For monotone aggregation dynamics on finite graphs with heterogeneous update rules, can one derive sharp phase transitions in consensus time as a function of network topology and the spectral gap of the opinion update operator?

A resolution would unify two largely separate bodies of work, namely algebraic and spectral graph theory on one side and nonlinear stochastic dynamics on the other. It would provide practical guarantees for distributed computing protocols and opinion formation models in social networks, give new tools for bounding mixing times of non-reversible interacting particle systems, and likely require developing a theory of nonlinear spectral gaps that could have independent consequences for functional inequalities on discrete spaces.