Semi-local ranking centrality (SLC)
The
semi-local ranking centrality
(SLC) quantifies node influence based on random walks within the \(l\)-neighborhood of each node [2]. A node is considered influential if the random walk frequently encounters other influential nodes in its neighborhood. The SLC measure applies a PageRank-like process to capture this local and semi-local influence.
At each step, the random walk either adds an immediate neighbor \(j\) of node \(i\) to the path with probability \(p_{ij}\), or stops. The probability of adding node \(j\) to the random walk starting from node \(i\) is defined as
\begin{equation*}
p_{ij} = a \Biggl[ λ \frac{d_i}{d_{\max}} + (1-λ) HP(i,j) \Biggr],
\end{equation*}
where \(d_i\) is the degree of node \(i\), \(d_{\max}\) is the maximum degree in the network, and
\[
HP(i,j) = \frac{|\mathcal{N}(i) \cap \mathcal{N}(j)|}{\min(d_i, d_j)}
\]
measures the similarity between nodes \(i\) and \(j\), with \(\mathcal{N}(i)\) denoting the set of immediate neighbors of node \(i\).
The parameter \(λ \in (0,1)\) balances the contributions of node degree and neighborhood similarity, and \(a \in (0,1)\) is a decay factor controlling the probability of continuing the random walk.
Let \(S_t(i)\) denote the set of nodes visited by the \(t\)-th random walker starting from node \(i\).
The semi-local ranking centrality of node \(i\) is defined as
\begin{equation*}
c_{\mathrm{SLC}}(i) = \sum_{t=1}^{T} |S_t(i)|,
\end{equation*}
where \(T\) is the total number of random walkers.
Dong et al. [2] suggest using \(T=100\), \(a=0.9\), and \(λ=0.85\), with the length of each random walk limited to the network diameter.