We next hypothesize that increased design connectivity will improve Δ𝐺 precision and that the expected number of edges for a connected design (Eq. 13) may be a tipping point for how 𝑘 should vary with 𝑛 to control error. To test this hypothesis, we ran numerical optimization simulations (see Methods and Materials) for A-optimal, D-optimal, and 'random' designs over varied parameter sets of (𝑛, 𝑘) with 200 replicates each (Fig. ). Here, we label designs we randomly generate until we find a sufficiently connected solution random designs (defined in Eq. 9). We performed simulations for 10, 15, 20, and 30 nodes and edge numbers ranging from 𝑛 -1 to 𝑘 𝑓 𝑢𝑙𝑙 . From this analysis, we found a rapid increase in the expected mean squared error of Δ𝐺, ⟨MSE(Δ𝐺)⟩, with decreasing edge count (Fig. ). In Fig. we show the ⟨MSE(Δ𝐺)⟩ values over the 200 simulations as points along curves at constant value of 𝑛, or 'isonode' curves, plotted with interpolation. Importantly, ⟨MSE(Δ𝐺)⟩ values worsen with increasing 𝑛 when linearly scaling the edge count at 1𝑛 or 2𝑛 edges.