Ongoing joint work with Bhaswar Bhattacharya, Ankan Ganguly and Giulio Zucal
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, the asymptotics of the log-partition function can be approached through an entropy-penalised variational problem of the form
$$ \psi(\boldsymbol\beta)=\sup_{\mathbf{W}}\left\{\sum_k \beta_k\, t(F_k,\mathbf{W})-I_\nu(\mathbf{W})\right\}. $$Questions under study
- How do coloured subgraph interactions determine the asymptotic free energy?
- Which probability graphons describe typical large edge-coloured networks?
- When are maximisers spatially homogeneous, and when do structured colour patterns emerge?
- What optimality equations and compactness properties govern the variational problem?
- How are extremal colouring problems connected to low-temperature Gibbs phases?
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.