Abstract
In a recent work we introduced a measure of importance for groups of vertices in a complex network. This centrality for groups is always between 0 and 1 and induces the eigenvector centrality over vertices. Furthermore, its value over any group is the fraction of all network flows intercepted by this group. Here we provide the rigorous mathematical constructions underpinning these results via a semi-commutative extension of a number theoretic sieve. We then established further relations between the eigenvector centrality and the centrality proposed here, showing that the latter is a proper extension of the former to groups of nodes. We finish by comparing the centrality proposed here with the notion of group-centrality introduced by Everett and Borgatti on two real-world networks: the Wolfe’s dataset and the protein-protein interaction network of the yeast Saccharomyces cerevisiae. In this latter case, we demonstrate that the centrality is able to distinguish protein complexes
Highlights
Context In our previous work on the subject, we argued the need to go beyond vertices when analysing complex networks
Another example is provided by the notion of protein essentiality, a property understood to be determined at the level of protein complexes, that is groups of proteins in the protein-protein interaction network (PPI) rather than at the level of individual proteins (Hart et al 2007; Ryan et al 2013)
The notion of group-centrality may be too coarse to perceived such features in the data, at least in the case of PPI. In this second work on the centrality c(.), we have rigorously established its meaning as a fraction of network flows intercepted by any chosen ensembles of nodes
Summary
Context In our previous work on the subject, we argued the need to go beyond vertices when analysing complex networks. 2. The precise value c(H) taken by the centrality on a subgraph H is the fraction of all network flows intercepted by H. We recall the definition of the centrality for cycles and subgraphs, introduced in (Giscard and Wilson 2017b).
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