Conditional McKean-Vlasov equations (CMVE) are nonlinear stochastic processes with applications to mean-field games with common noise, stochastic Lagrangian models, and entropic optimal transport. In this talk, I will discuss my recent work with Kavita Ramanan on a CMVE known as the Markov local-field equation (MLFE) that arises in the study of interacting particle systems on sparse networks. We identify a Lyapunov function for the MLFE that we call the sparse free energy and prove a corresponding H-theorem. We then show that the stationary distributions of the MLFE are exactly the marginals of extremal continuous Gibbs measures on regular trees and obtain rates of convergence.