We invite contributions focused on all aspects of mathematical, algorithmic, data analysis, and computational techniques in network science and its applications. Accepted submissions will be featured in the workshop as a 20-minute talk, 5-minute talk, or poster.

The school is meant for both final years undergraduate and graduate students who are intrigued by Applied Mathematics and Matrix Methods. The summer school takes place over the course of one entire month—in the two beautiful cities of Rome (Italy) and Moscow (Russia)—and thus it allows the students to really work over the topics that are discussed. Also it is a wonderful occasion to meet new people in the field of Applied Linear Algebra. I have been student of several editions of the school and strongly encourage participation. Please, feel free to contact me if you have questions.

Abstract:
We study the task of semi-supervised learning on multilayer graphs by taking into account both labeled and unlabeled observations together with the information encoded by each individual graph layer. We propose a regularizer based on the generalized matrix mean, which is a one-parameter family of matrix means that includes the arithmetic, geometric and harmonic means as particular cases. We analyze it in expectation under a Multilayer Stochastic Block Model and verify numerically that it outperforms state of the art methods.
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--- Poster that will be presented at NeurIPS19, by Pedro Mercado

Abstract:
We propose and analyse a general tensor-based framework for incorporating second order features into network measures. This approach allows us to combine traditional pairwise links with information that records whether triples of nodes are involved in wedges or triangles. Our treatment covers classical spectral methods and recently proposed cases from the literature, but we also identify many interesting extensions. In particular, we define a mutually-reinforcing (spectral) version of the classical clustering coefficient.
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--- Nodes with largest spectral clustering coefficient in the karate club network, for different tensors.

Abstract:
Being able to produce synthetic networks by means of generative random graph models and scalable algorithms is a recurring tool-of-the-trade in network analysis, as it provides a well founded basis for the statistical analysis of various properties in real-world networks. In this paper, we illustrate how to generate large random graphs having a power-law degree profile by means of the Chung-Lu model. In particular, we are concerned with the fulfillment of a fundamental hypothesis that must be placed on the model parameters, without which the generated graphs loose all the theoretical properties of the model, notably, the controllability of the expected node degrees and the absence of correlations between the degrees of two nodes joined by an edge.
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--- The power law degree distribution of random graphs in the Chung-Lu model.

Abstract:
This work is concerned with the computation of $\ell^p$-eigenvalues and eigenvectors of square tensors with $d$ modes. In the first part we propose two possible shifted variants of the popular (higher-order) power method for the computation of $\ell^p$-eigenpairs proving the convergence of both the schemes to the Perron $\ell^p$-eigenvector of the tensor, and the maximal corresponding $\ell^p$-eigenvalue, when the tensor is entrywise nonnegative and $p$ is strictly larger than the number of modes.
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Paper accepted on NeurIPS19

I am delighted to hear that our paper Generalized Matrix Means for Semi-Supervised Learning with Multilayer Graphs – with Pedro Mercado and Matthias Hein – has been accepted on the proceedings of this year’s NeurIPS conference.

Abstract:
Many graph mining tasks can be viewed as classification problems on high dimensional data. Within this class we consider the issue of discovering core-periphery structure, which has wide applications in the economic and social sciences. In contrast to many current approaches, we allow for weighted and directed edges and we do not assume that the overall network is connected. Our approach extends recent work on a relevant relaxed nonlinear optimization problem.
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--- Edge probability $p_{ij}(u)$ in the logistic core-periphery random graph model.

Abstract:
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-hard. Inspired by recent nonlinear modularity eigenvector approaches, we introduce the modularity total variation $TV_Q$ 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 $TV_Q$.
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The University of Strathclyde is hosting two major conferences this summer:
The 28th Biennial Numerical Analysis Conference (see also my previous post), where I will give the talk Networks core-periphery detection with nonlinear Perron eigenvectors and the 17th Workshop on Advances in Continuous Optimization, where I will give the talk Leading community detection in networks via total variation optimization.

Abstract:
Nonnegative tensors arise very naturally in many applications that involve large and complex data flows. Due to the relatively small requirement in terms of memory storage and number of operations per step, the (shifted) higher-order power method is one of the most commonly used technique for the computation of positive Z-eigenvectors of this type of tensors. However, unlike the matrix case, the method may fail to converge even for irreducible tensors.
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I am organizing a minisymposium and giving a talk:

Minisymposium:Matrix methods for networks
jointly organized with Francesca Arrigo Abstract: There is a strong relationship between network science and linear algebra, as complex networks can be represented and manipulated using matrices. Some popular tasks in network science, such as ranking nodes, identifying hidden structures, or classifying and labelling components in networks, can be tackled by exploiting the matrix representation of the data. In this minisymposium we sample some recent contributions that build on an algebraic representation of standard and higher-order networks to design models and algorithms to address a diverse range of network problems, including (but not limited to) core-periphery detection and centrality.

