C3

Diffusion models are central to generative modeling and have been adapted to graphs by diffusing adjacency matrix representations. The challenge of having up to $n!$ such representations for graphs with $n$ nodes is only partially mitigated by using permutation-equivariant learning architectures. Despite their computational efficiency, existing graph diffusion models struggle to distinguish certain graph families and their spectra, unless graph data are augmented with ad hoc features.
This shortcoming stems from enforcing the inductive bias within the learning architecture.
In this work, we leverage random matrix theory to analytically extract the spectral properties of the diffusion process, allowing us to push most of the inductive bias from the architecture into the dynamics. Building on this, we introduce the Dyson Diffusion Model, which employs Dyson's Brownian motion to capture the spectral dynamics of an Ornstein-Uhlenbeck process on the adjacency matrix.
Furthermore, conditioned on the spectral dynamics, we formulate a Lie group diffusion, appropriately modeling the remaining degrees of freedom. Strikingly, the resulting learning problem becomes permutation invariant at the Lie algebra level.
We demonstrate that the Dyson Diffusion Model learns graph spectra accurately and outperforms existing graph diffusion models.
This is joint work with Cai Dieball, Constantin Kogler, Renaud Lambiotte, Arnaud Doucet, Aljaz Godec, and George Deligiannidis.

Paper: Schwarz, T., Dieball, C., Kogler, C., Lambiotte, R., Doucet, A., Godec, A., Deligiannidis, G., 2026, Permutation-Invariant Spectral Learning via Dyson Diffusion, Accepted at NeurIPS 2026

Cell proliferation has been widely studied across experimental systems and mathematical modelling frameworks, from early embryonic development to regenerating tissues. A characteristic feature of early embryonic development is that initially synchronous cell populations progressively lose synchrony after a species-specific number of divisions. Here, we use an age-dependent branching process to determine how variability in cell-cycle duration controls the persistence of synchrony. We derive analytical expressions for the mean and variance of cell number and show how they identify the underlying generation-time distribution. For gamma-distributed generation times, this establishes a direct relationship between cell-cycle variability and the number of synchronous divisions. We use these results to infer generation-time statistics from population-level cell counts and investigate how systematic changes in cell-cycle duration can bias inference under a stationary model. Our results provide a quantitative framework linking the precision of individual cell-cycle clocks to observable population-level dynamics.