64 research outputs found
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
Inferring community structure in attributed hypergraphs using stochastic block models
Hypergraphs are a representation of complex systems involving interactions
among more than two entities and allow to investigation of higher-order
structure and dynamics in real-world complex systems. Community structure is a
common property observed in empirical networks in various domains. Stochastic
block models have been employed to investigate community structure in networks.
Node attribute data, often accompanying network data, has been found to
potentially enhance the learning of community structure in dyadic networks. In
this study, we develop a statistical framework that incorporates node attribute
data into the learning of community structure in a hypergraph, employing a
stochastic block model. We demonstrate that our model, which we refer to as
HyperNEO, enhances the learning of community structure in synthetic and
empirical hypergraphs when node attributes are sufficiently associated with the
communities. Furthermore, we found that applying a dimensionality reduction
method, UMAP, to the learned representations obtained using stochastic block
models, including our model, maps nodes into a two-dimensional vector space
while largely preserving community structure in empirical hypergraphs. We
expect that our framework will broaden the investigation and understanding of
higher-order community structure in real-world complex systems.Comment: 28 pages, 11 figures, 8 table
Consistency of Spectral Hypergraph Partitioning under Planted Partition Model
Hypergraph partitioning lies at the heart of a number of problems in machine
learning and network sciences. Many algorithms for hypergraph partitioning have
been proposed that extend standard approaches for graph partitioning to the
case of hypergraphs. However, theoretical aspects of such methods have seldom
received attention in the literature as compared to the extensive studies on
the guarantees of graph partitioning. For instance, consistency results of
spectral graph partitioning under the stochastic block model are well known. In
this paper, we present a planted partition model for sparse random non-uniform
hypergraphs that generalizes the stochastic block model. We derive an error
bound for a spectral hypergraph partitioning algorithm under this model using
matrix concentration inequalities. To the best of our knowledge, this is the
first consistency result related to partitioning non-uniform hypergraphs.Comment: 35 pages, 2 figures, 1 tabl
Exact Recovery for a Family of Community-Detection Generative Models
Generative models for networks with communities have been studied extensively
for being a fertile ground to establish information-theoretic and computational
thresholds. In this paper we propose a new toy model for planted generative
models called planted Random Energy Model (REM), inspired by Derrida's REM. For
this model we provide the asymptotic behaviour of the probability of error for
the maximum likelihood estimator and hence the exact recovery threshold. As an
application, we further consider the 2 non-equally sized community Weighted
Stochastic Block Model (2-WSBM) on -uniform hypergraphs, that is equivalent
to the P-REM on both sides of the spectrum, for high and low edge cardinality
. We provide upper and lower bounds for the exact recoverability for any
, mapping these problems to the aforementioned P-REM. To the best of our
knowledge these are the first consistency results for the 2-WSBM on graphs and
on hypergraphs with non-equally sized community
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Marchenko-Pastur law with relaxed independence conditions
We prove the Marchenko-Pastur law for the eigenvalues of sample
covariance matrices in two new situations where the data does not have
independent coordinates. In the first scenario - the block-independent model -
the coordinates of the data are partitioned into blocks in such a way that
the entries in different blocks are independent, but the entries from the same
block may be dependent. In the second scenario - the random tensor model - the
data is the homogeneous random tensor of order , i.e. the coordinates of the
data are all different products of variables chosen from a
set of independent random variables. We show that Marchenko-Pastur law
holds for the block-independent model as long as the size of the largest block
is and for the random tensor model as long as . Our main
technical tools are new concentration inequalities for quadratic forms in
random variables with block-independent coordinates, and for random tensors
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