Optimal Recurrent Network Topologies for Dynamical Systems Reconstruction

The paper’s claims about GeoHub’s construction and its qualitative effects are empirically well-supported, but not proved. In particular, Algorithm 2 implements (i) a target-first phase that gives readouts an additive in-degree bias while hidden nodes initially have zero weight, and (ii) a subsequent source phase that adds k/2 incoming edges to every node, which indeed favors readouts as in-degree hubs and yields small-world signatures in measurements (L not exceeding ER; high clustering; positive SWI) . However, the paper does not present a formal proof of these properties. The model’s solution correctly formalizes the in-degree hub bias under the stated algorithmic ordering, but overreaches elsewhere: the small-world bound is asserted (not derived) and, more critically, the universal-approximation step that underpins the “arbitrarily small Dstsp and DH” claim depends on a feedforward construction incompatible with the mean-centered PLRNN used in the paper (M is fixed, no pre-activation bias), so the existence argument does not follow for the stated architecture . The paper’s experimental observation that performance is insensitive to θ0 within a fixed mask is consistent with the model’s qualitative claim, but the latter’s formal existence proof is not established for the paper’s PLRNN and metrics (which are computed via binning/Gaussian smoothing of FFTs) .