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Knotty Centrality


Definition

A network measure called knotty-centrality is defined that quantifies the extent to which a given subset of a graph’s nodes constitutes a densely intra-connected topologically central connective core. Using this measure, the knotty centre of a network is defined as a sub-graph with maximal knotty-centrality.
Consider a directed graph G with N nodes. The knotty-centrality of a (non-empty, non-singleton) subset S of the nodes in G is given by
Knotty Centrality
where Es is the number of edges between nodes in S, and Ns is the number of nodes in S. bc(i) is the betweenness centrality of node i normalised with respect to the whole graph, such that
Knotty Centrality
where BC(i) is the (directed) betweenness centrality of node i. Knotty-centrality ranges from 0 to 1. It is 0 if none of the nodes in S is adjacent (Es=0): It is 1 if S is a clique and Knotty Centrality. If G is a clique then Knotty Centrality and KC(S) is undefined. The measure can be applied to either weighted or unweighted graphs by substituting weighted or unweighted variants of betweenness centrality into equation.

References

  • SHANAHAN, M. & WILDIE, M. 2012. Knotty-centrality: finding the connective core of a complex network. PloS one, 7, e36579.


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