10,835 research outputs found
Total variation based community detection using a nonlinear optimization approach
Maximizing the modularity of a network is a successful tool to identify an
important community of nodes. However, this combinatorial optimization problem
is known to be NP-complete. Inspired by recent nonlinear modularity eigenvector
approaches, we introduce the modularity total variation and show that
its box-constrained global maximum coincides with the maximum of the original
discrete modularity function. Thus we describe a new nonlinear optimization
approach to solve the equivalent problem leading to a community detection
strategy based on . The proposed approach relies on the use of a fast
first-order method that embeds a tailored active-set strategy. We report
extensive numerical comparisons with standard matrix-based approaches and the
Generalized RatioDCA approach for nonlinear modularity eigenvectors, showing
that our new method compares favourably with state-of-the-art alternatives
The Fiedler connection to the parametrized modularity optimization for community detection
This paper presents a comprehensive analysis of the generalized spectral
structure of the modularity matrix , which is introduced by Newman as the
kernel matrix for the quadratic-form expression of the modularity function
used for community detection. The analysis is then seamlessly extended to the
resolution-parametrized modularity matrix , where denotes
the resolution parameter. The modularity spectral analysis provides fresh and
profound insights into the -dynamics within the framework of modularity
maximization for community detection. It provides the first algebraic
explanation of the resolution limit at any specific value. Among the
significant findings and implications, the analysis reveals that (1) the maxima
of the quadratic function with as the kernel matrix always reside
in the Fiedler space of the normalized graph Laplacian or the null space of
, or their combination, and (2) the Fiedler value of the graph Laplacian
marks the critical value in the transition of candidate community
configuration states between graph division and aggregation. Additionally, this
paper introduces and identifies the Fiedler pseudo-set (FPS) as the de facto
critical region for the state transition. This work is expected to have an
immediate and long-term impact on improvements in algorithms for modularity
maximization and on model transformations.Comment: 11 pages, 3 figures, 1 tabl
Axioms for graph clustering quality functions
We investigate properties that intuitively ought to be satisfied by graph
clustering quality functions, that is, functions that assign a score to a
clustering of a graph. Graph clustering, also known as network community
detection, is often performed by optimizing such a function. Two axioms
tailored for graph clustering quality functions are introduced, and the four
axioms introduced in previous work on distance based clustering are
reformulated and generalized for the graph setting. We show that modularity, a
standard quality function for graph clustering, does not satisfy all of these
six properties. This motivates the derivation of a new family of quality
functions, adaptive scale modularity, which does satisfy the proposed axioms.
Adaptive scale modularity has two parameters, which give greater flexibility in
the kinds of clusterings that can be found. Standard graph clustering quality
functions, such as normalized cut and unnormalized cut, are obtained as special
cases of adaptive scale modularity.
In general, the results of our investigation indicate that the considered
axiomatic framework covers existing `good' quality functions for graph
clustering, and can be used to derive an interesting new family of quality
functions.Comment: 23 pages. Full text and sources available on:
http://www.cs.ru.nl/~T.vanLaarhoven/graph-clustering-axioms-2014
Generalized modularity matrices
Various modularity matrices appeared in the recent literature on network
analysis and algebraic graph theory. Their purpose is to allow writing as
quadratic forms certain combinatorial functions appearing in the framework of
graph clustering problems. In this paper we put in evidence certain common
traits of various modularity matrices and shed light on their spectral
properties that are at the basis of various theoretical results and practical
spectral-type algorithms for community detection
Community detection in networks via nonlinear modularity eigenvectors
Revealing a community structure in a network or dataset is a central problem
arising in many scientific areas. The modularity function is an established
measure quantifying the quality of a community, being identified as a set of
nodes having high modularity. In our terminology, a set of nodes with positive
modularity is called a \textit{module} and a set that maximizes is thus
called \textit{leading module}. Finding a leading module in a network is an
important task, however the dimension of real-world problems makes the
maximization of unfeasible. This poses the need of approximation techniques
which are typically based on a linear relaxation of , induced by the
spectrum of the modularity matrix . In this work we propose a nonlinear
relaxation which is instead based on the spectrum of a nonlinear modularity
operator . We show that extremal eigenvalues of
provide an exact relaxation of the modularity measure , however at the price
of being more challenging to be computed than those of . Thus we extend the
work made on nonlinear Laplacians, by proposing a computational scheme, named
\textit{generalized RatioDCA}, to address such extremal eigenvalues. We show
monotonic ascent and convergence of the method. We finally apply the new method
to several synthetic and real-world data sets, showing both effectiveness of
the model and performance of the method
Motif-based communities in complex networks
Community definitions usually focus on edges, inside and between the
communities. However, the high density of edges within a community determines
correlations between nodes going beyond nearest-neighbours, and which are
indicated by the presence of motifs. We show how motifs can be used to define
general classes of nodes, including communities, by extending the mathematical
expression of Newman-Girvan modularity. We construct then a general framework
and apply it to some synthetic and real networks
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