131 research outputs found

    A note on certain ergodicity coefficients

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    We investigate two ergodicity coefficients ϕ∥ ∥\phi_{\|\, \|} and τn−1\tau_{n-1}, 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 τn−1\tau_{n-1} and we show that, under mild conditions, it can be used to recast the eigenvector problem Ax=xAx=x as a particular M-matrix linear system, whose coefficient matrix can be defined in terms of the entries of AA. 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

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    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 pp-Laplacian

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    We consider the nonlinear graph pp-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 pp-Laplacian for any p≥1p\geq 1. While for p>1p>1 the bounds on the number of weak and strong nodal domains are the same as for the linear graph Laplacian (p=2p=2), the behavior changes for p=1p=1. We show that the bounds are tight for p≥1p\geq 1 as the bounds are attained by the eigenfunctions of the graph pp-Laplacian on two graphs. Finally, using the properties of the nodal domains, we prove a higher-order Cheeger inequality for the graph pp-Laplacian for p>1p>1. If the eigenfunction associated to the kk-th variational eigenvalue of the graph pp-Laplacian has exactly kk strong nodal domains, then the higher order Cheeger inequality becomes tight as p→1p\rightarrow 1

    An algebraic analysis of the graph modularity

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    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 MM, 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 MM; we point out several relations occurring between graph's communities and nonnegative eigenvalues of MM; and we derive a Cheeger-type inequality for the graph optimal modularity

    Community detection in networks via nonlinear modularity eigenvectors

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    Revealing a community structure in a network or dataset is a central problem arising in many scientific areas. The modularity function QQ 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 QQ 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 QQ unfeasible. This poses the need of approximation techniques which are typically based on a linear relaxation of QQ, induced by the spectrum of the modularity matrix MM. In this work we propose a nonlinear relaxation which is instead based on the spectrum of a nonlinear modularity operator M\mathcal M. We show that extremal eigenvalues of M\mathcal M provide an exact relaxation of the modularity measure QQ, however at the price of being more challenging to be computed than those of MM. 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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