Silent node rank (SNR)
Silent node rank
(SNR), also known as LurkerRank (LR) [2], is a spectral centrality measure designed to identify nodes that play a passive role in information networks, which are the nodes that consume more information than they produce and often remain unnoticed [3]. Such silent nodes typically have low connectivity and are common in many systems, including leechers in P2P networks or lurkers in online social networks.
While the in-degree/out-degree ratio provides a simple measure of silence, it ignores the influence of neighbors and often produces many nodes with identical scores. SNR improves upon this by incorporating neighbor behavior: a node is considered more silent if its in-neighbors are active and its out-neighbors are themselves silent. This network-aware approach yields a more accurate and diversified ranking of silent nodes.
Formally, the SNR score \(r_i\) of node \(i\) in a weighted, directed network is given by
\begin{equation*}
\begin{split}
r_i &= α \Bigg(
\frac{1}{d_i^{\mathrm{out}} + 1}
\sum_{j \in \mathcal{N}^{\mathrm{in}}(i)} w_{ji} \frac{d_j^{\mathrm{out}} {+} 1}{d_j^{\mathrm{in}} {+} 1} r_j
\Bigg)
\Bigg(
1 {+} \frac{d_i^{\mathrm{in}} {+} 1}{\sum_{j \in \mathcal{N}^{\mathrm{out}}(i)} (d_j^{\mathrm{in}} {+} 1)}
\sum_{j \in \mathcal{N}^{\mathrm{out}}(i)} w_{ij} \frac{d_j^{\mathrm{in}} {+} 1}{d_j^{\mathrm{out}} {+} 1} r_j
\Bigg) \\
&\quad + \frac{1{-}α}{N},
\end{split}
\end{equation*}
where \(w_{ij}\) is the edge weight, \(α\) is the damping factor, \(\mathcal{N}^{\mathrm{in}}(i)\) and \(\mathcal{N}^{\mathrm{out}}(i)\) are the in- and out-neighbors of \(i\), and \(d_j^{\mathrm{in}}\) and \(d_j^{\mathrm{out}}\) are the in- and out-degrees of node \(j\). The first term downweights the influence of node \(i\)’s in-neighbors based on its out-degree, while the second term boosts the score according to how silent its out-neighbors are, adjusted by \(i\)’s in-degree.
For undirected and unweighted networks, the SNR score simplifies to
\begin{equation*}
r_i = α \left( \frac{1}{d_i + 1} \sum_{j \in \mathcal{N}(i)} r_j \right)
\left( 1 + \frac{d_i + 1}{d_i + \sum_{j \in \mathcal{N}(i)} d_j} \sum_{j \in \mathcal{N}(i)} r_j \right)
+ \frac{1-α}{N}.
\end{equation*}