Shared Forecast Errors Erase Network Topology From Battery Fleet Dynamics

Referee: §4, Eq. (7) and Proposition 1: The O(1) bound on the topology-dependent factor requires the Laplacian spectrum to be uniformly bounded in N. The manuscript does not specify which Laplacian is used. For the unnormalized Laplacian, the star graph has λ_max = N, which eventually makes |a + κλ_max| ≥ 1, rendering the stationary variance undefined. If the normalized Laplacian is used, the argument is clean and Proposition 1 holds as stated.

Authors: The referee is correct on the substance. The manuscript uses the normalized Laplacian L_norm = D^{-1/2} L D^{-1/2} in the simulation code, whose eigenvalues lie in [0, 2] for all connected graphs, so the bound |a + κλ_m| < 1 is satisfied for all three tested topologies including the star, and Proposition 1 holds as stated. However, the manuscript does not say this anywhere — it refers only to 'the graph Laplacian' without specifying the normalization. This is a genuine omission that makes the analytical argument unverifiable from the text alone, and we will fix it. Specifically, we will: (1) state explicitly in §4 that L denotes the normalized Laplacian with spectrum in [0, 2]; (2) note that the O(1) bound on (1−(a+κλ_m)²)/(1−a²) follows immediately since λ_m ∈ [0, 2] for all connected graphs regardless of N; (3) add a remark that for the unnormalized Laplacian the argument would require bounded-degree graphs and would not cover the star topology, which is precisely why the normalized form is the appropriate choice here. We will also verify in the revised text that the simulation coupling parameter κ is chosen so that |a + κ·2| < 1, confirming the stability condition holds across all modes for all three topologies. revision: yes

Referee: §4–§5: The relationship between the linearized analysis (Eq. 5) and the simulation model (Eq. 2) is not fully specified. Eq. (2) describes a hybrid controller with mixing weight α, while Eq. (5) is a linearized recursion with self-dynamics coefficient a, coupling κ, and noise coupling b. The mapping between these parameterizations is not stated. Without knowing how a, κ, b, and α relate, the reader cannot verify that the simulation parameters fall within the regime where Proposition 1 applies.

Authors: The referee is right that this mapping is not stated and that it is load-bearing for connecting the analytical claim to the simulation results. We will add it. The linearized recursion (Eq. 5) is obtained by linearizing the hybrid controller (Eq. 2) about the operating point set by the time-of-use rule and the PV-derived price signal. The correspondence is: a is the linearized self-dynamics coefficient (from the agent's own state update under the time-of-use rule), κ = (1 − α)·κ_0 where κ_0 is the base neighbor-coupling gain and α is the mixing weight from Eq. (2), and b = α·b_0 where b_0 is the sensitivity of the action to the common price/feeder signal. Thus α = 1 gives κ = 0 (no neighbor coupling, pure common signal) and α = 0 gives b = 0 (no common channel, pure neighbor observation). We will add a paragraph in §4 or at the §4–§5 boundary stating this mapping explicitly, and we will report the numerical values of a, κ, and b used in the simulations along with the verification that |a + κλ_m| < 1 for all modes m and all three topologies. This will allow the reader to confirm that the simulations operate within the regime where Proposition 1 applies. revision: yes