Daniel L. Sussman

A Consistent Adjacency Spectral Embedding for Stochastic Blockmodel Graphs

We present a method to estimate block membership of nodes in a random graph generated by a stochastic blockmodel. We use an embedding procedure motivated by the random dot product graph model, a particular example of the latent position model. The embedding associates each node with a vector; these vectors are clustered via minimization of a square error criterion. We prove that this method is consistent for assigning nodes to blocks, as only a negligible number of nodes will be misassigned. We prove consistency of the method for directed and undirected graphs. The consistent block assignment makes possible consistent parameter estimation for a stochastic blockmodel. We extend the result in the setting where the number of blocks grows slowly with the number of nodes. Our method is also computationally feasible even for very large graphs. We compare our method with Laplacian spectral clustering through analysis of simulated data and a graph derived from Wikipedia documents.

CONSISTENT ADJACENCY-SPECTRAL PARTITIONING FOR THE STOCHASTIC BLOCK MODEL WHEN THE MODEL PARAMETERS ARE UNKNOWN ∗

For random graphs distributed according to a stochastic block model, we consider the inferential task of partioning vertices into blocks using spectral techniques. Spectral partioning using the normalized Laplacian and the adjacency matrix have both been shown to be consistent as the number of vertices tend to infinity. Importantly, both procedures require that the number of blocks and the rank of the communication probability matrix are known, even as the rest of the parameters may be unknown. In this article, we prove that the (suitably modified) adjacency-spectral partitioning procedure, requiring only an upper bound on the rank of the communication probability matrix, is consistent. Indeed, this result demonstrates a robustness to model mis-specification; an overestimate of the rank may impose a moderate performance penalty, but the procedure is still consistent. Furthermore, we extend this procedure to the setting where adjacencies may have multiple modalities and we allow for either directed or undirected graphs.

Perfect clustering for stochastic blockmodel graphs via adjacency spectral embedding

Vertex clustering in a stochastic blockmodel graph has wide applicability and has been the subject of extensive research. In thispaper, we provide a short proof that the adjacency spectral embedding can be used to obtain perfect clustering for the stochastic blockmodel and the degree-corrected stochastic blockmodel. We also show an analogous result for the more general random dot product graph model.

Universally consistent vertex classification for latent positions graphs

In this work we show that, using the eigen-decomposition of the adjacency matrix, we can consistently estimate feature maps for latent position graphs with positive definite link function

Spectral clustering for divide-and-conquer graph matching

\kappa

Consistent Latent Position Estimation and Vertex Classification for Random Dot Product Graphs

, provided that the latent positions are i.i.d. from some distribution F. We then consider the exploitation task of vertex classification where the link function

Association Between Visceral Adiposity and Colorectal Polyps on CT Colonography

\kappa

Fully automated adipose tissue measurement on abdominal CT

belongs to the class of universal kernels and class labels are observed for a number of vertices tending to infinity and that the remaining vertices are to be classified. We show that minimization of the empirical

Statistical Inference on Errorfully Observed Graphs

\varphi

Computing scalable multivariate glocal invariants of large (brain-) graphs

-risk for some convex surrogate