The Geometric Block Model and Applications

Galhotra, Sainyam; Pal, Soumyabrata; Mazumdar, Arya; Saha, Barna

doi:10.1109/allerton.2018.8635938

Cited by 7 publications

(27 citation statements)

References 31 publications

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“…This model captures stochastic block models as well as more complex models, e.g. random geometric graphs, where each vertex corresponds to a point in a metric space (selected randomly according to a particular distribution) and vertices share an edge if their points are sufficiently close [26,20,48,24].…”

Section: Introductionmentioning

confidence: 99%

Revealing Network Structure, Confidentially: Improved Rates for Node-Private Graphon Estimation

Borgs

Chayes

Smith

et al. 2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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Motivated by growing concerns over ensuring privacy on social networks, we develop new algorithms and impossibility results for fitting complex statistical models to network data subject to rigorous privacy guarantees. We consider the so-called node-differentially private algorithms, which compute information about a graph or network while provably revealing almost no information about the presence or absence of a particular node in the graph.We provide new algorithms for node-differentially private estimation for a popular and expressive family of network models: stochastic block models and their generalization, graphons. Our algorithms improve on prior work [15], reducing their error quadratically and matching, in many regimes, the optimal nonprivate algorithm [37]. We also show that for the simplest random graph models (G(n, p) and G(n, m)), node-private algorithms can be qualitatively more accurate than for more complex models-converging at a rate of 1 ε 2 n 3 instead of 1 ε 2 n 2 . This result uses a new extension lemma for differentially private algorithms that we hope will be broadly useful.

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Section: Introductionmentioning

confidence: 99%

Revealing Network Structure, Confidentially: Improved Rates for Node-Private Graphon Estimation

Borgs

Chayes

Smith

et al. 2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

View full text Add to dashboard Cite

show abstract

“…Finally, we compare in Figure 5 the accuracy of Algorithm 1 with the motif counting algorithms presented in references [8] and [9]. Those algorithms consist in counting the number of common neighbors, and clustering accordingly.…”

Section: Higher-order Spectral Clustering On 1-dimensional Gbmmentioning

confidence: 99%

“…We call Motif Counting 1 (resp. Motif Counting 2) the algorithm of reference [8] (resp. of reference [9]).…”

Section: Higher-order Spectral Clustering On 1-dimensional Gbmmentioning

confidence: 99%

“…For example on a GBM with n = 3000, r in = 0.08 and r out = 0.04, HOSC takes 8 seconds, while Motif Counting 1 takes 130 seconds and Motif Counting 2 takes 60 seconds on a laptop with 1.90GHz CPU and 15.5 GB memory. Motif Counting 1 corresponds to the algorithm described in [8] and Motif Counting 2 to the algorithm described in [9]. Results are averaged over 50 realisations, and error bars show the standard error (Color figure online)…”

Section: Higher-order Spectral Clustering On 1-dimensional Gbmmentioning

confidence: 99%

“…Then, the probability of an edge appearance between two nodes will depend both on the community labelling and on the positions of these nodes. Recent proposals of random geometric graphs with community structure include the Geometric Block Model (GBM) [8] and Euclidean random geometric graphs [2]. The nodes' interactions in geometric models are no longer independent: two interacting nodes are likely to have many common neighbors.…”

Section: Introductionmentioning

confidence: 99%

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Higher-Order Spectral Clustering for Geometric Graphs

Avrachenkov

Bobu

Dreveton

2021

J Fourier Anal Appl

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The present paper is devoted to clustering geometric graphs. While the standard spectral clustering is often not effective for geometric graphs, we present an effective generalization, which we call higher-order spectral clustering. It resembles in concept the classical spectral clustering method but uses for partitioning the eigenvector associated with a higher-order eigenvalue. We establish the weak consistency of this algorithm for a wide class of geometric graphs which we call Soft Geometric Block Model. A small adjustment of the algorithm provides strong consistency. We also show that our method is effective in numerical experiments even for graphs of modest size.

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Active Learning in the Geometric Block Model

Chien

Tulino

Llorca

2020

AAAI

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The geometric block model is a recently proposed generative model for random graphs that is able to capture the inherent geometric properties of many community detection problems, providing more accurate characterizations of practical community structures compared with the popular stochastic block model. Galhotra et al. recently proposed a motif-counting algorithm for unsupervised community detection in the geometric block model that is proved to be near-optimal. They also characterized the regimes of the model parameters for which the proposed algorithm can achieve exact recovery. In this work, we initiate the study of active learning in the geometric block model. That is, we are interested in the problem of exactly recovering the community structure of random graphs following the geometric block model under arbitrary model parameters, by possibly querying the labels of a limited number of chosen nodes. We propose two active learning algorithms that combine the use of motif-counting with two different label query policies. Our main contribution is to show that sampling the labels of a vanishingly small fraction of nodes (sub-linear in the total number of nodes) is sufficient to achieve exact recovery in the regimes under which the state-of-the-art unsupervised method fails. We validate the superior performance of our algorithms via numerical simulations on both real and synthetic datasets.

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The Geometric Block Model and Applications

Cited by 7 publications

References 31 publications

Revealing Network Structure, Confidentially: Improved Rates for Node-Private Graphon Estimation

Revealing Network Structure, Confidentially: Improved Rates for Node-Private Graphon Estimation

Higher-Order Spectral Clustering for Geometric Graphs

Active Learning in the Geometric Block Model

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