BG-index
The
BG-index
, also known as the \(β\)-measure [2, 3], is a social power index that quantifies a node's centrality based on its position within a network [4, 2]. Van den Brink and Gilles assume that the network represents a social structure, in which each node may dominate some nodes while being dominated by others.
The BG-index of node \(i\) measures the expected number of times it will be selected as a predecessor by its neighbors, assuming that each neighbor chooses one of its predecessors uniformly at random. Formally,
\begin{equation*}
c_{\mathrm{BG{-}index}}(i) = \sum_{j \in \mathcal{N}(i)} \frac{1}{|\{k : j \in \mathcal{N}(k)\}|} = \sum_{j \in \mathcal{N}(i)} \frac{1}{|\mathcal{N}(i)|} = \sum_{j \in \mathcal{N}(i)} \frac{1}{d_j} = \sum_{j=1}^{N} \frac{a_{ij}}{\sum_{k=1}^{N} a_{kj}},
\end{equation*}
where \(\mathcal{N}(i)\) denotes the set of neighbors of node \(i\), \(d_j\) is the degree of node \(j\) and \(a_{ij}\) are entries of the adjacency matrix \(A\).
For directed graphs, the BG-index has two versions: the positive \(β\)-measure (computed on the original graph \(G\)) and the negative \(β\)-measure (computed on the reverse graph of \(G\)) [3].