Extension of graphon duality beyond simple graphs

Extend the duality between graphons and grapheurs to limits of non-simple graphs, and characterize the relationship between the graphon cut metric and the grapheur W_square metric through this extended duality.

Background

The paper develops a duality between graphons and grapheurs in the setting of simple undirected graphs and proves convergence results using a random pairing. It notes that extending the framework to general grapheurs and more general kernels is technically more complicated.

The authors explicitly ask whether the bi-continuity conjecture holds, whether the duality extends to limits of non-simple graphs, and whether the two metric structures can be related through that duality. This entry records the concrete extension and metric-comparison questions rather than duplicating the separate bi-continuity conjecture.

References

Can the duality with graphons in Section~\ref{sec:dual_graphons} be extended to limits of non-simple graphs? Can we relate the cut metric and our $W_{\square}$-metric via this duality?

— Graph Limits via Quotients  (2512.23149 - Levin et al., 29 Dec 2025) in Section 6, “Conclusions and Future Directions,” item “Extending graphon duality”