Semi-local iterative algorithm (semi-IA)
The
semi-local iterative algorithm
(semi-IA) is an iterative centrality measure in which the importance of a node depends on the importance of other nodes, weighted by both the shortest-path distance and the number of shortest paths (NSPs) connecting them [2].
For a node \(i\), the update rule at iteration \(t+1\) is defined as
\[
X_i[t+1] = \sum_{j \in \mathcal{N}^{(\leq l)}(i)} \frac{n_{ij}^{γ}}{d_{ij}} X_j'[t], \quad X_i[0] = 1, \quad i = 1, \dots, N,
\]
where \(\mathcal{N}^{(\leq l)}(i)\) is the set of nodes within distance \(l\) from node \(i\), \(n_{ij}\) is the number of shortest paths between \(i\) and \(j\), \(0 \leq γ \leq 1\) is the NSP weighting factor, and \(X_j'[t]\) is the normalized influence of node \(j\) at iteration \(t\):
\[
X_j'[t] = \frac{X_j[t]}{\| X[t] \|}.
\]
Luan et al. [2] suggest setting the truncated radius \(l = 3\) for networks with fewer than 1000 nodes, and \(l = 5\) for larger networks, with \(γ = 0.2\). The semi-IA centrality of node \(i\) is the value \(\tilde{X}_i=\lim_{t \rightarrow \infty}X_i[t]\) at the steady state of the iteration.