Spectral Clustering on Spherical Coordinates Under the Degree-Corrected Stochastic Blockmodel

Passino, Francesco Sanna; Heard, Nicholas A.; Rubin-Delanchy, Patrick

doi:10.1080/00401706.2021.2008503

Cited by 14 publications

(21 citation statements)

References 34 publications

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“…For time windows corresponding to classroom time, for example, 09:00-10:00 and 15:00-16:00, the embedding forms rays of points in 10-dimensional space, with each ray broadly corresponding to a single school class. This is to be expected under a degree-corrected stochastic block model, and the distance along the ray is a measure of the node's activity level [20,25,33]. However, not all time windows exhibit this structure, for example, the different classes mix more during lunchtimes (time windows 12:00-13:00 and 13:00-14:00).…”

Section: Real Datamentioning

confidence: 97%

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Spectral embedding for dynamic networks with stability guarantees

Gallagher¹,

Jones²,

Rubin-Delanchy³

2021

Preprint

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We consider the problem of embedding a dynamic network, to obtain time-evolving vector representations of each node, which can then be used to describe the changes in behaviour of a single node, one or more communities, or the entire graph. Given this open-ended remit, we wish to guarantee stability in the spatio-temporal positioning of the nodes: assigning the same position, up to noise, to nodes behaving similarly at a given time (cross-sectional stability) and a constant position, up to noise, to a single node behaving similarly across different times (longitudinal stability). These properties are defined formally within a generic dynamic latent position model. By showing how this model can be recast as a multilayer random dot product graph, we demonstrate that unfolded adjacency spectral embedding satisfies both stability conditions, allowing, for example, spatio-temporal clustering under the dynamic stochastic block model. We also show how alternative methods, such as omnibus, independent or time-averaged spectral embedding, lack one or the other form of stability.

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Section: Real Datamentioning

confidence: 97%

“…Following recommendations regarding community detection under a degree-corrected stochastic block model [33], we analyse UASE using spherical coordinates Θ (t) ∈ [0, 2π) n×9 , for t ∈ [T ]. Since UASE demonstrates cross-sectional and longitudinal stability, we can combine the embeddings into a single point cloud Θ…”

Section: Clusteringmentioning

confidence: 99%

Spectral embedding for dynamic networks with stability guarantees

Gallagher¹,

Jones²,

Rubin-Delanchy³

2021

Preprint

Self Cite

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“…The standard adjustment, introduced by Ng et al [33], and employed extensively thereafter [37,29,28,41], is to project the spectral embeddings onto the unit sphere and subsequently perform clustering on these points. This projection step is intended to remove the ancillary effect of degree heterogeneity on the embedding.…”

Section: Introductionmentioning

confidence: 99%

“…One way or another, to correct a d-dimensional spectral embedding for node degree, a method will typically seek a projection of the nodes onto a d − 1-dimensional submanifold. This manifold is often, but not always [23], a sphere [33,37,29,28,41]. However, in geometry, the usual way of representing the space of lines through the origin is with a hyperplane in which each point represents the line going through it, known as projective space [27].…”

Section: Introductionmentioning

confidence: 99%

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Spectral clustering under degree heterogeneity: a case for the random walk Laplacian

Modell¹,

Rubin-Delanchy²

2021

Preprint

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This paper shows that graph spectral embedding using the random walk Laplacian produces vector representations which are completely corrected for node degree. Under a generalised random dot product graph, the embedding provides uniformly consistent estimates of degree-corrected latent positions, with asymptotically Gaussian error. In the special case of a degree-corrected stochastic block model, the embedding concentrates about K distinct points, representing communities. These can be recovered perfectly, asymptotically, through a subsequent clustering step, without spherical projection, as commonly required by algorithms based on the adjacency or normalised, symmetric Laplacian matrices. While the estimand does not depend on degree, the asymptotic variance of its estimate does -higher degree nodes are embedded more accurately than lower degree nodes. Our central limit theorem therefore suggests fitting a weighted Gaussian mixture model as the subsequent clustering step, for which we provide an expectation-maximisation algorithm.

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Community detection with nodal information: Likelihood and its variational approximation

Weng

Feng

2022

Stat

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Community detection is one of the fundamental problems in the study of network data. Most existing community detection approaches only consider edge information as inputs, and the output could be suboptimal when nodal information is available. In such cases, it is desirable to leverage nodal information for the improvement of community detection accuracy. Towards this goal, we propose a flexible network model incorporating nodal information and develop likelihood‐based inference methods. For the proposed methods, we establish favorable asymptotic properties as well as efficient algorithms for computation. Numerical experiments show the effectiveness of our methods in utilizing nodal information across a variety of simulated and real network data sets.

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Spectral Clustering on Spherical Coordinates Under the Degree-Corrected Stochastic Blockmodel

Cited by 14 publications

References 34 publications

Spectral embedding for dynamic networks with stability guarantees

Spectral embedding for dynamic networks with stability guarantees

Spectral clustering under degree heterogeneity: a case for the random walk Laplacian

Community detection with nodal information: Likelihood and its variational approximation

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