39,002 research outputs found
Covariate-assisted spectral clustering
Biological and social systems consist of myriad interacting units. The
interactions can be represented in the form of a graph or network. Measurements
of these graphs can reveal the underlying structure of these interactions,
which provides insight into the systems that generated the graphs. Moreover, in
applications such as connectomics, social networks, and genomics, graph data
are accompanied by contextualizing measures on each node. We utilize these node
covariates to help uncover latent communities in a graph, using a modification
of spectral clustering. Statistical guarantees are provided under a joint
mixture model that we call the node-contextualized stochastic blockmodel,
including a bound on the mis-clustering rate. The bound is used to derive
conditions for achieving perfect clustering. For most simulated cases,
covariate-assisted spectral clustering yields results superior to regularized
spectral clustering without node covariates and to an adaptation of canonical
correlation analysis. We apply our clustering method to large brain graphs
derived from diffusion MRI data, using the node locations or neurological
region membership as covariates. In both cases, covariate-assisted spectral
clustering yields clusters that are easier to interpret neurologically.Comment: 28 pages, 4 figures, includes substantial changes to theoretical
result
Hearing the clusters in a graph: A distributed algorithm
We propose a novel distributed algorithm to cluster graphs. The algorithm
recovers the solution obtained from spectral clustering without the need for
expensive eigenvalue/vector computations. We prove that, by propagating waves
through the graph, a local fast Fourier transform yields the local component of
every eigenvector of the Laplacian matrix, thus providing clustering
information. For large graphs, the proposed algorithm is orders of magnitude
faster than random walk based approaches. We prove the equivalence of the
proposed algorithm to spectral clustering and derive convergence rates. We
demonstrate the benefit of using this decentralized clustering algorithm for
community detection in social graphs, accelerating distributed estimation in
sensor networks and efficient computation of distributed multi-agent search
strategies
Fast Approximate Spectral Clustering for Dynamic Networks
Spectral clustering is a widely studied problem, yet its complexity is
prohibitive for dynamic graphs of even modest size. We claim that it is
possible to reuse information of past cluster assignments to expedite
computation. Our approach builds on a recent idea of sidestepping the main
bottleneck of spectral clustering, i.e., computing the graph eigenvectors, by
using fast Chebyshev graph filtering of random signals. We show that the
proposed algorithm achieves clustering assignments with quality approximating
that of spectral clustering and that it can yield significant complexity
benefits when the graph dynamics are appropriately bounded
Spectral density of the non-backtracking operator
The non-backtracking operator was recently shown to provide a significant
improvement when used for spectral clustering of sparse networks. In this paper
we analyze its spectral density on large random sparse graphs using a mapping
to the correlation functions of a certain interacting quantum disordered system
on the graph. On sparse, tree-like graphs, this can be solved efficiently by
the cavity method and a belief propagation algorithm. We show that there exists
a paramagnetic phase, leading to zero spectral density, that is stable outside
a circle of radius , where is the leading eigenvalue of the
non-backtracking operator. We observe a second-order phase transition at the
edge of this circle, between a zero and a non-zero spectral density. That fact
that this phase transition is absent in the spectral density of other matrices
commonly used for spectral clustering provides a physical justification of the
performances of the non-backtracking operator in spectral clustering.Comment: 6 pages, 6 figures, submitted to EP
Higher-Order Spectral Clustering for Geometric Graphs
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.Comment: 23 pages, 6 figure
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