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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.
From: Hsin-Lun Li [view email]
[v1]
Thu, 25 Dec 2025 15:47:28 UTC (5,934 KB)
[v2]
Tue, 23 Jun 2026 11:45:15 UTC (1,985 KB)
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