Speakers:

Francesca Arrigo (Strathclyde)

Mihai Cucuringu (Oxford)

Gissell Estrada-Rodriguez (Heriot-Watt)

Dario Fasino (Udine)

Philip Knight (Strathclyde)

Francesco Tudisco (Strathclyde)

My talk will be on Networks core-periphery detection with nonlinear Perron eigenvectors

Abstract:
Signed graphs encode positive (attractive) and negative (repulsive) relations between nodes. We extend spectral clustering to signed graphs via the one-parameter family of Signed Power Mean Laplacians, defined as the matrix power mean of normalized standard and signless Laplacians of positive and negative edges. We provide a thorough analysis of the proposed approach in the setting of a general Stochastic Block Model that includes models such as the Labeled Stochastic Block Model and the Censored Block Model.
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--- SBM spectral embeddings for different power mean Laplacians

Abstract:
We introduce a ranking model for temporal multi-dimensional weighted and directed networks based on the Perron eigenvector of a multi-homogeneous order-preserving map. The model extends to the temporal multilayer setting the HITS algorithm and defines five centrality vectors: two for the nodes, two for the layers, and one for the temporal stamps. Nonlinearity is introduced in the standard HITS model in order to guarantee existence and uniqueness of these centrality vectors for any network, without any requirement on its connectivity structure.
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ICIAM19: International Congress on Industrial and Applied Mathematics

I am organizing a 16 speakers (super-)minisymposium and giving a talk at the next ICIAM19 conference in Valencia (Spain), July 15-19.

Abstract: The analysis of complex networks is a rapidly growing field with applications in many diverse areas. A typical computational paradigm is to reduce the system to a set of pairwise relationships modeled by a graph (matrix) and employ tools within this framework. However, many real-world networks feature temporally evolving structures and higher-order interactions. Such components are often missed when using static and lower-order methods. This minisymposium explores recent advances in models, theory, and algorithms for dynamic and higher-order interactions and data, spanning a broad range of topics including persistent homology, tensor analysis, random walks with memory, and higher-order network analysis.

Group1 – Community detection and clustering

Christine Klymko, LLNL
Improving seed set expansion with semi-supervised information

Tim La Fond, LLNL
Representing the Evolution of Communities in Dynamic Networks

Nate Veldt, Purdue
Algorithmic Advances in Higher-Order Correlation Clustering

Marya Bazzi, ATI
Community structure in temporal multilayer networks

Group2 – Simplicial complexes

Heather Harrington, Oxford
Topological data analysis for investigation of dynamics and biological networks

Alice Patania, Indiana
Null hypothesis for simplicial complexes

Braxton Osting, Utah
Spectral Sparsification of Simplicial Complexes for Clustering and Label Propagation

Austin Benson, Cornell
Simplicial closure and higher-order link prediction.

Group3 – Tensor methods and high-performance computing

Francesca Arrigo
Eigenvector-based Centrality Measures in Multilayer Networks

Orly Alter, Utah
Multi-Tensor Decompositions for Personalized Cancer Diagnostics, Prognostics, and Therapeutics.

Chunxing Yin, GA Tech
A New Algorithm Model for Massive-Scale Streaming Graph Analysis

Tahsin Reza, UBC
Distributed Algorithms for Exact and Fuzzy Graph Pattern Matching

Group4 – Higher-order random walks

Eisha Nathan, LLNL
Nonbacktracking Walks in Dynamic Graphs

Michael Schaub, MIT
Random walks on simplicial complexes and the normalized Hodge Laplacian

Keita Iwabuchi, LLNL

Francesco Tudisco, Strathclyde
Higher-order ergodicity coefficients

This event is part of the research project MAGNET for which I would like to acknowledge support from the Marie Curie individual fellowship scheme.

Abstract:
We derive and analyse a new iterative algorithm for detecting network core–periphery structure. Using techniques in nonlinear Perron-Frobenius theory, we prove global convergence to the unique solution of a relaxed version of a natural discrete optimization problem. On sparse networks, the cost of each iteration scales linearly with the number of nodes, making the algorithm feasible for large-scale problems. We give an alternative interpretation of the algorithm from the perspective of maximum likelihood reordering of a new logistic core–periphery
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--- Top ten core train stations in the city of London

ALAMA NLA 2019

I have accepted the kind invitation of José Mas and Fernando de Terán Vergara to give a two-hour lecture in June (17-19) at the ALAMA (Spanish Society for Linear Agebra, Matrix Analysis and Applications) 2019 workshop at the Polytechnic University of Valencia.

