Extension to undirected configuration models with heterogeneous degrees

Extend the scaling results for the voter model on random d-regular graphs to the undirected configuration model with prescribed heterogeneous vertex degrees by deriving explicit formulas for the diffusion constant and the profile function that govern the evolution of the expected fraction of discordant edges, analogous to the constants θ_d and the function f_d(t) identified in the regular case.

Background

For random regular graphs, the voter model exhibits three time scales and admits a precise description of the expected fraction of discordant edges via a profile function f_d(t) tied to random-walk meeting times on the infinite d-regular tree, and a diffusion constant θ_d that enters the Wright–Fisher scaling limit. These results are encapsulated in Theorems 2.3 (Theorem 1) and 2.3 (Theorem 1plus).

The paper explicitly notes that extending these results beyond regular graphs to the undirected configuration model with heterogeneous degrees is currently unresolved; moreover, even the form of the corresponding θ and f functions is unknown. In contrast, for directed configuration models, the authors cite work that successfully identifies analogous quantities under mild conditions, underscoring the challenge specific to the undirected heterogeneous-degree case.