Fig. 6.

Spectrum of the nonbacktracking matrix in the complex plane for some real networks commonly used as benchmarks for community detection, taken from refs. 30–35. The radius of the circle is the square root of the largest eigenvalue, which is a heuristic estimate of the bulk of the spectrum. The overlap is computed using the signs of the second eigenvector for the networks with two communities, and using k means for those with three and more communities. The nonbacktracking operator detects communities in all these networks, with an overlap comparable to the performance of other spectral methods. As in the case of synthetic networks generated by the stochastic block model, the number of real eigenvalues outside the bulk appears to be a good indicator of the number q of communities.