Asymptotic Stability of Conservative Convex-Combination Dynamics on Multilayer Graphs

We study discrete-time consensus dynamics on multilayer networks in which each layer evolves via a time-varying doubly stochastic interaction matrix, and inter-layer coupling is introduced through two mechanisms: (i) distribute-then-average and (ii) average-then-distribute. These define conservative redistribution processes that preserve total mass across all layers and can be viewed as stochastic averaging driven by products of time-inhomogeneous stochastic matrices with structured coupling.
For both mechanisms, we construct quadratic Lyapunov functionals that form nonnegative supermartingales, yielding almost sure convergence. The analysis combines martingale arguments with dissipation identities and connectivity properties of induced interaction graphs. Under recurrent connectivity conditions on subgraphs of the time-varying interaction structure, we prove asymptotic consensus to the global average determined by the initial total mass.
This provides a unified framework for multilayer averaging dynamics, extending classical consensus results for products of stochastic matrices to settings with explicit inter-layer coupling. As corollaries, we specialize the general framework to the multilayer garbage disposal dynamics, thereby establishing convergence guarantees under natural connectivity conditions on the underlying graphs.