Fig. 1: Cycle ratios of nodes in an example network.

a An example network with cycle ratios of nodes. Here the number in each node is its label, the value next to it is its cycle ratio and nodes of the same color have the same cycle ratio. b The cycle number matrix of example network in (a) and how to calculate the cycle ratio of node 1. Here the element \(c_{ij}\) in cycle number matrix is the number of shortest cycles that pass through both nodes \(i\) and \(j\) if \(i\, \ne\, j\). If \(i = j\), \(c_{ii}\) is the number of shortest cycles that contain node \(i\). For node 1 the non-zero elements in the green square in the matrix are neighbors with common shortest cycles with node 1, and each value (\(c_{1j}\), where \(j = 1,2,3,4\,{{{{{{{\mathrm{and}}}}}}}}\,5\)) represents the number of these cycles. The elements in the red square (\(c_{jj}\), where \(j = 1,2,3,4\,{{{{{{{\mathrm{and}}}}}}}}\,5\)) are the number of shortest cycles of each neighbor. The sum of the ratios of \(c_{1j}\) and \(c_{jj}\) is the cycle ratio of node 1. c Every node’s associated cycles in \(S\), degree, H-index, coreness¹⁰ and cycle ratio. Here \(S\) is the set of all shortest cycles of example network in (a) and \(S = \{ \{ 1,2,3\} ,\{ 1,2,4\} ,\{ 1,2,5\} ,\{ 1,3,4\} ,\{ 2,3,4\} ,\{ 3,6,7,8\} \}\).