## Francesco Tudisco

Assistant Professor (RTDb)

School of Mathematics
GSSI Gran Sasso Science Institute
Viale Francesco Crispi 7 — 67100 — L’Aquila (Italy)
email: francesco dot tudisco at gssi dot it
phone: (+39) 3926535300; (+44) 7501110581

## Minitutorial @ SIAM LA 2021

I am very excited I will be giving a minitutorial on Applied Nonlinear Perron–Frobenius Theory at the SIAM conference on Applied Linear Algebra (LA21).
I will present the tutorial together with Antoine Gautier.

We will introduce the concept of multihomogeneous operators and we will present the state-of-the-art version of the nonlinear Perron-Frobenius theorem for nonnegative nonlinear mappings. We will discuss several numerical optimization implications connected to nonlinear and higher-order versions of the Power and the Sinkhorn methods and several open challenges, both from the theoretical and the computational viewpoints. We will also discuss numerous problems in data mining, machine learning and network science which can be cast in terms of nonlinear eigenvalue problems with eigenvector nonlinearities and we will show how the nonlinear Perron-Frobenius theory can help solve them.

## Editor for SIAM Review

I have accepted an invite to serve as associate editor in the Survey & Review section of SIAM Review (SIREV), the flagship section of one of the highest impact applied math journal. Excited and looking forward to starting!

## Paper accepted on SIAM J Math of Data Science

Excited that our paper Ergodicity coefficients for higher-order stochastic processes, joint work with Dario Fasino, has been accepted on the SIAM Journal on Mathematics of Data Science

## Talk @ SIAM Imaging Science Conference

Last day of the first virtual SIAM Imaging Science conference today. I am presenting a talk at the minisymposium Nonlinear Spectral Analysis with Applications in Imaging and Data Science organized by Leon Bungert (Friedrich-Alexander Universitaet Erlangen-Nuernberg, Germany), Guy Gilboa (Technion Israel Institute of Technology, Israel) and Ido Cohen (Israel Institute of Technology, Israel).

These are title and abstract of my talk:

Nodal Domain Theorem for the p-Laplacian on Graphs and the Related Multiway Cheeger Inequality
We consider the p-Laplacian on discrete graphs, a nonlinear operator that generalizes the standard graph Laplacian (obtained for p=2). We consider a set of variational eigenvalues of this operator and analyze the nodal domain count of the corresponding eigenfunctions. In particular, we show that the famous Courant’s nodal domain theorem for the linear Laplacian carries over almost unchanged to the nonlinear case. Moreover, we use the nodal domains to prove a higher-order Cheeger inequality that relates the k-way graph cut to the k-th variational eigenvalue of the p-Laplacian.

Below you can find my slides, in case you wish to have a look at them

Link to slideshare presentation: Nodal Domain Theorem for the p-Laplacian on Graphs and the Related Multiway Cheeger Inequality

## SIAM NS happening virtually on July 9 and 10

I will preset my first ever Virtual Poster at the first official virtual SIAM Network Science workshop!

Free registration — Tweet feed #SIAMNS20 — More info and schedule: https://ns20.cs.cornell.edu/

Des Higham will present our work on higher-order eigenvector-based network coefficients on July 10, 9am Pacific Time (5pm UK, 6pm EU)

My poster session room will be on nonlinear eigenevector centralities and will be on for 45 min starting at 4pm Pacific Time (midnight UK, 1am EU). Lots of coffee is planned for that day. You may wish to have a look at my poster:

## MSCA Day @ GSSI

We are organizing a “Marie Skłodowska Curie Action Day” virtual event to illustrate some fundamental aspects of Horizon 2020 MSC fellowships. We will discuss some of the application rules, evaluation criteria, how do we think a successful application should be written and we will share personal experiences as recipients and supervisors of MSC individual fellowships.

This event has been promoted and coordinated by my amazing colleague Elisabetta Baracchini

The event will be held virtually on July 2, 9am — 1pm (Italian CET time) via this Zoom meeting room. Details on the program can be found here. Participation is open to everyone and is totally free.

## Virtual minisymposium @ SIAM MDS

Michael Schaub, Santiago Segarra and I are organizing a virtual minisymposium on Learning from data on networks within the SIAM Conference on Mathematics of Data Science 2020, happening virtually during the whole month of June. See also the conference’s virtual program.

