NETWORKS Lecture Series

Besides the regular lectures given by our researchers and professors within the IMT School's PhD program in Systems Science and other PhD tracks, NETWORKS hosts special lecture series and workshops by our Visiting Professors & Guest Scholars

23/11/2023, IMT School of Advanced Studies Lucca, San Francesco complex

A random day on random graphs and random walks

The NETWORKS Unit hosts two seminars by mathematicians Rajat Hazra and Luca Avena

Rajat Hazra

Associate Professor, University of Leiden, The Netherlands

Luca Avena

Associate Professor, University of Florence, Italy

Seminar by Rajat Hazra: Spectra of inhomogeneous random graphs

23/11/2024, 9:30-11:00 (Sagrestia, San Francesco complex).

Also online at

Abstract: I will describe some results on the bulk of the spectrum of sparse and dense inhomogeneous random graphs and how the two spectrums are related. In the second part of the talk, I will focus on the edge on the spectrum and describe properties of the largest eigenvalue, eigenvectors and some related large deviation results.

Seminar by Luca Avena: Meetings of random walks & consensus dynamics on sparse random digraphs

23/11/2023, 16:00-17:30 (Aula 1, San Francesco complex).

Also online at

Abstract: I will first discuss the general relation between meeting time of random walks and consensus time in Markovian opinion dynamics on finite networks. I will then focus on sparse random directed graphs and give an account on a recent result in which we characterize in such directed setup how random walks meet and related implications for opinion dynamics. I will in particular discuss how the directed degree structure of the underlying network may speed up or slow down meetings of different random walks.

14/04/2023 - 26/05/2023, IMT School of Advanced Studies Lucca 

(also online at

Dynamics, heterogeneity and fluctuations

Prof. Roberto Livi

IMT Visiting Professor & Honorary Professor, University of Florence; 

President of the Italian Statistical Physics Society (SIFS).

This series aims at introducing the audience to the theory of dynamical systems, encompassing deterministic, disordered, stochastic and irreversible systems, providing a unified description in terms of various statistical indicators, typically associated with the solution of spectral problems. The series consists of four self-contained daily blocks, where each block has a first 2-hour introductory lecture intended for a broader audience (11:00 - 13:00) and a second 2-hour lecture providing a more detailed illustration of useful methods and applications (14:00 - 16:00). 

Useful resources at this link.

Block 1. Deterministic dynamical systems: tools and methods

14/04/2023, 11:00-13:00 (Sagrestia, San Francesco complex) and 14:00-16:00 (Sagrestia, San Francesco complex). 

Also online at

The first block is devoted to the description of the basic mathematical tools, needed to study determinisitic dynamical systems. As applications we discuss how to characterize the dynamics by statistical methods (symbolic representation, etc) and by its spectrum of Lyapunov exponents, making use of analytical and numerical techniques. Topics include:

Block 2. Dynamical evolutions in spatially extended systems

28/04/2023, 11:00-13:00 (Sagrestia, San Francesco complex) and 14:00-16:00 (Sagrestia, San Francesco complex).

Also online at

The second block deals with deterministic dynamical systems made of many degrees of freedom, organized on lattices and networks. Applications will be focused on the relation of hydrodynamics with the Lyapunov spectrum in conservative systems, on the spectral method for solving transport problems in disordered harmonic chains and on suitable methods (stable chaos, heterogeneous mean-field) for models of neural networks. Topics include:

Block 3. Stochastic dynamics: basic ingredients

12/05/2023, 11:00-13:00 (Sagrestia, San Francesco complex) and 14:00-16:00 (Conference Room, San Ponziano complex). 

Also online at

The third block provides an introduction to stochastic processes in discrete-time (Markov chains) and continuous time (Langevin, Fokker-Planck). For what concerns applications we focus on simple examples of Markov processes (random walker on a ring, Monte-Carlo algorithms) and on models of anomalous diffusion. Topics include:

Block 4. Stochastic processes, reversibility and fluctuation relations

26/05/2023, 11:00-13:00 (Sagrestia, San Francesco complex) and 14:00-16:00 (Sagrestia, San Francesco complex). 

Also online at

The fourth block deals with the problem of “irreversibility” in stochastic processes in the framework of the Chapman Kolmogorov equation and on the basic aspects of fluctuation-dissipation relations. Applications are devoted to outline the general approach to solve “first passage problems” and also to describe Jarzinsky, Crooks and Gallavotti-Cohen relations, associated with fluctuating thermodynamics. The final comments will be about linear response theory and coupled transport. Topics include:

25/11/2019,  27/11/2019, IMT School of Advanced Studies Lucca.

Mathematical theory of complex networks

Dr. Luca Avena

IMT Visiting Professor & Assistant Professor, University of Leiden.

This is a series of two mini-workshops. Each workshop begins with a longer lecture by the visiting professor, followed by a series of shorter seminars by IMT professors and PhD students that work on related themes. There will be plenty of time devoted to informal discussions, aimed at establishing potential lines of joint research between Dr. Avena and IMT researchers and students. 

To keep focus, each mini-workshop has a specific theme: "Network coarse-graining" (25 November, 9:00-13:00) and "Brain networks and metastability" (27 November, 9:00-13:00). Both workshops are open to all IMT members, to maximize the number and diversity of potential collaborations generated by the event. Whoever is interested in the topics discussed will have a chance of further discussing with the visiting professor during the week.

Workshop 1. Network coarse-graining

27/11/2019, 9:00-13:00 (Aula 2, San Francesco complex)

Network coarse-graining refers to the reduction of a larger network to a smaller version of it, by keeping certain properties preserved. This procedure is needed in a diverse range of disciplines, especially when the original network is too big to be dealt with mathematically, computationally or experimentally. The mini-workshop aims at exposing different techniques that have been proposed across disciplines (most prominently mathematics, physics and computer science), and to highlight the open challenges that require a multidisciplinary approach.


Workshop 2. Brain networks and metastability

27/11/2019, 9:00-13:00 (Aula 2, San Francesco complex)

Metastability is the property of systems that, before eventually reaching a completely stable state, visit a number of intermediate states that are temporarily and transiently stable. This workshop aims at introducing some mathematical aspects of metastability and discussing the potential applications to the empirical analysis of brain networks, especially in relation to the presence of possible transient neural states and/or modular functional structures.