Godfather index
The
Godfather index
measures a node's brokerage and coordination capital by counting the number of pairs of a node’s neighbors who are not connected to each other [2]. The centrality of node \(i\), denoted \(c_{\mathrm{GF}}(i)\), can be expressed as
\[
c_{\mathrm{GF}}(i) = \sum_{k>j} a_{ik} a_{ij} (1 - a_{kj}) = \sum_{k > j \in \mathcal{N}(i)} (1 - a_{kj}),
\]
where \(a_{ij}\) is the adjacency matrix of the network and \(\mathcal{N}(i)\) is the set of neighbors of node \(i\).
Jackson [2] shows that the Godfather Index is inversely related to the clustering coefficient \(c_{\mathrm{cl}}(i)\), weighted by the number of neighbor pairs:
\[
c_{\mathrm{GF}}(i) = (1 - c_{\mathrm{cl}}(i)) \frac{d_i (d_i - 1)}{2},
\]
where \(d_i\) is the degree of node \(i\).
The Godfather Index is also related to the redundancy coefficient \(c_r(i)\) as
\[
c_{\mathrm{GF}}(i) = \frac{d_i (d_i - 1)}{2} - \frac{d_i}{2} c_r(i).
\]
This formulation highlights that nodes with many unconnected neighbor pairs (low redundancy) have higher Godfather centrality, reflecting their potential as brokers or coordinators in the network.