131 research outputs found
A note on certain ergodicity coefficients
We investigate two ergodicity coefficients and ,
originally introduced to bound the subdominant eigenvalues of nonnegative
matrices.
The former has been generalized to complex matrices in recent years and
several properties for such generalized version have been shown so far.
We provide a further result concerning the limit of its powers. Then we
propose a generalization of the second coefficient and we show
that, under mild conditions, it can be used to recast the eigenvector problem
as a particular M-matrix linear system, whose coefficient matrix can be
defined in terms of the entries of . Such property turns out to generalize
the two known equivalent formulations of the Pagerank centrality of a graph
Modularity bounds for clusters located by leading eigenvectors of the normalized modularity matrix
Nodal theorems for generalized modularity matrices ensure that the cluster
located by the positive entries of the leading eigenvector of various
modularity matrices induces a connected subgraph. In this paper we obtain lower
bounds for the modularity of that set of nodes showing that, under certain
conditions, the nodal domains induced by eigenvectors corresponding to highly
positive eigenvalues of the normalized modularity matrix have indeed positive
modularity, that is they can be recognized as modules inside the network.
Moreover we establish Cheeger-type inequalities for the cut-modularity of the
graph, providing a theoretical support to the common understanding that highly
positive eigenvalues of modularity matrices are related with the possibility of
subdividing a network into communities
A nodal domain theorem and a higher-order Cheeger inequality for the graph -Laplacian
We consider the nonlinear graph -Laplacian and its set of eigenvalues and
associated eigenfunctions of this operator defined by a variational principle.
We prove a nodal domain theorem for the graph -Laplacian for any .
While for the bounds on the number of weak and strong nodal domains are
the same as for the linear graph Laplacian (), the behavior changes for
. We show that the bounds are tight for as the bounds are
attained by the eigenfunctions of the graph -Laplacian on two graphs.
Finally, using the properties of the nodal domains, we prove a higher-order
Cheeger inequality for the graph -Laplacian for . If the eigenfunction
associated to the -th variational eigenvalue of the graph -Laplacian has
exactly strong nodal domains, then the higher order Cheeger inequality
becomes tight as
An algebraic analysis of the graph modularity
One of the most relevant tasks in network analysis is the detection of
community structures, or clustering. Most popular techniques for community
detection are based on the maximization of a quality function called
modularity, which in turn is based upon particular quadratic forms associated
to a real symmetric modularity matrix , defined in terms of the adjacency
matrix and a rank one null model matrix. That matrix could be posed inside the
set of relevant matrices involved in graph theory, alongside adjacency,
incidence and Laplacian matrices. This is the reason we propose a graph
analysis based on the algebraic and spectral properties of such matrix. In
particular, we propose a nodal domain theorem for the eigenvectors of ; we
point out several relations occurring between graph's communities and
nonnegative eigenvalues of ; and we derive a Cheeger-type inequality for the
graph optimal modularity
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
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