Source
- Li, Chen, Han, Zeng, Liu & Tang (2025), LIT-676, read in full in NOTE-529: Observations 3–4, Figs. 4–5.
- Zhou, Zhang & Tsang (2023), LIT-666, read in full in NOTE-517: §§IV–VII, Figs. 5–9, Theorems 1–2.
Both readings name this as a candidate account and hold it back as not yet filed (NOTE-529 and NOTE-517, Bearing on the record).
What was actually shown
Graphs (LIT-676). On contextual stochastic block model graphs, with two Gaussian feature classes and in-class and out-class edge probabilities, the linear barrier between GCNs trained from different seeds falls as density and feature separability rise, and is highest at middling homophily, with strongly homophilous and strongly heterophilous graphs better connected (Observation 4, Fig. 5). On a given graph, GCN, GraphSAGE and GAT give similar barriers while an MLP on the same features differs markedly (Observation 3, Fig. 4). Graphs from similar domains give similar interpolation curves (Observation 2).
Federated averaging (LIT-666). Ten clients, Dirichlet label skew with α ∈ {0.1, 0.2, 0.5}. FedAvg global models from one initialisation, trained under different heterogeneity, disagree on more than 20% of inputs, show a line barrier with an accuracy drop of about 10% that is larger for (0.1, 0.2) than for (0.1, 0.5) and grows over rounds, and are joined by a one-bend polygonal chain without barrier (Figs. 5–7). In a two-layer mean-field model with heterogeneity as Gaussian noise of variance βh(α) on each neuron’s update, two global models are ε_C-connected by a 7-segment path with ε_C ∝ √h(α)((√log N + z)/√N + …) (Theorem 2). In MNIST runs the line barrier falls with width from N = 200 to N = 6400 (Fig. 9).
What could have come out otherwise. Barriers could have tracked the architecture rather than the data, or stayed flat as heterogeneity rose. In both settings they moved with the data.
What this does not say
- That data structure is the only thing that sets the barrier. The federated models also differ in random start and training history; no IID control is reported.
- That the bounds hold. LIT-676’s barrier bound (through the spectral gap), its generalisation bound and its graph distance cite an appendix that holds no proofs, and use undefined constants. LIT-666’s Lemma 1 requires h(α) ≤ C₅ and Theorem 1, which uses it, states h(α) ≥ C₅; its theory is for two-layer networks, squared loss and one-pass data, its experiments for VGG11 and cross-entropy.
- That the CSBM results cover heterophily as defined. The paper’s homophily definition requires p_in > p_out, so h > ½, yet heterophilous results are reported (NOTE-529).
- That the real graphs match the synthetic trend. The “match” is asserted, and the real-graph barriers are given only as plots.
Connections
- THEORY-115 holds that from a shared, already-stable start, linear connectivity holds only when the objectives share structure. This account is the other side: one objective, no shared stable start, and the data’s structure moving the barrier. THEORY-115 already cites both papers as limits on its own scope.
- THEORY-557 takes LIT-676’s result that the curved path removes the barrier.
- THEORY-109. LIT-666’s Theorem 2 applies Kuditipudi et al.’s route, dropout stability implies a short low-loss path (LIT-671), with heterogeneity worsening the stability.
- QUESTION-069 asks whether connectedness is a property of the solutions or of the regime they were trained in. On this account the data are part of the regime.