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

D. Fasino

Dario Fasino

Francesco Tudisco

K. Rohe

M. E. J. Newman

M. Fiedler

P. Tarazaga

R. R. Nadakuditi

V. Nikiforov

English

arXiv

summary:We propose a new localization result for the leading eigenvalue and eigenvector of a symmetric matrix $A$. The result exploits the Frobenius inner product between $A$ and a given rank-one landmark matrix $X$. Different choices for $X$ may be used, depending on the problem under investigation. In particular, we show that the choice where $X$ is the all-ones matrix allows to estimate the signature of the leading eigenvector of $A$, generalizing previous results on Perron-Frobenius properties of matrices with some negative entries. As another application we consider the problem of community detection in graphs and networks. The problem is solved by means of modularity-based spectral techniques, following the ideas pioneered by Miroslav Fiedler in mid-'70s. \endgraf We show that a suitable choice of $X$ can be used to provide new quality guarantees of those techniques, when the network follows a stochastic block model

Fasino, Dario

Tudisco, Francesco

Institute of Mathematics AS CR, v. v. i.

Localization of dominant eigenpairs and planted communities by means of Frobenius inner products

Institutional Research Information System

Crossref

Czechoslovak Mathematical Journal

TUDISCO, FRANCESCO

Archivio istituzionale della ricerca - Università di Padova

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 detectio

FASINO, Dario

Archivio istituzionale della ricerca - Università degli Studi di Udine

We propose a new localization result for the leading eigenvalue and eigenvector of a symmetric matrix A. The result exploits the Frobenius inner product between A and a given rank-one landmark matrix X. Different choices for X may be used, depending on the problem under investigation. In particular, we show that the choice where X is the all-ones matrix allows to estimate the signature of the leading eigenvector of A, generalizing previous results on Perron-Frobenius properties of matrices with some negative entries. As another application we consider the problem of community detection in graphs and networks. The problem is solved by means of modularity-based spectral techniques, following the ideas pioneered by Miroslav Fiedler in mid-'70s

Czech digital mathematics library

Linear Algebra and its Applications

https://air.uniud.it/bitstream/11390/1099522/1/cmj66_3_23.pdf

Generalized modularity matrices

Abstract

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Institute of Mathematics AS CR, v. v. i.

Institutional Research Information System

Crossref

Archivio istituzionale della ricerca - Università di Padova

Archivio istituzionale della ricerca - Università degli Studi di Udine

Archivio istituzionale della ricerca - Università di Padova

Archivio istituzionale della ricerca - Università degli Studi di Udine

Czech digital mathematics library

Crossref