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Abstract: We propose a metric for vulnerability of labeled graphs that has the following two properties: (1) when the labeled graph is considered as an unlabeled one, the metric reduces to the corresponding metric for an unlabeled graph; and (2) the metric has the same value for differently labeled fully connected graphs, reflecting the notion that any arbitrarily labeled fully connected topology is equally vulnerable as any other. A vulnerability analysis of two real-world networks, the power grid of the European Union, and an autonomous system network, has been performed. The networks have been treated as graphs with node labels. The analysis consists of calculating characteristic path lengths between labels of nodes and determining largest connected cluster size under two node and edge attack strategies. Results obtained are more informative of the networks’ vulnerability compared to the case when the networks are modeled with unlabeled graphs. [ABSTRACT FROM AUTHOR]

Abstract: Rich-club and page-club coefficients and their null models are introduced for directed graphs. Null models allow for a quantitative discussion of the rich-club and page-club phenomena. These coefficients are computed for four directed real-world networks: Arxiv High Energy Physics paper citation network, Web network (released from Google), Citation network among US Patents, and email network from a EU research institution. The results show a high correlation between rich-club and page-club ordering. For journal paper citation network, we identify both rich-club and page-club ordering, showing that “elite” papers are cited by other “elite” papers. Google web network shows partial rich-club and page-club ordering up to some point and then a narrow declining of the corresponding normalized coefficients, indicating the lack of rich-club ordering and the lack of page-club ordering, i.e. high in-degree (PageRank) pages purposely avoid sharing links with other high in-degree (PageRank) pages. For UC patents citation network, we identify page-club and rich-club ordering providing a conclusion that “elite” patents are cited by other “elite” patents. Finally, for email communication network we show lack of both rich-club and page-club ordering. We construct an example of synthetic network showing page-club ordering and the lack of rich-club ordering. [Copyright &y& Elsevier]