Minicourse abstract

Michael Bronstein - Foundations of deep learning on graphs and manifolds

In the past decade, deep learning methods have achieved unprecedented performance on a broad range of problems in various fields from computer vision to speech recognition. So far research has mainly focused on developing deep learning methods for Euclidean-structured data. However, many important applications have to deal with non-Euclidean structured data, such as graphs and manifolds. Such data are becoming increasingly important in computer graphics and 3D vision, sensor networks, drug design, biomedicine, high energy physics, recommendation systems, and social media analysis. The adoption of deep learning in these fields has been lagging behind until recently, primarily since the non-Euclidean nature of objects dealt with makes the very definition of basic operations used in deep networks rather elusive. In this tutorial, I will introduce the mathematical foundations of deep learning on graphs and manifolds, overview existing methods and outline the key difficulties and future research directions.

Talks abstracts

• Michael Bronstein - Geometric deep learning: from Large Hadron Collider to protein design

Geometric deep learning, a novel class of methods extending neural network architectures to deal with non-Euclidean structured data such as graphs and manifolds, has recently become a hot topic in the ML community. In this talk, I will showcase the applications of this promising research field on problems from the domains of 3D computer vision and graphics, medical imaging, drug design, protein science, high energy physics, and social science.

• Elisenda Feliu - Applied algebra in the analysis of biochemical reaction networks

In the context of (bio)chemical reaction networks, the dynamics of the concentrations of the chemical species over time are often modelled by a system of parameter-dependent ordinary differential equations, which are typically polynomial or described by rational functions. The polynomial structure of the system allows the use of techniques from algebra to study properties of the system around steady states, for arbitrary parameter values.

In this talk I will start by presenting the formalism of the theory of reaction networks, and how applied algebra plays a role in the study of relevant questions like the number and stability of steady states, and the existence of periodic solutions. I will proceed to present new results tackling these questions using, among other examples, a ubiquitous and challenging network from cell signaling (the dual futile cycle) as a case example. For this network, which is relatively small, several basic questions, such as the existence of oscillations or the determination of the parameter region of multiple steady states, remain unresolved.

The results I will present arise from different joint works involving Conradi, Kaihnsa, Mincheva, Torres, Yürük, Wiuf and de Wolff.

• Kathryn Hess Bellwald - Topological analysis of networks

Over the past decade or so, graph theory has proved to be extremely useful for analyzing network structure and function, for networks arising in brain imaging, power grids, social networks, and more. More recently, the tools of algebraic topology have been successfully applied to characterizing and quantifying network structure and function, particularly in networks of neurons and brain regions.

In this talk I will present both the general framework for such topological analyses and a number of proof-of-concept case studies, illustrating the utility of these methods.

• Philippe Jacquod - Spectral geometry and dynamical properties of deterministic systems on complex graphs

Graphs/networks are ubiquitous in modern sciences. Roughly speaking, networked systems of interest fall in two classes.

The first class is made up of systems where agents randomly diffuse through the network. The second class includes deterministic dynamical systems defined on graphs, and whose dynamics is determined by well-defined differential equations.

My talk will focus on the second class. I will discuss several popular examples of such systems, such as consensus algorithms, coupled oscillator systems and electric power grids. My interest will be to, first, determine and characterize the steady-state solutions to the corresponding systems of differential equations and, second, to investigate the dynamical properties of those systems once they are perturbed away from those steady-states. I will connect several dynamical quantities following a system disturbance, such as the transient excursion away from the steady-state, the propagation of disturbance waves, the location of the most influential network nodes or the escape probability from the steady-state basin of attraction, to the spectral properties of the coupling graph matrix of relevance.

• Ioan Manolescu - Percolation and other statistical mechanics models

In this talk I will give a basic introduction to the theory of percolation, stating with percolation on trees (aka the Galton-Watson model), moving on to the complete graph (aka the Erdös–Rényi random graph) and finally finishing with the hypercubic lattice Z^d. The aim of the presentation is to familiarise the audience with the phenomena encountered in statistical mechanics, using percolation as an example.

If time permits, I will briefly mention other statistical mechanics models such as the preferential attachement model or self-avoiding walk.

• Toshiyuki Nakagaki - Adaptive development of biological network based on use-and-growth rule

A kind of huge amoeboid organism named Physarum plasmodium constructs an intricate network of veins for circulating nutrients and signals over the entire body. The network shape (topology of connectivity, and sequence of branching in vein network, for instance) is drastically re-organized within hours in response to external conditions. The past studies showed that the network shape was optimized to maximize possibility of survival, in some senses. So we may extract an algorithm for optimal design of functional network from the primitive organism. The key thing is adaptive dynamics of current-reinforcement rule: each vein of network becomes thicker when current is large enough through the vein itself, while it becomes thinner and dies out otherwise. We propose the equations of motion for this simple rule, and functions and formation of transport network is analyzed. We will show that the rule is applicable to the other bio-systems: (1) social dynamics of public transportation, (2) formation of network structure in sponge bone (bone remodeling in other words). A tractable perspective to think similarly of a variety of bio-network is given from the viewpoint of current-reinforcement rule (or, in general, use-and-growth rule)

• Alan Newell - Natural patterns and their defects

Patterns with almost periodic or quasi periodic structures are ubiquitous in nature and in the laboratory. We see them as sand ripples, in cloud streets, in convecting fluids, on optical beams, and as phyllotactic structures with Fibonacci connections on the meristems of plants. We discuss their origins, their properties and in particular their universal nature and their defects, which carry both a topological charge and which store energy, and some intriguing consequences. The lecture should be accessible to all.

• Gerd Schröder-Turk - Morphometric and topological measures for spatial networks

My interest is in spatial networks that represent the geometry and topology of network-like porous solids, where the solid phase or the void phase can be represented as a 3D network. Examples are the pore networks in rocks that determine transport properties, the nanostructures in insect nanostructures that facilitate photonic effects, the strut network structure in soap froths that determine the dynamics and the energy of the froth, or the contact networks in granular materials that determine how forces are transmitted. Often, in these cases the geometric and topological structure of these networks provides clues or insight into the physical properties of these materials. I will provide examples of this type of research, which I often refer to as ‘materials geometry’, as the materials science of matter where the geometry determines the physics and hence the geometric structure analysis is crucial. I will specifically spend time on discussing some metrics which we have dubbed ‘Minkowski tensors’ which straddle the questions of network geometry, structural anisotropy and topology and which we have used to describe physical materials with.

• Robert Smith? - When zombies attack! Mathematical modelling of an outbreak of zombie infection

Zombies are a popular figure in pop culture/entertainment and they are usually portrayed as being brought about through an outbreak or epidemic. Consequently, we model a zombie attack, using biological assumptions based on popular zombie movies. We introduce a basic model for zombie infection, determine equilibria and their stability, and illustrate the outcome with numerical solutions. We then refine the model to introduce a latent period of zombification, whereby humans are infected, but not infectious, before becoming undead. We then modify the model to include the effects of possible quarantine or a cure. Finally, we examine the impact of regular, impulsive reductions in the number of zombies and derive conditions under which eradication can occur. We show that only quick, aggressive attacks can stave off the doomsday scenario: the collapse of society as zombies overtake us all.