Fig. 8: Accuracy of semiwalk-based approximations relative to cycle-based Degree of Balance (DoB) estimates.

Cycle-based DoB scores were computed using methods from ref. ¹⁹, and congruence was assessed based on Pearson correlation coefficients, r, and relative error \(\bar{\epsilon }=\left\langle \left\vert \frac{x-y}{y}\right\vert \right\rangle\). a “Grayscale” (k-balance) measures based on semiwalks (MSB) and proper cycles in small networks studied in this paper (see “Methods: New Guinea Highlands tribes” and “Sampson’s Monastery dataset”). In this case, DoB values are reported for all possible cycle lengths and both MSB and cycle-based estimates are exact. b Pearson correlations and relative errors for different cycle lengths calculated over co-sponsorship networks from the U.S. Congress (“Methods: Co-sponsorship relations in the U.S. Congress”). In this case, MSB approximations are based on m = 10 leading eigenvalues, and cycle-based estimates are approximated using sampling based on 10,000 samples. Only cycles of length up to 15 were considered.