SAMBa Conference 2026

Title: From Graphs to Hypergraphs: Geometry-Aware Generative Modelling by Heat Diffusion

Abstract: Graphs and hypergraphs provide natural representations for complex relational systems, from molecules and social networks to biological interaction systems, co-authorship data and knowledge structures. Yet generating new realistic graphs or hypergraphs from training data remains challenging: these objects are discrete, combinatorial, permutation-symmetric and strongly structured. Generative modelling asks how we can learn from observed examples and then produce new examples with similar statistical and structural properties. Recent developments in machine learning, including diffusion and flow-based generative models, have transformed generative modelling for continuous data, but their direct application to relational objects can obscure the geometry that makes graphs and hypergraphs distinctive.

In this talk, I will describe a geometry-aware approach to graph generation based on continuous-time diffusion processes defined directly from graph operators. The first part of the talk will focus on simple graphs represented by adjacency matrices. I will introduce a generator-matching framework in which the graph Laplacian and its heat kernel define a topology-aware forward noising process. This heat diffusion progressively smooths graph structure in a manner determined by the graph itself. The associated infinitesimal generator can then be learned by a neural surrogate and reversed, using the probability flow ODE, to generate new graphs.

The second part extends this perspective from pairwise graphs to higher-order relational data. Here, I will consider a structured stochastic diffusion model for hypergraph generation defined directly on incidence matrices. Using a two-sided heat operator acting across both nodes and hyperedges, together with an Ornstein–Uhlenbeck component, we obtain a forward process with a tractable Gaussian terminal law. This process preserves hypergraph-aware structure near the data while enabling reverse-time generation from a universal base distribution.

Taken together, these works show how the operators that describe relational structure can also be used to design generative models, leading to graph and hypergraph generators with clearer geometry, stronger inductive bias and a more principled connection between structure and stochastic dynamics.

Title: Reproducibility crises in machine learning: root causes, potential solutions and the road to the clinic

Abstract: Machine learning has a reproducibility problem. Across published work it is routinely difficult to reproduce reported results even with the code, the models and the data to hand. The consequences run from wasted effort to unsafe conclusions. I will argue this is no accident: each link in the chain from data, through code and models, to papers and peer review is systematically compromised, and I will trace several of the root causes. I will look in particular at missing data and imputation, where standard quality metrics such as mean squared error can be actively misleading and a distributional view (e.g. the Wasserstein distance) is more faithful. I will then turn to what reproducibility demands once we want to use a model rather than merely publish it, drawing on our work translating an AI algorithm for coronary OCT imaging from a research prototype towards a regulated clinical tool. The mathematics, software engineering and the incentives all have to line up and quite often don’t.

Title: The dynamics of working with industry

Abstract: In this talk I shall share my personal experience how, as an academic with a background in applied dynamical systems, in the second half of my career I have developed an approach to working with industry and external partners. I shall illustrate via a few examples of projects I have worked on at the industrial/academic interface. I shall also describe some of the current mechanisms and opportunities for knowledge exchange, coordinated by the UK KE Hub for the mathematical sciences. But, most of all, I shall try to explain how working on applied projects is a 'people sport' that involves a lot of listening and networking. I have also found that while knowledge exchange rarely requires your precise area ofexpertise, it does not necessarily mean selling your mathematical soul. Often external partners look to work with mathematical scientists because we are blue-sky thinkers, people who can spot connections between seemingly disparate topics. Nevertheless, in my experience, working on genuinely applied problems sometimes has the uncanny benefit of to feedback into cool new areas of fundamental research.

Title: A Moving boundary problem for Brownian particles with singular forward-backward interactions

Abstract: In this talk, we shall consider a system of Brownian particles, each of which is absorbed upon hitting an associated moving boundary. The dynamics of the boundaries are determined by the conditional probabilities of the particles being absorbed before some final time horizon, given the current knowledge of the system. While the particles evolve forward in time, the conditional probabilities must be resolved backward in time, thus giving rise to a system of singular forward-backward SDEs coupled through hitting times. We will see that its analysis leads to a novel type of tiered moving boundary problem, with each level corresponding to a different configuration of unabsorbed particles. We shall discuss the well-posedness of this problem and relate it rigorously to the original forward-backward system. Moreover, we will discuss an application to the study of contagion in financial networks, and we will give some illustrations of how the system behaves.

Title: Modelling and control of opinion dynamics on evolving networks

Abstract: The field of opinion dynamics has recently seen a large interest from the mathematics community, both from modelling and control perspectives. Most works focus on the Hegselmann-Krause model, a bounded confidence model that assumes everyone can communicate with everyone else as long as their opinions are close enough. Typical results focus on analysing whether the system achieves consensus, and control strategies aim to steer a population towards consensus (or make it so more quickly) by using controls that act directly on agents.

In this talk, after an overview on my research and experience, I will explore different ways of making these models more realistic, from (a) having opinions evolve within a social network that constrains communication between agents (b) restricting controls to a subset of the network, and (c) considering one-to-one Boltzmann-type interactions with different types of agents who can lie to steer the system, with the overall goal of optimising the system dynamics to achieve consensus.