Research visit @ Uni of Udine

This week I am visiting the Department of Mathematics, Computer Science and Physics of the University of Udine (Italy), invited by Dario Fasino (thanks Dario!). We are planning to work hard to finalize our work on Higher-order ergodicity coefficients for tensors. I will also give a seminar, you can see the details in the flyer below. I am looking forward to exciting days of math and some authentic frico!

Brief lecture course at Rome-Moscow school 2018

I will be visiting the Department of Mathematics of University of Rome “Tor Vergata”, my alma mater, from Monday September 17 to Friday 21 and in occasion of the Rome-Moscow school on Matrix Methods and Applied Linear Algebra. See also my previous post. I will teach a brief course on nonlinear spectral methods for higher-order centrality and core-periphery detection in networks. I hope to finish my notes and the associated Julia Notebooks soon, and post them here. A quick look to the weather forecast: today the temperature in Rome is more than twice (30C) the one here in Scotland (14C).

Abstract:
With the notion of mode-$j$ Birkhoff contraction ratio, we prove a multilinear version of the Birkhoff-Hopf and the Perron-Fronenius theorems, which provide conditions on the existence and uniqueness of a solution to a large family of systems of nonlinear equations of the type $$f_i(x_1,…,x_ν)= λ_i x_i, ,$$ being $x_i$ and element of a cone $C_i$ in a Banach space $V_i$. We then consider a family of nonlinear integral operators $f_i$ with positive kernel, acting on product of spaces of continuous real valued functions.
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Visiting KTH Royal Institute of Technology

I look forward to visit and give a seminar talk at the Numerical Analysis group of the Department of Mathematics of KTH in Stockholm, Sweden. I will stay there for two weeks: Sunday August 26 to Saturday September 8. Thanks Elias for the kind invitation and for hosting me!

Two conferences in June

I am participating at two exciting conferences next June 2018:

The program of both events looks very exciting, I am really looking forward for them.

At SIAM IS I will present my work on Community detection with nonlinear modularity (based on this paper) at the minisymposium Optimization for Imaging and Big Data (see here) organized by Francesco Rinaldi and Margherita Porcelli.

At IMA NLA OPT I will present my work on Small updates of function of adjacency matrices (based on this paper) at the minisymposium Matrix Functions and
Quadrature Rules with Applications to Complex Network (see here) organized by Francesca Arrigo and Stefano Pozza.

I am flying to Hong Kong today to attend the SIAM Applied Linear Algebra conference 2018 where I am organizing a minisymposium on Nonlinear Perron-Frobenius theory and applications and giving a talk titled A new Perron-Frobenius theorem for nonnegative tensors. See also my previous post.

Abstract:
Multilayer graphs encode different kind of interactions between the same set of entities. When one wants to cluster such a multilayer graph, the natural question arises how one should merge the information form different layers. We introduce in this paper a one-parameter family of matrix power means for merging the Laplacians from different layers and analyze it in the stochastic block model. We show that this family allows to recover ground truth clusters under different settings and verify this in real world data.
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Abstract:
We provide explicit expressions for the eigenvalues and eigenvectors of matrices that can be written as the Hadamard product of a block partitioned matrix with constant blocks and a rank one matrix. Such matrices arise as the expected adjacency or modularity matrices in certain random graph models that are widely used as benchmarks for community detection algorithms.
Please cite this work as: @article{fasino2018expected, title={The expected adjacency and modularity matrices in the degree corrected stochastic block model}, author={Fasino, Dario and Tudisco, Francesco}, journal={Special Matrices}, volume={6}, pages={110--121}, year={2018} }

Visiting University of Padua

I will be visiting the Department of Mathematics of University of Padua from Monday 5 to Friday 9 February. Very excited to meet some of my past colleagues and to have the chance to work over various ongoing collaborations.
Moreover, the Italian workshop “Due Giorni” on Numerical Linear Algebra will take place in the Department on Thursday 8 and Friday 9. According to the list of participants and their abstracts a lot of interesting work will be presented. I will give a talk on Multi-dimensional nonlinear Perron-Frobenius
theorem and its application to network centrality. Here is my abstract.

