Continuous and robust clustering coefficients for weighted and directed networks

We introduce new clustering coefficients for weighted networks. They are continuous and robust against edge weight changes. Recently, generalized clustering coefficients for weighted and directed networks have been proposed. These generalizations have a common property, that their values are not continuous. They are sensitive with edge weight changes, especially at zero weight. With these generalizations, if vanishingly low weights of edges are truncated to weight zero for some reason, the coefficient value may change significantly from the original value. It is preferable that small changes of edge weights cause small changes of coefficient value. We call this property the continuity of generalized clustering coefficients. Our new coefficients admit this property. In the past, few studies have focused on the continuity of generalized clustering coefficients. In experiments, we performed comparative assessments of existing and our generalizations. In the case of a real world network dataset (C. Elegans Neural network), after adding random edge weight errors, though the value of one discontinuous generalization was changed about 436 %, the value of proposed one was only changed 0.2 %.