The structure of weighted small-world networks

The small-world property, vertices are highly clustered yet the path length between them is small, has been widely studied in unweighted graphs. In many real-world networks, the connections have associated weights that represent the strength of the interactions among vertices. This important feature, however, has not drawn a lot of attention until recently because the weighted networks are more difficult to analyze than their unweighted counterparts. In this study, the structure of the weighted small-world networks is investigated. Two weighted scientific collaboration networks, whose unweighted versions have been shown to have small-world properties, were analyzed. We generalized the clustering coefficient, and the average shortest path to combine the weight and the topological structure of the network. Furthermore, the nested community structure was also studied. By zooming in the small-world networks, we showed that the sub-networks are still small-world networks.