Joint work with Bhaswar B. Bhattacharya, Ankan Ganguly and Giulio Zucal.
Preprint: Colorful Exponential Random Graph Models, arXiv:2608.31130v1, submitted 31 August 2026.
Exponential random graph models assign Gibbs weights to networks according to selected graph statistics. In the edge-coloured setting each edge takes one of $q$ colours, and the interaction may reward or penalise specified coloured subgraphs.
A probability graphon for $q$ colours is a symmetric measurable map into the simplex,
$$ \mathbf{W}(x,y)=\bigl(W_1(x,y),\ldots,W_q(x,y)\bigr)\in\Delta_{q-1}, \qquad W_a(x,y)\ge 0,\quad \sum_{a=1}^{q} W_a(x,y)=1 . $$For an edge-decorated graph $F$ carrying a colour label $\ell(ij)$ on each edge, its homomorphism density in $\mathbf{W}$ is
$$ t(F,\mathbf{W})=\int_{[0,1]^{V(F)}}\ \prod_{ij\in E(F)} W_{\ell(ij)}(x_i,x_j)\ \mathrm{d}\mathbf{x}. $$Building on the large-deviation principle for probability graphons, we express the limiting normalised log-partition function through an entropy-penalised variational problem of the form
$$ \psi(\boldsymbol\beta)=\sup_{\mathbf{W}}\left\{\sum_k \beta_k\, t(F_k,\mathbf{W})-I_\nu(\mathbf{W})\right\}. $$Here $I_\nu$ is the entropy cost relative to the reference colour distribution $\nu$. Typical large colourings concentrate around the set of maximisers.
Main results
- Replica symmetry: we identify families of models with constant variational maximisers, corresponding to asymptotically independent edge colours.
- High temperature: Euler–Lagrange fixed-point equations yield a general criterion for a unique constant maximiser.
- Zero temperature: the dominant energy first selects ground states; the remaining energy terms and entropy then select among them.
- Structured phases: induced-wedge and rainbow-triangle models connect this selection principle to extremal combinatorics and exhibit symmetry breaking at finite temperature.
Numerical experiments complement the rigorous results. The proved regimes do not give a complete finite-temperature phase diagram.
Public presentation
Presented at IMPMS 2026 (Palermo) in the contributed session Asymptotics of Random Graphs, which I co-organised with Elena Magnanini:
Exponential Random Edge-Coloured Graphs via Probability Graphons: Free Energies and Extremal Colorings.