and 𝐀 = 𝐂 𝑖 -𝐂 𝑓 𝑢𝑙𝑙 . For 10, 20, and 40 𝑛, we generated random similarity scores ranging from 0 to 1. We generated random similarities via the NumPy random_integers() function and then symmetrized the similarity array. We then varied 𝑘 from 𝑛 -1 to 𝑛(𝑛 -1)∕2 and generated random designs for each value of 𝑛. For each replicate, random designs were generated until a solution was found that satisfies (Eq. 9) so that the design is minimally connected and invertible. For 40 nodes, an edge count below 42 did not randomly generate an invertible solution, so we omitted designs with less than 42 edges from the analysis. We then calculated ||𝐀|| 𝐹 according to Eq. 25 and repeated this process for 20 trials for each 𝑘 and 𝑛 combination. The similarity arrays were held constant for each value of 𝑛. Finally, we calculated the standard deviations in the Frobenius distances for the error in Fig. . The standard deviations are the errors shown in Fig. with light smoothing using a 1-D Gaussian filter with a standard deviation for the Gaussian kernel of 1.2.