Francesco Tudisco

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

An Efficient Multilinear Optimization Framework for Hypergraph Matching

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Node and Layer Eigenvector Centralities for Multiplex Networks

, 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 and Laplacian matrices. In this paper we analyze certain spectral properties of modularity matrices, which are related to the community detection problem. In particular, we propose a nodal domain theorem for the eigenvectors of

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

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The expected adjacency and modularity matrices in the degree corrected stochastic block model

; we point out several relations occurring between the graphs communities and nonnegative eigenvalues of

Euler-Richardson method preconditioned by weakly stochastic matrix algebras: a potential contribution to Pagerank computation

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A note on certain ergodicity coeflcients

; and we derive a Cheeger-type inequality for the graph modularity.

Generalized modularity matrices

Hypergraph matching has recently become a popular approach for solving correspondence problems in computer vision as it allows the use of higher-order geometric information. Hypergraph matching can be formulated as a third-order optimization problem subject to assignment constraints which turns out to be NP-hard. In recent work, we have proposed an algorithm for hypergraph matching which first lifts the third-order problem to a fourth-order problem and then solves the fourth-order problem via optimization of the corresponding multilinear form. This leads to a tensor block coordinate ascent scheme which has the guarantee of providing monotonic ascent in the original matching score function and leads to state-of-the-art performance both in terms of achieved matching score and accuracy. In this paper we show that the lifting step to a fourth-order problem can be avoided yielding a third-order scheme with the same guarantees and performance but being two times faster. Moreover, we introduce a homotopy type method which further improves the performance.

On complex power nonnegative matrices

Eigenvector-based centrality measures are among the most popular centrality measures in network science. The underlying idea is intuitive and the mathematical description is extremely simple in the framework of standard, mono-layer networks. Moreover, several efficient computational tools are available for their computation. Moving up in dimensionality, several efforts have been made in the past to describe an eigenvector-based centrality measure that generalizes Bonacich index to the case of multiplex networks. In this work, we propose a new definition of eigenvector centrality that relies on the Perron eigenvector of a multi-homogeneous map defined in terms of the tensor describing the network. We prove that existence and uniqueness of such centrality are guaranteed under very mild assumptions on the multiplex network. Extensive numerical studies are proposed to test the newly introduced centrality measure and to compare it to other existing eigenvector-based centralities.

Clustering signed networks with the geometric mean of Laplacians

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.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.