6,662 research outputs found
Extended Bernoulli and Stirling matrices and related combinatorial identities
In this paper we establish plenty of number theoretic and combinatoric
identities involving generalized Bernoulli and Stirling numbers of both kinds.
These formulas are deduced from Pascal type matrix representations of Bernoulli
and Stirling numbers. For this we define and factorize a modified Pascal matrix
corresponding to Bernoulli and Stirling cases.Comment: Accepted for publication in Linear Algebra and its Application
Optimization via Low-rank Approximation for Community Detection in Networks
Community detection is one of the fundamental problems of network analysis,
for which a number of methods have been proposed. Most model-based or
criteria-based methods have to solve an optimization problem over a discrete
set of labels to find communities, which is computationally infeasible. Some
fast spectral algorithms have been proposed for specific methods or models, but
only on a case-by-case basis. Here we propose a general approach for maximizing
a function of a network adjacency matrix over discrete labels by projecting the
set of labels onto a subspace approximating the leading eigenvectors of the
expected adjacency matrix. This projection onto a low-dimensional space makes
the feasible set of labels much smaller and the optimization problem much
easier. We prove a general result about this method and show how to apply it to
several previously proposed community detection criteria, establishing its
consistency for label estimation in each case and demonstrating the fundamental
connection between spectral properties of the network and various model-based
approaches to community detection. Simulations and applications to real-world
data are included to demonstrate our method performs well for multiple problems
over a wide range of parameters.Comment: 45 pages, 7 figures; added discussions about computational complexity
and extension to more than two communitie
Sparse random graphs: regularization and concentration of the Laplacian
We study random graphs with possibly different edge probabilities in the
challenging sparse regime of bounded expected degrees. Unlike in the dense
case, neither the graph adjacency matrix nor its Laplacian concentrate around
their expectations due to the highly irregular distribution of node degrees. It
has been empirically observed that simply adding a constant of order to
each entry of the adjacency matrix substantially improves the behavior of
Laplacian. Here we prove that this regularization indeed forces Laplacian to
concentrate even in sparse graphs. As an immediate consequence in network
analysis, we establish the validity of one of the simplest and fastest
approaches to community detection -- regularized spectral clustering, under the
stochastic block model. Our proof of concentration of regularized Laplacian is
based on Grothendieck's inequality and factorization, combined with paving
arguments.Comment: Added reference
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