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