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… 

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References

SHOWING 1-10 OF 51 REFERENCES

Regularized Spectral Clustering under the Degree-Corrected Stochastic Blockmodel

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

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

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

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

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

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

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

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

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

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.
...