Spectral Methods Complex Systems

List of basic topics to be covered:

• Graph Theory
• Linear Algebra: from spectra to matrix functions
• Advanced Linear Algebra: from Matrix equations to probabilistic methods for graphs and eigenvalues
• Introduction to Spectral and Pseudospectral Theory of Linear operators
• Random Walks: discrete and continuous
• Iterative Methods for linear systems
• Eigenvalue inequalities
• Transfer operator theory
• Random Matrices

List of applications to be covered:

• Statistical Mechanics
• Complex Networks
• Dynamical systems
• Econophysics and Finance
• Machine learning
• Information theory
• Ecology and Populations
• Biophysics and Biology
• Stochastic thermodynamics
• Control
• Quantum Mechanics

Detailed Abstracts

Wessel Bruinsma and James Requeima

Topics

* Introduction to probabilistic modelling and Gaussian processes

* Kernel approximation

- Random Fourier Features (RFFs)

- RFFs for Gaussian processes: the Sparse Spectrum Approximation (SSA)

- RFFs for Deep Gaussian Processes (DGPs)

* Kernel design

- Spectral Mixture Kernel (SMK)

- Multi-Output Spectral Mixture Kernel (MOSMK)

- Generalised Spectral Mixture Kernel (GSM)

- Gaussian Process Convolution Model (GPCM)

* Variational inference

- The inducing points approximation for Gaussian processes

- Inter-domain inducing points

- Variational Fourier Features (VFFs)

- Rates of convergence

* Spectrum estimation

- Bayesian Nonparametric Spectral Estimation (BNSE)

Daniele Marinazzo

• Connectivity and graph theory
• Large scale modelling in the brain

Jacopo Grilli

Lotka and Volterra were among the first to attempt to mathematize the dynamics of interacting populations. While their work had a profound influence on ecology, leading to many of the results that were covered in the preceding chapters, their approach is difficult to generalize to the case of many interacting species. When the number of species in a community is sufficiently large, there is little hope of obtaining analytical results by carefully studying the system of dynamical equations describing their interactions. I will introduce an approach based on the theory of random matrices that exploits the very large number of species to derive cogent mathematical results. I will review basic concepts in random matrix theory by illustrating their applications to the study of multispecies systems. I will introduce tools that can be used to yield new insights into community ecology, and conclude with a list of open problems.

Anatoly Zlotnik

• Control of linear systems
• Carleman linearization and approximation
• Pseudospectral control

Daniele Marinazzo

The topic closest to spectral methods is the application of the Kernel trick to Granger causality

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.100.144103

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.77.056215

The application to complex systems lies in the fact that a system is characterized by joint dynamics, and that this is a way of inferring the structure of the system looking at its dynamics. It would also be a convenient way to introduce the concept of Granger causality and information transfer which is not universally known across the complex systems community.

Joseph Lizier

Introduction to information theory

Fabio Caccioli

1 - global cascades in complex networks:

2 - stability of financial networks, general introduction to financial networks

-pathways and portfolio stability

Matteo Polettini

SPECTRAL METHODS FOR THERMODYNAMICS (5h)

The course will teach how to handle techniques to compute the statistics of current-type observables of continuous-time Markov chains (CTMC). The course will cover:

- Definition and characterization of CTMCs.

- Definition of state vs. current-like observables, physical interpretation.

- Level 2 large deviations: calculation of the joint rate function, prefactor.

- Level 1 large deviations for the currents: definition of the Scaled-Cumulant Generating Function of the currents (SCGF), derivation of the tilted generator, Perron-Frobenius theorem, SCGF as dominant eigenvalue

- Symmetries of the SCGF under marginal time-reversal, fluctuation theorems, laws of thermodynamics.

- Examples: 4-state system; interacting-particle model.

David Métivier

To study the dynamics of a finite dimensional system, the first step is in general to study some of its spectral propreties. For example, a set of Ordinary Differential Equations (ODE) can be analyzed close to an equilibrium by a finite number of eigenvalues.

When facing similar analyze in infinite dimensional system like Partial Differential Equations (PDE), new pieces of spectrum can appear like the Continuous Spectrum. The dynamical effect of this component is less clear than a standard eigenvalue.

In this lecture, I will start with an introduction to dynamical systems with examples in population dynamics and economy. Then, I will introduce the synchronization phenomenon.

After some mathematical definitions for infinite dimensional systems, we will explore through examples some properties of the Continuous Spectrum with a particular focus on the mixing phenomenon.

I will conclude with application of these concepts to the synchronization phenomenon in the Kuramoto model.

Overview:

- (Quick) Introduction to dynamical systems

- Spectral analysis in infinite dimensional systems

- Mixing phenomenon (and Landau damping)

- Synchronization

Alexis Tantet

Transfer operators govern the decay of correlations, or mixing, and their eigenvalues are associated with resonances in the power spectrum, giving insights on the sensitivity of the system to perturbation. They are thus particularly well suited to the study of the intrinsic variability of chaotic or stochastic systems and their response to forcing.

