Graph-Coupled Oscillator Networks

The paper’s Proposition 3.3 equates “oversmoothing” in continuous time with exponential stability of constant states (c,0) and sketches a bidirectional implication; its intent is clear but the proof is terse and informal about quantifiers and required stability notions. The candidate gives a more structural dynamical-systems argument: decomposing into consensus vs. disagreement directions, tying Dirichlet energy to distance from the consensus manifold, and explaining why exponential decay of disagreement is a transverse stability statement. For Proposition 3.4, the paper proves that under ReLU and α ≥ 1/2, the constant states (c,0) are not exponentially stable, leveraging a linearized energy identity; the candidate supplies a simpler and stronger explanation that no single (c,0) can be exponentially stable for any α ≥ 0 because these equilibria form a continuum (yielding a zero eigenvalue in the consensus block). Minor frictions remain: the paper phrases Prop. 3.3 in terms of “some c” and basins of attraction, whereas the model describes oversmoothing as decay to the set of consensus states and momentarily states a stronger “every (c,0)” stability requirement; still, both reach the same substantive conclusions and no logical contradiction arises. See the statements and proofs as given in the main text and supplement for Prop. 3.3 and 3.4, and the energy identity (C.1) used by the paper’s authors.