Non-backtracking centrality
Non-backtracking (NB) centrality
is a spectral measure designed to mitigate localization effects, where a hub with high centrality artificially inflates the centrality of its neighbors, which in turn feed back and further exaggerate the hub's centrality [2].
The NB centrality is defined using the \(L \times L\) non-symmetric
non-backtracking matrix
\(B\), where each row and column corresponds to a directed edge \((i,j)\), with elements
\[
B_{(k,l),(i,j)} = δ_{jk} (1 - δ_{il}),
\]
and \(δ_{jk}\) is the Kronecker delta. The element \(v_{(j,i)}\) of the leading eigenvector of \(B\) represents the centrality of node \(j\) ignoring contributions from node \(i\). The full non-backtracking centrality of node \(i\) is then
\begin{equation*}
c_{NB}(i) = \sum_{j \in \mathcal{N}(i)} a_{ji} \, v_{(j,i)}.
\end{equation*}
The non-backtracking centrality can be efficiently computed as the first \(N\) elements of the leading eigenvector of the \(2N \times 2N\) matrix
\[
M = \begin{bmatrix} A & I - D \\ I & 0 \end{bmatrix},
\]
where \(A\) is the adjacency matrix, \(I\) is the \(N \times N\) identity matrix, and \(D\) is the diagonal matrix of node degrees.