Our mini will take place on June 30, staring at 10:00 am Eastern time (Boston)
[7am California, 9am Texas, 3pm UK, 4pm EU, 10pm China]

For more details and to register to join the event online (free of charge), please see the minisymposium webpage.

Abstract

Modern societies increasingly depend on complex networked systems to support our daily routines. Electrical energy is delivered by the power grid; the Internet enables almost instantaneous world-wide interactions; our economies rest upon a complex network of inter-dependencies spanning the globe. Networks are ubiquitous in complex biological, social, engineering, and physical systems. Understanding structures and dynamics defined over such networks has thus become a prevalent challenge across many disciplines. A recurring question which appears in a wide variety of problems is how one can exploit the interplay between the topological structure of the system and available measurements at the nodes (or edges) of the networks. The goal of this minisymposium is to bring together researchers from different mathematical communities – from network science, machine learning, statistics, signal processing and optimization – to discuss and highlight novel approaches to understand and learn from data defined on networks.

Speakers:

• Michael Schaub — Learning from graphs and data on networks: overview and outlook
• Caterina De Bacco — Incorporating node attributes in community detection for multilayer networks
• Danai Koutra — The Power of Summarization in Network Representation Learning (and beyond)
• Ekaterina Rapinchuk — Applications of Auction Dynamics to Data Defined on Networks
• David Gleich — Nonlinear processes on networks
• Jan Overgoor — Choosing To Grow a Graph: Modeling Network Formation as Discrete Choice

New paper out

Abstract: Label spreading is a general technique for semi-supervised learning with point cloud or network data, which can be interpreted as a diffusion of labels on a graph. While there are many variants of label spreading, nearly all of them are linear models, where the incoming information to a node is a weighted sum of information from neighboring nodes. Here, we add nonlinearity to label spreading through nonlinear functions of higher-order structure in the graph, namely triangles in the graph. ... Read more

--- Accuracy of Nonlinear Higher-order Label Spreading on synthetic stochastic block models. Table entries are the average accuracy over 10 random instances.

## Data Science Open Day @ Uni of Padua

Excited to take part today at the Open House event for the Master’s Degree in Data Science at the Department of Mathematics of the University of Padova. I will give a high-level introduction to the problem of link prediction in networks and how to use PageRank eigenvectors to compute a mathematically informed prediction. The live streaming of the event is available on youtube.

New paper out

#### Ergodicity coefficients for higher-order stochastic processes

Abstract: The use of higher-order stochastic processes such as nonlinear Markov chains or vertex-reinforced random walks is significantly growing in recent years as they are much better at modeling high dimensional data and nonlinear dynamics in numerous application settings. In many cases of practical interest, these processes are identified with a stochastic tensor, and their stationary distribution is a tensor Z-eigenvector. However, fundamental questions such as the convergence of the process towards a limiting distribution and the uniqueness of such a limit are still not well understood and are the subject of rich recent literature. ... Read more

## One World seminars

The COVID19 pandemic resulted in the mass cancellation of in-person conferences and seminars across the globe. Wonderful initiatives have resulted as a response to this unfortunate situation. For example, many scientific communities worldwide have started “One World” online seminar series and several conference committees are working in order to put forward online versions of traditional meetings.

Here I would like to list the initiatives related to my research interests that I am aware of. If you know of any other online meeting I have missed, please do let me know!

Acronym Title When Platform
OWML One World Seminar Series on the Mathematics of Machine Learning Wednesdays @ 12 noon ET (UTC-4) Zoom
OWSP One World Signal Processing Seminar Fridays Zoom
MADS Mathematical Methods for Arbitrary Data Sources Mondays @ 2pm CET (UTC+2) Zoom
E-NLA Online seminar series on Numerical Linear Algebra Wednesdays @ 4pm CET (UTC+2) Zoom
MINDS One World Mathematics of INformation, Data, and Signals Seminar Thursdays @ 2:30pm EDT (UTC-4)
OPT One World Optimization Seminar Mondays @ 3pm CEST (UTC+2) Zoom
IMAGINE Imaging & Inverse Problems Wednesdays @ 4pm CET (UTC+2) Zoom
GAMENET One World Mathematical Game Theory Seminar Mondays @ 3pm CEST (UTC+2) Zoom
PROB One World Probability Seminar Weekends @ 3-4pm CEST (UTC+2) Zoom

## Paper accepted on SIAM Applied Mathematics

Our paper Total variation based community detection using a nonlinear optimization approach, joint work with Andrea Cristofari and Francesco Rinaldi from the University of Padua, has been accepted on the SIAM Journal on Applied Mathematics

## Visit and Talk @ Uni Kent

I am traveling today to visit and give a talk at the pure, applicable and numerical mathematics seminar at University of Kent, Canterbury (UK). Thanks Bas Lemmens and Marina Iliopoulou for the invitation and for hosting me!

