155,374 research outputs found
Concentration of random graphs and application to community detection
Random matrix theory has played an important role in recent work on
statistical network analysis. In this paper, we review recent results on
regimes of concentration of random graphs around their expectation, showing
that dense graphs concentrate and sparse graphs concentrate after
regularization. We also review relevant network models that may be of interest
to probabilists considering directions for new random matrix theory
developments, and random matrix theory tools that may be of interest to
statisticians looking to prove properties of network algorithms. Applications
of concentration results to the problem of community detection in networks are
discussed in detail.Comment: Submission for International Congress of Mathematicians, Rio de
Janeiro, Brazil 201
Community Detection in Hypergraphs, Spiked Tensor Models, and Sum-of-Squares
We study the problem of community detection in hypergraphs under a stochastic
block model. Similarly to how the stochastic block model in graphs suggests
studying spiked random matrices, our model motivates investigating statistical
and computational limits of exact recovery in a certain spiked tensor model. In
contrast with the matrix case, the spiked model naturally arising from
community detection in hypergraphs is different from the one arising in the
so-called tensor Principal Component Analysis model. We investigate the
effectiveness of algorithms in the Sum-of-Squares hierarchy on these models.
Interestingly, our results suggest that these two apparently similar models
exhibit significantly different computational to statistical gaps.Comment: In proceedings of 2017 International Conference on Sampling Theory
and Applications (SampTA
Detection and localization of change points in temporal networks with the aid of stochastic block models
A framework based on generalized hierarchical random graphs (GHRGs) for the
detection of change points in the structure of temporal networks has recently
been developed by Peel and Clauset [1]. We build on this methodology and extend
it to also include the versatile stochastic block models (SBMs) as a parametric
family for reconstructing the empirical networks. We use five different
techniques for change point detection on prototypical temporal networks,
including empirical and synthetic ones. We find that none of the considered
methods can consistently outperform the others when it comes to detecting and
locating the expected change points in empirical temporal networks. With
respect to the precision and the recall of the results of the change points, we
find that the method based on a degree-corrected SBM has better recall
properties than other dedicated methods, especially for sparse networks and
smaller sliding time window widths.Comment: This is an author-created, un-copyedited version of an article
accepted for publication/published in Journal of Statistical Mechanics:
Theory and Experiment. IOP Publishing Ltd is not responsible for any errors
or omissions in this version of the manuscript or any version derived from
it. The Version of Record is available online at
http://dx.doi.org/10.1088/1742-5468/2016/11/11330
Matrix concentration inequalities with dependent summands and sharp leading-order terms
We establish sharp concentration inequalities for sums of dependent random
matrices. Our results concern two models. First, a model where summands are
generated by a -mixing Markov chain. Second, a model where summands are
expressed as deterministic matrices multiplied by scalar random variables. In
both models, the leading-order term is provided by free probability theory.
This leading-order term is often asymptotically sharp and, in particular, does
not suffer from the logarithmic dimensional dependence which is present in
previous results such as the matrix Khintchine inequality.
A key challenge in the proof is that techniques based on classical cumulants,
which can be used in a setting with independent summands, fail to produce
efficient estimates in the Markovian model. Our approach is instead based on
Boolean cumulants and a change-of-measure argument.
We discuss applications concerning community detection in Markov chains,
random matrices with heavy-tailed entries, and the analysis of random graphs
with dependent edges.Comment: 69 pages, 4 figure
Community Detection in High-Dimensional Graph Ensembles
Detecting communities in high-dimensional graphs can be achieved by applying
random matrix theory where the adjacency matrix of the graph is modeled by a
Stochastic Block Model (SBM). However, the SBM makes an unrealistic assumption
that the edge probabilities are homogeneous within communities, i.e., the edges
occur with the same probabilities. The Degree-Corrected SBM is a generalization
of the SBM that allows these edge probabilities to be different, but existing
results from random matrix theory are not directly applicable to this
heterogeneous model. In this paper, we derive a transformation of the adjacency
matrix that eliminates this heterogeneity and preserves the relevant
eigenstructure for community detection. We propose a test based on the extreme
eigenvalues of this transformed matrix and (1) provide a method for controlling
the significance level, (2) formulate a conjecture that the test achieves power
one for all positive significance levels in the limit as the number of nodes
approaches infinity, and (3) provide empirical evidence and theory supporting
these claims.Comment: 8 pages, 3 figure
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