Range-limited Centrality


Definition

This is an efficient method that generates for every node and every edge its betweenness centrality based on shortest paths of lengths not longer than $l=1,..., L$ in case of non-weighted networks, and for weighted networks the corresponding quantities based on minimum weight paths with path weights not larger than $w_l=l\Delta, l=1,2,...,L=R/\Delta$. These measures provide a systematic description on the positioning importance of a node (edge) with respect to its network neighborhoods 1-step out, 2-steps out, etc. up to including the whole network.
In order to define range-limited betweenness centralities, let $b_l(j)$ denote the $BC$ of a node $j$ for all-pair shortest directed paths of $fixed, exact$ length $l$. Then Range-limited Centrality defined as follow:

$$B_L(i)={\underset{l=1}{\overset{L}{\sum}}} b_l(j)$$

were $B_L(i)$ represents the betweenness centrality of node $j$ obtained from paths not longer than $L$.


This is more informative than traditional centrality measures, as network transport typically happens on all length-scales, from transport to nearest neighbors to the farthest reaches of the network.



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