Figure 4

Real world performance.

The dots are the eigenvalues of the respective matrices. The black circle is the approximate analytical bound of the bulk eigenvalues for the non-backtracking and flow matrices, respectively (Ref. 20) and (Ref. 21), where c is degree and 〈^.〉 is an average. These bounds were derived for the stochastic block model, so are used here as an heuristic guide for the distribution of eigenvalues resulting from the real-world networks and computed using their degree distribution. n denotes the number of nodes in the network. m denotes the number of undirected edges in the network. All the random walk operators are square matrices of order 2m. Values in legends are NMI from Table 1.