Short-Range Interaction Centrality (SRIC)
The
Short-Range Interaction Centrality (SRIC)
index, originally called the key-borrower index (KBI), is a power index based on the concept of individual and group influence of nodes in a network [2, 3, 4]. Each node \(i\) is assumed to have an individual threshold of influence \(q_i\), which represents the level at which this node becomes affected. This threshold can be specified externally based on domain knowledge or determined from the network structure, for instance, as a function of the degree of each node. A group of nodes \(\Omega(i) \subset \mathcal{N}\) is called
critical
for node \(i\) if their collective influence exceeds the threshold \(q_i\), i.e.,
\begin{equation*}
\sum_{k \in \Omega(i)} a_{ki} \geq q_i.
\end{equation*}
A node \(k\) is termed
pivotal
for the group \(\Omega(i)\) if its removal renders the group non-critical. The set of pivotal members of \(\Omega(i)\) is denoted by \(\Omega^{p}(i)\).
The SRIC index considers only direct and indirect influence through one intermediate node, i.e., short-range influence. Formally, the initial adjacency matrix \(A\) is transformed into a matrix of direct influence \(P\), where \(D\) is the out-degree matrix. The indirect influence \(p_{ihj}\) of node \(i\) on node \(j\) via an intermediate node \(h\) is defined as
\begin{equation*}
p_{ihj} =
\begin{cases}
\frac{\min(a_{ih}, a_{hj})}{D_{kk}}, & \text{if } a_{ih} > 0, \ a_{hj} > 0, \ i \neq j \neq h, \\
0, & \text{otherwise}.
\end{cases}
\end{equation*}
The SRIC centrality of node \(i\) is defined as the average of the normalized short-range influence \(χ_i(j)\) of node \(i\) on all nodes \(j \in \mathcal{N}\). This is evaluated by considering all critical groups \(\Omega_k(j)\) in which node \(i\) is pivotal:
\begin{equation*}
c_{SRIC}(i) = \frac{1}{N} \sum_{j \in \mathcal{N}} \frac{χ_i(j)}{\sum_{h \in \mathcal{N}} χ_h(j)},
\end{equation*}
where
\begin{equation*}
χ_i(j) = \sum_{k: i \in \Omega^{p}_k(j)} \frac{p_{ij} + \sum_{h \in \Omega_k(j)} p_{ihj}}{|\Omega_k(j)|}.
\end{equation*}