Pivotal index
The
pivotal index
is a power index that quantifies both the individual and collective influence of nodes within a network [2]. It is based on the assumption that each node \(i\) possesses an individual
influence threshold
\(q_i\), which specifies the level of accumulated influence required for the node to become affected (e.g., \(q_i = 3\)).
A subset of nodes \(\Omega(i) \subset \mathcal{N}\) is called
critical
for node \(i\) if the total influence exerted by the members of \(\Omega(i)\) on \(i\) meets or exceeds its threshold value:
\begin{equation*}
\sum_{k \in \Omega(i)} w_{ki} \geq q_i,
\end{equation*}
where \(w_{ki}\) denotes the influence weight of node \(k\) on node \(i\).
A node \(k\) is said to be
pivotal
for the group \(\Omega(i)\) if its exclusion from this group makes the group non-critical. The set of all pivotal members within \(\Omega(i)\) is denoted by \(\Omega^{p}(i)\). The
pivotal influence index
\(c_{PI}(i)\) of node \(i\) is defined as the total number of pivotal groups for that node:
\[
c_{PI}(i) = \sum_{\Omega(i) \subseteq \mathcal{N}(i)} |\Omega(i)| \cdot |\Omega^{p}(i)|.
\]
Since the number of possible critical groups can grow exponentially with network size, resulting in high computational complexity, a practical variant of the pivotal index considers only subsets of size up to \(k\). Aleskerov and Yakuba [2] also proposed an
order-\(k\) extension
of the pivotal index that accounts for indirect influences. In this extension, influence is evaluated over \((k{+}1)\)-hop neighborhoods, where link strength is defined as the maximum bottleneck capacity among all \((k{+}1)\)-length paths.
For unweighted networks, the pivotal influence index \(c_{PI}(i)\) of node \(i\) simplifies to
\[
c_{PI}(i) = \bigl(\lceil q_i \rceil\bigr)^2 \binom{d_i}{\lceil q_i \rceil},
\]
where \(d_i\) denotes the degree of node \(i\), and \(\lceil \cdot \rceil\) denotes the ceiling function.
The pivotal index has been applied in various contexts, including the identification of influential countries in global food trade networks [3] and oil trade networks [4], analysis of trade relations among economic sectors across countries [5], and bibliometric studies of publications on Parkinson’s disease [6].