Abstract:
We introduce the concept of shape partition of a tensor and formulate a general tensor eigenvalue problem that includes all previously studied eigenvalue problems as special cases. We formulate irreducibility and symmetry properties of a nonnegative tensor $T$ in terms of the associated shape partition. We recast the eigenvalue problem for $T$ as a fixed point problem on a suitable product of projective spaces. This allows us to use the theory of multihomogeneous order-preserving maps to derive a new and unifying Perron–Frobenius theorem for nonnegative tensors which either implies earlier results of this kind or improves them, as weaker assumptions are required.
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I am organizing a minisymposium and giving a talk at next SIAM Applied Linear Algebra conference in Hong Kong. Below are some detail of my contributions.
This event is part of the research project MAGNET for which I would like to acknowledge support from the Marie Curie individual fellowship scheme.

Minisymposium:Nonlinear Perron-Frobenius theory and applications
jointly organized with Antoine Gautier Abstract: Nonlinear Perron-Frobenius theory addresses problems such as existence, uniqueness and maximality of positive eigenpairs of different types of nonlinear and order-preserving mappings.In recent years tools from this theory have been successfully exploited to address problems arising from a range of diverse applications and various areas, such as graph and hypergraph analysis, machine learning, signal processing, optimization and spectral problems for nonnegative tensors. This minisymposium sample some recent contributions in this field, covering advances in both the theory and the applications of Perron-Frobenius theory for nonlinear mappings. Speakers:

Antoine Gautier (Saarland University, Germany)

Francesca Arrigo (University of Strathclyde, UK)

Shmuel Friedland (University of Chicago, USA)

Jiang Zhou (Harbin Engineering University, China)

Contributed talk: A new Perron-Frobenius theorem for nonnegative tensors Abstract: Based on the concept of dimensional partition we consider a general tensor spectral problem that includes all known tensor spectral problems as special cases. We formulate irreducibility and symmetry properties in terms of the dimensional partition and use the theory of multi-homogeneous order-preserving maps to derive a general and unifying Perron-Frobenius theorem for nonnegative tensors that either includes previous results of this kind or improves them by weakening the assumptions there considered.

I am very happy to hear from Carmine di Fiore that the 6th edition of the Rome Moscow summer school on Matrix Methods and Applied Linear Algebra will take place next summer, between August 25 and September 23, 2018.
This summer school is very peculiar because is long (one month!) and thus allows students to really work over the topics that are discussed. Also it is a wonderful occasion to meet new people in the field of Applied Linear Algebra. I have been student of several editions of the school and strongly encourage participation.

ZiF Final Conference

Getting ready for the ZiF final conference in Bielefeld.
I will talk about the nodal domains of the $p$-Laplacian operator on discrete graphs.

Precisely, this is the abstract of my talk:
The number of nodal domains induced by the eigenfunctions of the Laplacian operator has been completely described both for graphs and for continuous domains. For $p\geq 1$, the $p$-Laplacian is a nonlinear operator which reduces to the standard Laplacian when $p=2$. This nonlinear operator has gained popularity in recent years as, for instance, it can be used to improve data clustering algorithms. We consider a set of variational eigenvalues of the $p$-Laplacian on discrete graphs and analyze the nodal domain structure of the associated eigenfunctions. We show that when $p>1$, the upper bound in the linear nodal domain theorem carries over unchanged to the nonlinear setting, whereas some properties are lost when $p=1$. We also discuss an higher-order Cheeger inequality that can be obtained by exploiting the nodal structure of the $p$-Laplacian.

Abstract:
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.
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Abstract:
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.
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Abstract:
In a graph or complex network, communities and anti-communities are node sets whose modularity attains extremely large values, positive and negative, respectively. We consider the simultaneous detection of communities and anti-communities, by looking at spectral methods based on various matrix-based definitions of the modularity of a vertex set. Invariant subspaces associated to extreme eigenvalues of these matrices provide indications on the presence of both kinds of modular structure in the network.
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--- Two communities and one anti-community coexist in the ‘‘Small World’’ citation network

Abstract:
Identifying important components in a network is one of the major goals of network analysis. Popular and effective measures of importance of a node or a set of nodes are defined in terms of suitable entries of functions of matrices $f(A)$. These kinds of measures are particularly relevant as they are able to capture the global structure of connections involving a node. However, computing the entries of $f(A)$ requires a significant computational effort.
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Abstract:
Revealing a community structure in a network or dataset is a central problem arising in many scientific areas. The modularity function $Q$ 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 module and a set that maximizes $Q$ is thus called leading module. Finding a leading module in a network is an important task, however the dimension of real-world problems makes the maximization of $Q$ unfeasible.
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--- Bipartition obtained by the linear (left) and nonlinear (right) spectral methods. Networks shown, from left to right: Electronic2, Drugs, and YeastS.