New paper out

#### Convergence of the nonlinear power method for matrix norms with application to the log-Sobolev constant of Markov chains

Abstract: We analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate $\ell^p$ matrix norms. In particular, exploiting the Birkoff–Hopf contraction ratio of nonnegative matrices, we obtain novel and explicit global convergence guarantees for a range of matrix norms whose computation has been recently proven to be NP-hard in the general case, including the case of mixed-subordinate norms induced by the vector norms made by the sum of different $\ell^p$-norms of subsets of entries. ... Read more

## Seminar talk @ RAL

I am in Oxford (UK) today, giving a talk at the Rutherford Appleton Lab and Uni of Oxford’s Numerical Analysis group joint seminar on Computational Mathematics and Applications. Thank you Michael Wathen and Tyrone Rees for the invitation!

## Paper accepted on Proc Royal Society A

Our paper A framework for second order eigenvector centralities and clustering coefficients, joint work with Francesca Arrigo and Des Higham, has been accepted in the Proceedings of the Royal Society Series A

## Doctoral course @ Uni Padua

Starting from March 1, I will be visiting the University of Padua to teach the doctoral course Eigenvector methods for learning from data on networks for the PhD program in Computational Mathematics. You can use this link if you wish to enroll for my course. Thanks Michela for the invitation!

## Plenary talk @ HHXXI

I have been invited to give a plenary talk this summer at the Householder Symposium XXI. You can read the abstract of my talk from the book of abstracts. Looking forward for this exciting opportunity!

New paper out

#### Nonlocal PageRank

Abstract: In this work we introduce and study a nonlocal version of the PageRank. In our approach, the random walker explores the graph using longer excursions than just moving between neighboring nodes. As a result, the corresponding ranking of the nodes, which takes into account a long-range interaction between them, does not exhibit concentration phenomena typical for spectral rankings taking into account just local interactions. We show that the predictive value of the rankings obtained using our proposals is considerably improved on different real world problems. ... Read more

Konstantin successfully passed today his preliminary PhD exam. Congratulations!

Our paper Generalized Matrix Means for Semi-Supervised Learning with Multilayer Graphs is being presented today by Pedro Mercado at NeurIPS 2019. You may wish to have a look at the poster:

## SIAM Workshop on Network Science 2020

Excited to be part of the Program Committee of the SIAM Workshop on Network Science!

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 workshop is co-located with the Second Joint SIAM/CAIMS Annual Meeting, the SIAM Conference on Imaging Science (IS20), and the Canadian Symposium on Fluid Dynamics.

## Rome-Moscow school 2020

The 7th edition of the Rome Moscow summer school on Matrix Methods and Applied Linear Algebra is in preparation! This is the 10th anniversary of this exciting series of summer schools. The tentative dates for the school are:

• Moscow: Aug 22 – Sept 5
• Rome: Sept 6 – Sept 20

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.

New paper out

#### Generalized matrix means for semisupervised learning with multilayer graphs

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. ... Read more

--- Poster that will be presented at NeurIPS19, by Pedro Mercado

New paper out

#### A framework for second order eigenvector centralities and clustering coefficients

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. ... Read more

--- Nodes with largest spectral clustering coefficient in the karate club network, for different tensors.

New paper out

#### Generating large scale-free networks with the Chung-Lu random graph model

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. ... Read more

--- The power law degree distribution of random graphs in the Chung-Lu model.

New paper out

#### Shifted and extrapolated power methods for tensor $\ell^p$-eigenpairs

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. ... Read more

## 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.

New paper out

#### A fast and robust kernel optimization method for core–periphery detection in directed and weighted graphs

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. ... Read more

--- Edge probability $p_{ij}(u)$ in the logistic core-periphery random graph model.