66,979 research outputs found
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
Equality of Lifshitz and van Hove exponents on amenable Cayley graphs
We study the low energy asymptotics of periodic and random Laplace operators
on Cayley graphs of amenable, finitely generated groups. For the periodic
operator the asymptotics is characterised by the van Hove exponent or zeroth
Novikov-Shubin invariant. The random model we consider is given in terms of an
adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph.
The asymptotic behaviour of the spectral distribution is exponential,
characterised by the Lifshitz exponent. We show that for the adjacency
Laplacian the two invariants/exponents coincide. The result holds also for more
general symmetric transition operators. For combinatorial Laplacians one has a
different universal behaviour of the low energy asymptotics of the spectral
distribution function, which can be actually established on quasi-transitive
graphs without an amenability assumption. The latter result holds also for long
range bond percolation models
Exchangeable Random Networks
We introduce and study a class of exchangeable random graph ensembles. They
can be used as statistical null models for empirical networks, and as a tool
for theoretical investigations. We provide general theorems that carachterize
the degree distribution of the ensemble graphs, together with some features
that are important for applications, such as subgraph distributions and kernel
of the adjacency matrix. These results are used to compare to other models of
simple and complex networks. A particular case of directed networks with
power-law out--degree is studied in more detail, as an example of the
flexibility of the model in applications.Comment: to appear on "Internet Mathematics
The emergence of coherence in complex networks of heterogeneous dynamical systems
We present a general theory for the onset of coherence in collections of
heterogeneous maps interacting via a complex connection network. Our method
allows the dynamics of the individual uncoupled systems to be either chaotic or
periodic, and applies generally to networks for which the number of connections
per node is large. We find that the critical coupling strength at which a
transition to synchrony takes place depends separately on the dynamics of the
individual uncoupled systems and on the largest eigenvalue of the adjacency
matrix of the coupling network. Our theory directly generalizes the Kuramoto
model of equal strength, all-to-all coupled phase oscillators to the case of
oscillators with more realistic dynamics coupled via a large heterogeneous
network.Comment: 4 pages, 1 figure. Published versio
Bipartite Mixed Membership Distribution-Free Model. A novel model for community detection in overlapping bipartite weighted networks
Modeling and estimating mixed memberships for un-directed un-weighted
networks in which nodes can belong to multiple communities has been well
studied in recent years. However, for a more general case, the bipartite
weighted networks in which nodes can belong to multiple communities, row nodes
can be different from column nodes, and all elements of adjacency matrices can
be any finite real values, to our knowledge, there is no model for such
bipartite weighted networks. To close this gap, this paper introduces a novel
model, the Bipartite Mixed Membership Distribution-Free (BiMMDF) model. As a
special case, bipartite signed networks with mixed memberships can also be
generated from BiMMDF. Our model enjoys its advantage by allowing all elements
of an adjacency matrix to be generated from any distribution as long as the
expectation adjacency matrix has a block structure related to node memberships
under BiMMDF. The proposed model can be viewed as an extension of many previous
models, including the popular mixed membership stochastic blcokmodels. An
efficient algorithm with a theoretical guarantee of consistent estimation is
applied to fit BiMMDF. In particular, for a standard bipartite weighted network
with two row (and column) communities, to make the algorithm's error rates
small with high probability, separation conditions are obtained when adjacency
matrices are generated from different distributions under BiMMDF. The behavior
differences of different distributions on separation conditions are verified by
extensive synthetic bipartite weighted networks generated under BiMMDF.
Experiments on real-world directed weighted networks illustrate the advantage
of the algorithm in studying highly mixed nodes and asymmetry between row and
column communities.Comment: 33 pages, 12 figures, 4 table
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