Unweighted network measures are commonly used to analyze real-world networks due to their simplicity and intuitiveness. This motivated the search for generalizations of unweighted network measures that take weights into account. We propose a new generalization methodology that captures how focused are the interactions over edges. The less focused the interaction (more uniform over edges) the closer is our generalization to the original unweighted measure. None of the previously developed generalizations capture this aspect of weighted networks. We analyze several real-world networks using our generalizations of the degree and the clustering coefficient. The analysis shows that our generalizations reveal interesting observations.
@cite_1: Abstract Ties often have a strength naturally associated with them that differentiate them from each other. Tie strength has been operationalized as weights. A few network measures have been proposed for weighted networks, including three common measures of node centrality: degree, closeness, and betweenness. However, these generalizations have solely focused on tie weights, and not on the number of ties, which was the central component of the original measures. This paper proposes generalizations that combine both these aspects. We illustrate the benefits of this approach by applying one of them to Freeman’s EIES dataset.
The most recent attempt to generalize the degree (along with betweenness and closeness) relied on mixing total weights (strength) with the cardinality (degree) @cite_1 . The generalized degree based on this method is the @math -degree that we described earlier: @math -degree = @math . Although the generalization methodology were applied to the betweenness and closeness centrality measures, only the degree generalization was evaluated. More importantly, the generalization has a tuning parameter without clear guidelines regarding how to set it, unlike our methodology which is parameterless. Also the connection to the original unweighted measure does not depend on the weights, but on the tuning parameter (when @math equals 0). As a result, the @math -generalization is not really a generalization, but rather a new class of weighted measures.