# Spectral redemption in clustering sparse networks

@article{Krzakala2013SpectralRI, title={Spectral redemption in clustering sparse networks}, author={Florent Krzakala and Cristopher Moore and Elchanan Mossel and Joe Neeman and Allan Sly and Lenka Zdeborov{\'a} and Pan Zhang}, journal={Proceedings of the National Academy of Sciences}, year={2013}, volume={110}, pages={20935 - 20940} }

Significance Spectral algorithms are widely applied to data clustering problems, including finding communities or partitions in graphs and networks. We propose a way of encoding sparse data using a “nonbacktracking” matrix, and show that the corresponding spectral algorithm performs optimally for some popular generative models, including the stochastic block model. This is in contrast with classical spectral algorithms, based on the adjacency matrix, random walk matrix, and graph Laplacian…

## 574 Citations

### Spectral community detection in sparse networks

- Computer ScienceArXiv
- 2013

This work makes use of a relaxation method to derive a spectral community detection algorithm that works well even in the sparse regime where other methods break down, Interestingly, however, the matrix at the heart of the method is not exactly the non-backtracking matrix, but a variant of it with a somewhat different definition.

### A unified framework for spectral clustering in sparse graphs

- Computer ScienceJ. Mach. Learn. Res.
- 2021

It is demonstrated that a conveniently parametrized form of regularized Laplacian matrix can be used to perform spectral clustering in sparse networks, without suffering from its degree heterogeneity.

### Optimized Deformed Laplacian for Spectrum-based Community Detection in Sparse Heterogeneous Graphs

- Computer ScienceArXiv
- 2019

This article study spectral clustering based on the deformed Laplacian matrix $D-rA$ for sparse heterogeneous graphs (following a two-class degree-corrected stochastic block model) shows that, unlike competing methods such as the Bethe Hessian or non-backtracking operator approaches, clustering is insensitive to the graph heterogeneity.

### Consistency of spectral clustering in stochastic block models

- Computer Science
- 2015

It is shown that, under mild conditions, spectral clustering applied to the adjacency matrix of the network can consistently recover hidden communities even when the order of the maximum expected degree is as small as $\log n$ with $n$ the number of nodes.

### Semi-Supervised Clustering of Sparse Graphs: Crossing the Information-Theoretic Threshold

- Computer ScienceArXiv
- 2022

It is proved that with arbitrary fraction of the labels revealed, the detection problem is feasible throughout the parameter domain, and two eﬃcient algorithms are introduced, one combinatorial and one based on optimization, to integrate label information with graph structures.

### Consistency of Spectral Clustering on Hierarchical Stochastic Block Models

- Computer Science
- 2020

A recursive bi-partitioning algorithm is developed that divides the network into two communities based on the Fiedler vector of the unnormalized graph Laplacian and repeats the split until a stopping rule indicates no further community structures.

### Nonbacktracking spectral clustering of nonuniform hypergraphs

- Computer Science, MathematicsArXiv
- 2022

This work studies spectral clustering for nonuniform hypergraphs based on the hypergraph nonbacktracking operator and proves a theorem of Ihara-Bass type to enable faster computation of eigenpairs and proposes an alternating algorithm for inference in a hypergraph stochastic blockmodel via linearized belief-propagation.

### Community detection in multilayer graphs using spectral methods and core-finding

- Computer Science
- 2017

A heuristic algorithm inspired by Belief Propagation and identifying core communities is proposed and discussed, and a finite-time bound on the misclassification rate is proved.

### An Adaptive Spectral Algorithm for the Recovery of Overlapping Communities in Networks

- Computer Science
- 2015

Combinatorial spectral clustering, a simple spectral algorithm designed to identify overlapping communities in networks, is presented and is shown to perform well on simulated data and on real-world graphs with known overlapping communities.

### A Spectral Algorithm with Additive Clustering for the Recovery of Overlapping Communities in Networks

- Computer ScienceALT
- 2016

An adaptive version of the algorithm, that does not require the knowledge of the number of hidden communities, is proved to be consistent under the SBMO when the degrees in the graph are (slightly more than) logarithmic.

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