Comparison of Different Generalizations of Clustering Coefficient and Local Efficiency for Weighted Undirected Graphs

Binary undirected graphs are well established, but when these graphs are constructed, often a threshold is applied to a parameter describing the connection between two nodes. Therefore, the use of weighted graphs is more appropriate. In this work, we focus on weighted undirected graphs. This implies that we have to incorporate edge weights in the graph measures, which require generalizations of common graph metrics. After reviewing existing generalizations of the clustering coefficient and the local efficiency, we proposed new generalizations for these graph measures. To be able to compare different generalizations, a number of essential and useful properties were defined that ideally should be satisfied. We applied the generalizations to two real-world networks of different sizes. As a result, we found that not all existing generalizations satisfy all essential properties. Furthermore, we determined the best generalization for the clustering coefficient and local efficiency based on their properties and the performance when applied to two networks. We found that the best generalization of the clustering coefficient is C-M,C-hm, defined in Miyajima and Sakuragawa (2014), while the best generalization of the local efficiency is E-loc(P), proposed in this letter. Depending on the application and the relative importance of sensitivity and robustness to noise, other generalizations may be selected on the basis of the properties investigated in this letter.