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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 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 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 . If G is a clique then 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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