SpectralRank (SR)
SpectralRank
(SR) is a parameter-free extension of LeaderRank designed to evaluate node propagation capability in networks [2]. Similar to LeaderRank, SR introduces a ground node \(N+1\) that connects bidirectionally to all nodes in the network \(G\). Each node \(i\) is assigned a score \(s_i[t]\) at discrete time \(t\), representing its propagation potential.
The initial scores are
\[
s_{N+1}[0] = 0 \quad \text{for the ground node,} \qquad s_i[0] = 1 \quad \text{for all other nodes } i \in \mathcal{N}.
\]
At each time step, the score of node \(i\) is updated based on the scores of its neighbors:
\begin{align*}
\tilde{s}_i[t+1] &= c \sum_{j=1}^{N+1} a_{ij} \, s_j[t], \\
s_i[t+1] &= \frac{\tilde{s}_i[t+1]}{\max_j \tilde{s}_j[t+1]},
\end{align*}
where \(c = 1 / λ_{\max}\), and \(λ_{\max}\) is the leading eigenvalue of the augmented adjacency matrix
\[
\tilde{A} =
\begin{bmatrix}
A & \mathbf{1} \\
\mathbf{1}^T & 0
\end{bmatrix},
\]
which includes the ground node. Here, \(A\) is the original \(N \times N\) adjacency matrix, and \(\mathbf{1}\) is a column vector of ones. The SpectralRank of node \(i\) is defined as the steady-state score
\[
\Tilde{s}_i = \lim_{t \to \infty} s_i[t],
\]
which quantifies the node’s long-term propagation influence in the network.