# Consistency of spectral clustering in stochastic block models

@article{Lei2015ConsistencyOS, title={Consistency of spectral clustering in stochastic block models}, author={Jing Lei and Alessandro Rinaldo}, journal={Annals of Statistics}, year={2015}, volume={43}, pages={215-237} }

We analyze the performance of spectral clustering for community extraction in stochastic block models. We show 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. This result applies to some popular polynomial time spectral clustering algorithms and is further extended to degree corrected stochastic…

## 482 Citations

### Strong Consistency of Spectral Clustering for Stochastic Block Models

- Computer Science, MathematicsIEEE Transactions on Information Theory
- 2020

Under some weak conditions, the K-means algorithm applied to the eigenvectors of the graph Laplacian associated with its first few largest eigenvalues can classify all individuals into the true community uniformly correctly almost surely.

### Strong Consistency, Graph Laplacians, and the Stochastic Block Model

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

It is proved that spectral clustering is able to achieve exact recovery of the planted community structure under conditions that match the information-theoretic limits.

### Randomized Spectral Clustering in Large-Scale Stochastic Block Models

- Computer ScienceJournal of Computational and Graphical Statistics
- 2022

It turns out that, under mild conditions, the randomized spectral clustering algorithms lead to the same theoretical bounds as those of the original spectral clusters algorithm, and this work extends the results to degree-corrected stochastic block models.

### On Consistency of Compressive Spectral Clustering

- Computer Science2018 IEEE International Symposium on Information Theory (ISIT)
- 2018

The effects of sparsity, dimensionality and filter approximation error on the consistency of the algorithm in recovering planted clusters in the stochastic block model is characterized.

### 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.

### 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.

### On Consistency of Compressive Spectral Clustering On Consistency of Compressive Spectral Clustering

- Computer Science
- 2018

The effects of sparsity, dimensionality and filter approximation error on the consistency of the algorithm in recovering planted clusters is characterized.

### 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.

### 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.

### On the efficacy of higher-order spectral clustering under weighted stochastic block models

- Computer Science
- 2022

It turns out that when the network is dense with weak signal of weights, higher-order spectral clustering can really lead to the performance gain in clustering.

## References

SHOWING 1-10 OF 51 REFERENCES

### Regularized Spectral Clustering under the Degree-Corrected Stochastic Blockmodel

- Computer ScienceNIPS
- 2013

The paper characterizes and justifies several of the variations of the spectral clustering algorithm in terms of the Degree-Corrected Stochastic Blockmodel and the Extended Planted Partition model, two statistical models that allow for highly heterogeneous degrees.

### Role of normalization in spectral clustering for stochastic blockmodels

- Computer Science
- 2015

This paper theoretically shows that normalization shrinks the spread of points in a class by a constant fraction under a broad parameter regime and obtains sharp deviation bounds of empirical principal eigenvalues of graphs generated from a stochastic blockmodel.

### Spectral redemption in clustering sparse networks

- Computer ScienceProceedings of the National Academy of Sciences
- 2013

A way of encoding sparse data using a “nonbacktracking” matrix, and it is shown that the corresponding spectral algorithm performs optimally for some popular generative models, including the stochastic block model.

### Spectral clustering and the high-dimensional stochastic blockmodel

- Computer Science
- 2011

The asymptotic results in th is paper are the first clustering results that allow the number of clusters in the model to grow with theNumber of nodes, hence the name high-dimensional.

### Noise Thresholds for Spectral Clustering

- Computer ScienceNIPS
- 2011

The performance of a spectral algorithm for hierarchical clustering is analyzed and it is shown that on a class of hierarchically structured similarity matrices, this algorithm can tolerate noise that grows with the number of data points while still perfectly recovering the hierarchical clusters with high probability.

### A Consistent Adjacency Spectral Embedding for Stochastic Blockmodel Graphs

- Computer Science, Mathematics
- 2011

It is proved that this method to estimate block membership of nodes in a random graph generated by a stochastic blockmodel is consistent for assigning nodes to blocks, as only a negligible number of nodes will be misassigned.

### Spectral Clustering of Graphs with General Degrees in the Extended Planted Partition Model

- Computer Science, MathematicsCOLT
- 2012

A spectral clustering algorithm for similarity graphs drawn from a simple random graph model, where nodes are allowed to have varying degrees, is examined, and guarantees on the performance are shown that it outputs the correct partition under a wide range of parameter values.

### Consistent Adjacency-Spectral Partitioning for the Stochastic Block Model When the Model Parameters Are Unknown

- Mathematics, Computer ScienceSIAM J. Matrix Anal. Appl.
- 2013

This article proves that the (suitably modified) adjacency-spectral partitioning procedure, requiring only an upper bound on the rank of the communication probability matrix, is consistent and demonstrates a robustness to model mis-specification.

### Clustering Sparse Graphs

- Computer ScienceNIPS
- 2012

We develop a new algorithm to cluster sparse unweighted graphs - i.e. partition the nodes into disjoint clusters so that there is higher density within clusters, and low across clusters. By sparsity…

### Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2011

This paper uses the cavity method of statistical physics to obtain an asymptotically exact analysis of the phase diagram of the stochastic block model, a commonly used generative model for social and biological networks, and develops a belief propagation algorithm for inferring functional groups or communities from the topology of the network.