Wasserstein ergodicity and phase transitions in iterated function systems on sparse random graphs

We study Wasserstein ergodicity and phase transitions for networked iterated function systems (IFS) on sparse graphs. Our main results are: under split-Lipschitz contractivity of the node maps with constants (αself,βnbr) and a graph-level spectral-gap certificate gap(P)≥δ>0 for the random-walk matrix P, the global Markov operator on P2(Rnd) has a unique invariant law and exhibits exponential W2-mixing with explicit rate κ(δ)1/2, where κ(δ)=αself2+βnbr2(1−δ)<1. Conversely, when the graph is disconnected the kernel decomposes componentwise, and global uniqueness of the invariant measure generally fails. The proofs combine synchronous couplings with W2-contractivity and graph-aware spectral estimates. We give finite-size bounds explicit in (αself,βnbr,gap(P)) and, for linear–Gaussian dynamics, an exact Gaussian stationary law via a discrete Lyapunov (Kronecker) formula. Specializing to Erdős–Rényi graphs, the connectivity scale pn=clognn (c>1) ensures a positive spectral gap, thereby transferring the quenched bounds to typical instances at that scale. Finally, we quantify the dependence on gap(P) and present empirical evidence (together with conjectures) for the scaling of mixing times near connectivity, delineating how graph sparsity, local contractivity, and spectral geometry jointly govern ergodic behavior.