This paper studies how a common, but noisy, price or forecast signal can mask the influence of the communication network among many batteries that try to coordinate charging. The authors show both with equations and computer experiments that when agents act on a shared forecast whose errors are correlated across the fleet, the fleet’s collective behavior collapses onto a single “consensus” pattern. In that situation the detailed topology of who talks to whom leaves almost no detectable fingerprint on the fleet’s effective dynamics.

The setup is simple. Each battery mixes two inputs when choosing charging power: a shared price-like signal and an average of neighbors’ actions. A mixing weight α sets how much an agent trusts the common signal (α=1) versus neighbor observations (α=0). Forecast errors are modeled so that each agent’s error equals a shared shock plus a private error. The shared part is weighted by a correlation parameter ρ that runs from zero (errors independent) to one (errors identical across agents). The authors sweep ρ from 0 to 1 and test three network shapes: a line, a star, and a Watts–Strogatz small-world graph.

Analytically they linearize the dynamics and use a mode decomposition tied to the graph Laplacian. The key math fact is that the correlated part of the forecast error projects entirely onto the single consensus mode (the equal-weights vector), while topology affects only transverse modes. Because the population mean forecast error has variance that tends to ρσ2 as fleet size grows, any nonzero ρ produces an O(1) common forcing that does not average away. As ρ times the fleet size grows, the consensus mode’s variance dominates and the fleet behaves as if it were one degree of freedom, regardless of the wiring.

Simulations support this picture. The authors measure w1, the fraction of total variance in the dominant collective mode, and a participation ratio PR that counts effective collective dimensions. As ρ increases, w1 rises smoothly from about 0.44 to about 0.96 and PR falls from about 4.3 to about 1.1. The dominant empirical mode aligns with the consensus vector very closely; at ρ=1 the alignment is about 0.998 for all tested topologies. They also quantify detectability: the small differences from switching topology (topology spread) are smaller than the variation caused simply by redrawing the forecast noise. For example, at a mixed rule (α=0.5) the topology spread in w1 is about 0.004, while noise variability on a single graph is about 0.03.