SYNCHRONIZATION IN RANDOM NETWORKS OF IDENTICAL PHASE OSCILLATORS: A GRAPHON APPROACH

The paper’s Theorem 2.3 proves L∞ convergence at a fixed time from the sampled dynamics (SDS) to the continuum dynamics (CDS) under W ∈ C^1, Lipschitz f and D, and α_n = ω((log n)/n), by first comparing SDS to an averaged system (ADS), then ADS to CDS, and combining the two bounds; see the stated result and proof roadmap, including Propositions 3.1 and 3.2 and the scaling requirement on α_n . The candidate solution follows the same structure (SDS→DDS/ADS via Bernstein concentration and a degree bound; DDS/ADS→CDS via an O(1/n) quadrature error using W ∈ C^1) and closes with Grönwall, matching the paper’s assumptions and conclusion. Differences are stylistic (e.g., a time-net in the candidate vs. a squared-error energy method in the paper), not substantive.