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*}s̃_i[t+1] &= c \sum_{j=1}^{N+1} a_{ij} \, s_j[t], \\s_i[t+1] &= \frac{s̃_i[t+1]}{\max_j s̃_j[t+1]},\end{align*}where \(c = 1 / λ_{\max}\), and \(λ_{\max}\) is the leading eigenvalue of the augmented adjacency matrix\[Ã =\begin{bmatrix}A & \mathbf{1} \\\mathbf{1}^T & 0\end{bmatrix},\]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\[s̃_i = \lim_{t \to \infty} s_i[t],\]which quantifies the node’s long-term propagation influence in the network.