Starting at Strathclyde

Today is the first day of my new academic position as Marie Curie Fellow at the department of Mathematics and Statistics of University of Strathclyde.

Abstract:
Let $S$ be a column stochastic matrix with at least one full row. Then $S$ describes a Pagerank-like random walk since the computation of the Perron vector $x$ of $S$ can be tackled by solving a suitable M-matrix linear system $Mx = y$, where $M = I − \tau A$, $A$ is a column stochastic matrix and $\tau$ is a positive coefficient smaller than one. The Pagerank centrality index on graphs is a relevant example where these two formulations appear.
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Abstract:
Hypergraph matching has recently become a popular approach for solving correspondence problems in computer vision as it allows to integrate higher-order geometric information. Hypergraph matching can be formulated as a third-order optimization problem subject to the 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.
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Abstract:
We consider the nonlinear graph $p$-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 $p$-Laplacian for any $p\geq 1$. While for $p>1$ the bounds on the number of weak and strong nodal domains are the same as for the linear graph Laplacian ($p=2$), the behavior changes for $p=1$. We show that the bounds are tight for $p\geq 1$ as the bounds are attained by the eigenfunctions of the graph $p$-Laplacian on two graphs.
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Abstract:
The Perron-Frobenius theory for nonnegative matrices has been generalized to order-preserving homogeneous mappings on a cone and more recently to nonnegative multilinear forms. We unify both approaches by introducing the concept of order-preserving multi-homogeneous mappings, their associated nonlinear spectral problems and spectral radii. We show several Perron-Frobenius type results for these mappings addressing existence, uniqueness and maximality of nonnegative and positive eigenpairs. We prove a Collatz-Wielandt principle and other characterizations of the spectral radius and analyze the convergence of iterates of these mappings towards their unique positive eigenvectors.
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Abstract:
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.
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Abstract:
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.
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Abstract:
Signed networks allow to model positive and negative relationships. We analyze existing extensions of spectral clustering to signed networks. It turns out that existing approaches do not recover the ground truth clustering in several situations where either the positive or the negative network structures contain no noise. Our analysis shows that these problems arise as existing approaches take some form of arithmetic mean of the Laplacians of the positive and negative part.
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Abstract:
We prove that the Bernoulli numbers satisfy some special lower triangular Toeplitz systems of linear equations. One of these systems has a strong link with eleven Ramanujan’s linear equations satisfied by the first eleven Bernoulli numbers.
Please cite this work as: @article{difiore2016lower, title={Lower triangular Toeplitz--Ramanujan systems whose solution yields the Bernoulli numbers}, author={Di Fiore, Carmine and Tudisco, Francesco and Zellini, Paolo}, journal={Linear Algebra and its Applications}, volume={496}, pages={510--526}, year={2016}, publisher={Elsevier} }

Abstract:
We investigate two ergodicity coefficients $\phi_{|| \cdot ||}$ and $\tau_{n-1}$, 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 $\tau_{n-1}$ and we show that, under mild conditions, it can be used to recast the eigenvector problem $Ax = x$ as a particular $M$-matrix linear system, whose coefficient matrix can be defined in terms of the entries of $A$.
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Abstract:
In this paper we study adaptive $L(k)QN$ methods, involving special matrix algebras of low complexity, to solve general (non-structured) unconstrained minimization problems. These methods, which generalize the classical BFGS method, are based on an iterative formula which exploits, at each step, an ad hoc chosen matrix algebra $L(k)$. A global convergence result is obtained under suitable assumptions on $f$.
Please cite this work as: @article{cipolla2015adaptive, title={Adaptive matrix algebras in unconstrained minimization}, author={Cipolla, Stefano and Di Fiore, Carmine and Tudisco, Francesco and Zellini, Paolo}, journal={Linear Algebra and its Applications}, volume={471}, pages={544--568}, year={2015}, publisher={Elsevier} }

Abstract:
Power nonnegative matrices are defined as complex matrices having at least one nonnegative integer power. We exploit the possibility of deriving a Perron–Frobenius-like theory for these matrices, obtaining three main results and drawing several consequences. We study, in particular, the relationships with the set of matrices having eventually nonnegative powers, the inverse of M-type matrices and the set of matrices whose columns (rows) sum up to one.
Please cite this work as: @article{tudisco2015complex, title={On complex power nonnegative matrices}, author={Tudisco, Francesco and Cardinali, Valerio and Di Fiore, Carmine}, journal={Linear Algebra and its Applications}, volume={471}, pages={449--468}, year={2015}, publisher={Elsevier} }