of detecting edge correlation between two Erdős-Rényi random graphs on n unlabeled nodes can be formulated as a hypothesis testing problem: under the null hypothesis, the two graphs are sampled independently; under the alternative, the two graphs are independently sub-sampled from a parent graph which is Erdős-Rényi G(n, p) (so that their marginal distributions are the same as the null). We establish a sharp information-theoretic threshold when p = n −α+o(1) for α ∈ (0, 1] which sharpens a constant factor in a recent work by Wu, Xu and Yu. A key novelty in our work is an interesting connection between the detection problem and the densest subgraph of an Erdős-Rényi graph.

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Sharp threshold for alignment of graph databases with Gaussian weights

Ganassali¹

2020

Preprint

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We study the fundamental limits for reconstruction in weighted graph (or matrix) database alignment. We consider a model of two graphs where π * is a planted uniform permutation and all pairs of edge weights (Ai,j, B π * (i),π * (j) ) 1≤i 0, there is an estimator π -namely the MAP estimator -based on the observation of databases A, B that achieves exact reconstruction with high probability. Conversely, if nρ 2 ≤ 4 log n − log log n − ω(1), then any estimator π verifies π = π with probability o(1).This result shows that the information-theoretic threshold for exact recovery is the same as the one obtained for detection in a recent work by Y. Wu, J. Xu and S. Yu [WXY20]: in other words, for Gaussian weighted graph alignment, the problem of reconstruction is not more difficult than that of detection.Though the reconstruction task was already well understood for vector-shaped database alignment (that is taking signal of the form (ui, v π * (i) ) 1≤i≤n where (ui, v π * (i) ) are i.i.d. pairs in R du × R dv ), its formulation for graph (or matrix) databases brings a drastically different problem for which the hard phase is conjectured to be huge.The study is based on the analysis of the MAP estimator, and proofs rely on proper use of the correlation structure of energies of permutations.

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Testing correlation of unlabeled random graphs

Cited by 5 publications

References 42 publications

Settling the Sharp Reconstruction Thresholds of Random Graph Matching

Settling the Sharp Reconstruction Thresholds of Random Graph Matching

Detection threshold for correlated Erdős-Rényi graphs via densest subgraphs

Sharp threshold for alignment of graph databases with Gaussian weights

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