75,356 research outputs found
Consistency of community detection in networks under degree-corrected stochastic block models
Community detection is a fundamental problem in network analysis, with
applications in many diverse areas. The stochastic block model is a common tool
for model-based community detection, and asymptotic tools for checking
consistency of community detection under the block model have been recently
developed. However, the block model is limited by its assumption that all nodes
within a community are stochastically equivalent, and provides a poor fit to
networks with hubs or highly varying node degrees within communities, which are
common in practice. The degree-corrected stochastic block model was proposed to
address this shortcoming and allows variation in node degrees within a
community while preserving the overall block community structure. In this paper
we establish general theory for checking consistency of community detection
under the degree-corrected stochastic block model and compare several community
detection criteria under both the standard and the degree-corrected models. We
show which criteria are consistent under which models and constraints, as well
as compare their relative performance in practice. We find that methods based
on the degree-corrected block model, which includes the standard block model as
a special case, are consistent under a wider class of models and that
modularity-type methods require parameter constraints for consistency, whereas
likelihood-based methods do not. On the other hand, in practice, the degree
correction involves estimating many more parameters, and empirically we find it
is only worth doing if the node degrees within communities are indeed highly
variable. We illustrate the methods on simulated networks and on a network of
political blogs.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1036 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org). With Correction
A spectral method for community detection in moderately-sparse degree-corrected stochastic block models
We consider community detection in Degree-Corrected Stochastic Block Models
(DC-SBM). We propose a spectral clustering algorithm based on a suitably
normalized adjacency matrix. We show that this algorithm consistently recovers
the block-membership of all but a vanishing fraction of nodes, in the regime
where the lowest degree is of order log or higher. Recovery succeeds even
for very heterogeneous degree-distributions. The used algorithm does not rely
on parameters as input. In particular, it does not need to know the number of
communities
Non-Backtracking Spectrum of Degree-Corrected Stochastic Block Models
Motivated by community detection, we characterise the spectrum of the
non-backtracking matrix in the Degree-Corrected Stochastic Block Model.
Specifically, we consider a random graph on vertices partitioned into two
equal-sized clusters. The vertices have i.i.d. weights
with second moment . The intra-cluster connection probability for
vertices and is and the inter-cluster
connection probability is .
We show that with high probability, the following holds: The leading
eigenvalue of the non-backtracking matrix is asymptotic to . The second eigenvalue is asymptotic to when , but asymptotically bounded by
when . All the remaining eigenvalues are
asymptotically bounded by . As a result, a clustering
positively-correlated with the true communities can be obtained based on the
second eigenvector of in the regime where
In a previous work we obtained that detection is impossible when meaning that there occurs a phase-transition in the sparse regime of the
Degree-Corrected Stochastic Block Model.
As a corollary, we obtain that Degree-Corrected Erd\H{o}s-R\'enyi graphs
asymptotically satisfy the graph Riemann hypothesis, a quasi-Ramanujan
property.
A by-product of our proof is a weak law of large numbers for
local-functionals on Degree-Corrected Stochastic Block Models, which could be
of independent interest
Joint Spectral Clustering in Multilayer Degree-Corrected Stochastic Blockmodels
Modern network datasets are often composed of multiple layers, either as
different views, time-varying observations, or independent sample units,
resulting in collections of networks over the same set of vertices but with
potentially different connectivity patterns on each network. These data require
models and methods that are flexible enough to capture local and global
differences across the networks, while at the same time being parsimonious and
tractable to yield computationally efficient and theoretically sound solutions
that are capable of aggregating information across the networks. This paper
considers the multilayer degree-corrected stochastic blockmodel, where a
collection of networks share the same community structure, but
degree-corrections and block connection probability matrices are permitted to
be different. We establish the identifiability of this model and propose a
spectral clustering algorithm for community detection in this setting. Our
theoretical results demonstrate that the misclustering error rate of the
algorithm improves exponentially with multiple network realizations, even in
the presence of significant layer heterogeneity with respect to degree
corrections, signal strength, and spectral properties of the block connection
probability matrices. Simulation studies show that this approach improves on
existing multilayer community detection methods in this challenging regime.
Furthermore, in a case study of US airport data through January 2016 --
September 2021, we find that this methodology identifies meaningful community
structure and trends in airport popularity influenced by pandemic impacts on
travel
Bayesian stochastic blockmodels for community detection in networks and community-structured covariance selection
Networks have been widely used to describe interactions among objects in diverse fields. Given the interest in explaining a network by its structure, much attention has been drawn to finding clusters of nodes with dense connections within clusters but sparse connections between clusters. Such clusters are called communities, and identifying such clusters is known as community detection. Here, to perform community detection, I focus on stochastic blockmodels (SBM), a class of statistically-based generative models. I present a flexible SBM that represents different types of data as well as node attributes under a Bayesian framework. The proposed models explicitly capture community behavior by guaranteeing that connections are denser within communities than between communities.
First, I present a degree-corrected SBM based on a logistic regression formulation to model binary networks. To fit the model, I obtain posterior samples via Gibbs sampling based on Polya-Gamma latent variables. I conduct inference based on a novel, canonically mapped centroid estimator that formally addresses label non-identifiability and captures representative community assignments. Next, to accommodate large-scale datasets, I further extend the degree-corrected SBM to a broader family of generalized linear models with group correction terms. To conduct exact inference efficiently, I develop an iteratively-reweighted least squares procedure that implicitly updates sufficient statistics on the network to obtain maximum a posteriori (MAP) estimators. I demonstrate the proposed model and estimation on simulated benchmark networks and various real-world datasets.
Finally, I develop a Bayesian SBM for community-structured covariance selection. Here, I assume that the data at each node are Gaussian and a latent network where two nodes are not connected if their observations are conditionally independent given observations of other nodes. Under the context of biological and social applications, I expect that this latent network shows a block dependency structure that represents community behavior. Thus, to identify the latent network and detect communities, I propose a hierarchical prior in two levels: a spike-and-slab prior on off-diagonal entries of the concentration matrix for variable selection and a degree-corrected SBM to capture community behavior. I develop an efficient routine based on ridge regularization and MAP estimation to conduct inference
Community Detection and Classification Guarantees Using Embeddings Learned by Node2Vec
Embedding the nodes of a large network into an Euclidean space is a common
objective in modern machine learning, with a variety of tools available. These
embeddings can then be used as features for tasks such as community
detection/node clustering or link prediction, where they achieve state of the
art performance. With the exception of spectral clustering methods, there is
little theoretical understanding for other commonly used approaches to learning
embeddings. In this work we examine the theoretical properties of the
embeddings learned by node2vec. Our main result shows that the use of k-means
clustering on the embedding vectors produced by node2vec gives weakly
consistent community recovery for the nodes in (degree corrected) stochastic
block models. We also discuss the use of these embeddings for node and link
prediction tasks. We demonstrate this result empirically, and examine how this
relates to other embedding tools for network data
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