Linear threshold centrality (LTC)
The Linear threshold centrality (LTC), also called Linear Threshold Rank (LTR), quantifies the influence of nodes based on the linear threshold (LT) model [2]. In the LT model, a node \(i\) becomes active once the weighted sum of its active neighbors exceeds its individual threshold \(θ_i\).The LTC centrality of node \(i\) is defined as the fraction of nodes that can be activated when \(i\) and its neighbors are taken as the seed set:\[c_{LTC}(i) = \frac{|F(\{i\} \cup \mathcal{N}(i))|}{N},\]where \(\mathcal{N}(i)\) is the set of neighbors of \(i\) and \(F(X)\) is the set of nodes eventually activated by the linear threshold process starting from the seed set \(X\). Formally,\begin{equation*}F(X) = \bigcup_{t=0}^{k} F_t(X) = F_0(X) \cup F_1(X) \cup \dots \cup F_k(X),\end{equation*}where\begin{equation*}k = \min \{ t \in \mathbb{N} \mid F_t(X) = F_{t+1}(X) \} \le N.\end{equation*}At each time step \(t > 0\), a node \(j \notin F_{t-1}(X)\) becomes active if the weighted fraction of its active neighbors exceeds its threshold, i.e.,\begin{equation*}j \in F_t(X) \quad \text{if} \quad \sum_{l \in F_{t-1}(X) \cap \mathcal{N}^{\mathrm{in}}(j)} w_{lj} \ge θ_j.\end{equation*}LTC thus captures both the local influence of a node through its immediate neighborhood and its potential to trigger wider cascades in the network. Nodes with high LTC centrality are those that, together with their immediate neighbors, can activate a large fraction of the network, indicating strong local influence and high cascading potential.