We present two applications of the study of the spectrum of transfer operators. The first one is concerned with nonlinear oscillators perturbed by noise. The study of the geometry of the underlying system, together with small-noise expansions relying on Floquet theory, allow to reveal the phenomenon of phase diffusion and to quantify the sensitivity of the system to different perturbations. For a large noise, the mixing spectrum may be approximated from reduced transfer operators. This approach is used to study the response to stochastic forcing of the dominant mode of climate variability, namely, El Nino-Southern Oscillation.

Second, critical slowing down at the approach of a chaotic attractor crisis is discussed in terms of spectral gap of the transfer operators. A numerical analysis of a chaotic attractor crisis for the Lorenz flow is presented, allowing to stress the distinct role played by stable and unstable resonances in response to perturbations. The applicability of the reduction to some high-dimension systems is demonstrated with a chaotic attractor crisis in a climate model.

Henk Dijkstra

The spectrum of the transfer operator can be related to the decay of correlations of state variables in a stochastic dynamical system. Transfer operator techniques have recently found application in climate dynamics. In this webinar, I will first give a short introduction into the spectral analysis of transfer operators for elementary stochastic dynamical systems. Next, I will present applications of these techniques in understanding transitions in atmospheric flows and irregular behavior of the El Nino-Southern Oscillation phenomenon.

Sarah de Nigris: 1-1.5 hours

Anomalous diffusion on networks: theoretical aspects and an application to machine learning

Anomalous diffusion processes, both in the superdiffusive regime and in the subdiffusive ones, have spurred a lot of theoretical research effort, along with experimental validation, for decades now. Their description, however, strongly relies on the existence of a metric in continuous space. Complex networks lack an intrinsic metric definition and, in this talk, I will tackle some theoretical "recipes" to work around this issue and recover such regimes on networks as well. More on the applied side, some machine learning algorithms exploit diffusion, for classification tasks as an example, and enhanced diffusion regimes can address and correct some shortcomings of those algorithms.

Thomas Perron - 1, 1.5 hours

Spectra of random networks in the weak clustering regime

The asymptotic behavior of dynamical processes in networks can be expressed as a function of spectral properties of the corresponding adjacency and Laplacian matrices. Although many theoretical results are known for the spectra of traditional configuration models, networks generated through these models fail to describe many topological features of real-world networks, in particular non-null values of the clustering coefficient. Here we investigate effects of cycles of order three (triangles) in network spectra. By using recent advances in random matrix theory, we determine the spectral distribution of the network adjacency matrix as a function of the average number of triangles attached to each node for networks without modular structure and degree-degree correlations.Implications to network dynamics are discussed. Our findings can shed light in the study of how particular kinds of subgraphs influence network dynamics.

Petter Holme

Title: Temporal networks of human interaction

Abstract:

Since the turn of the millennium, networks have become a universal paradigm for simplifying large-scale complex systems, and for studying their system-wide functionalities. At the same time, there is considerable evidence that temporal structures, such as the burst-like behavior of human activity, affect dynamic systems on the network. These two lines of research come together in the study of temporal networks. Over the last five years, there has been a growing interest in how to analyze and model datasets in which we not only know which units interact (like in a traditional, static network), but also when the interactions take place. Just like static network analysis, the development of temporal network theory has been accelerated by the availability of new datasets. It should be noted that temporal networks are more than just extensions of static networks—they are e.g. (unlike simple, directed, weighted and multiplex networks) not transitive. In other words, if A and B are connected, and B and C are also connected, this does not imply that A and C are connected. Perhaps for this reason, temporal network theory has focused less on structural measures and studies of simple evolutionary models, and more on randomization studies and the simulation of spreading on empirical data. I will describe the state of the field, my own contributions (mostly about how temporal contact patterns affect infectious disease spreading), and discuss some future challenges.

Naoki Masuda

“Random walks on networks”

In the lecture, I will cover random walks on networks and their applications to model dynamics and mine information from networks. I will first spend sufficient time to cover basics such as the stationary density, relaxation time, mode decomposition, exit time and so on. Then, I will cover several applications such as the PageRank, pairwise comparison, respondent-driven sampling and opinion dynamics.

Alioscia Hamma Exact solution of the Quantum Ising model in 1d

Massimiliano Lupo Pasini see attached

Francesco Caravelli

• Brief introduction to matrices and eigenvalues
• Graph theory, Laplacians and their properties
• giant component, stability of a fixed point and maximum eigenvalue of a Jacobian
• Markov chains and mapping to resistance networks
• A series of trickeries for the evaluation of eigenvectors/eigenvalues