New Evidential Centrality (NEC)
New evidential centrality
(NEC) is a hybrid measure based on Dempster-Shafer evidence theory that combines a node's degree with the global network structure, as quantified by shortest path distances [2].
NEC addresses limitations of the existing evidential centrality (EVC), which was originally designed for weighted networks [3].
The concept of evidential centrality is analogous to multi-attribute decision making (MADM), in which multiple factors are combined to obtain a final ranking of nodes.
For each node \(i\), NEC computes two basic probability assignments (BPAs): one based on degree and one based on shortest paths.
The degree-based BPA is defined as
\begin{equation*}
M_k(i) = \bigl(m_{ki}(h), m_{ki}(l), m_{ki}(θ)\bigr),
\end{equation*}
where
\[
m_{ki}(h) = \frac{d_i - d_{\min}}{d_{\max} - d_{\min} + μ}, \quad
m_{ki}(l) = \frac{d_{\max} - d_i}{d_{\max} - d_{\min} + μ}, \quad
m_{ki}(θ) = 1 - m_{ki}(h) - m_{ki}(l),
\]
with \(d_i\) the degree of node \(i\), \(d_{\min}\) and \(d_{\max}\) the minimum and maximum degrees in the network, \(μ \in (0,1)\) a small constant, and \(θ = \{h,l\}\) the frame of discernment. Here, \(m_{ki}(h)\) and \(m_{ki}(l)\) represent the degrees of belief that node \(i\) has high or low influence based on its degree, while \(m_{ki}(θ)\) captures the remaining uncertainty.
The shortest path-based BPA is defined as
\begin{equation*}
M_d(i) = \frac{1}{N} \sum_{j=1}^{N}
\bigl(m^j_{di}(h), m^j_{di}(l), m^j_{di}(θ)\bigr),
\end{equation*}
where the mass functions for each target node \(j\) are
\begin{align*}
m^j_{di}(h) &= \frac{d_{ij} - \min_k(d_{ik})}
{\max_k(d_{ik}) - \min_k(d_{ik}) + ε},\\
m^j_{di}(l) &= \frac{\max_k(d_{ik}) - d_{ij}}
{\max_k(d_{ik}) - \min_k(d_{ik}) + ε},\\
m^j_{di}(θ) &= 1 - m^j_{di}(h) - m^j_{di}(l),
\end{align*}
with \(d_{ij}\) denoting the shortest path distance from node \(i\) to node \(j\),
\(ε \in (0,1)\), and \(k\) ranging over all nodes in the network.
Wei et al.\ [3] use \(μ = ε = 0.5\).
The combined influence of node \(i\) is obtained by merging the degree- and shortest path-based BPAs using a modified Dempster's rule of combination \(\oplus\):
\begin{equation*}
M(i) = M_k(i) \oplus M_d(i) = \bigl(m_i(h), m_i(l), m_i(θ)\bigr),
\end{equation*}
where \(m_i(h)\), \(m_i(l)\), and \(m_i(θ)\) denote the resulting masses assigned to hypotheses \(h\), \(l\), and \(θ\), respectively. This fusion of \(M_k(i)\) and \(M_d(i)\) produces a single BPA for node \(i\), integrating information from both local connectivity (degree) and global position (shortest paths).
Finally, the NEC centrality of node \(i\) is defined as
\begin{equation*}
c_{\mathrm{NEC}}(i) = m_i(h) - m_i(l).
\end{equation*}
An extension of NEC, called
Multi-Evidence Centrality
(MeC), was proposed by Mo and Deng in [4].
MeC integrates four centrality measures: degree, betweenness, harmonic, and correlation within the evidential framework.
Each measure contributes a basic probability assignment (BPA) reflecting different aspects of node importance, and the BPAs are fused using Dempster's rule of combination to compute a single, comprehensive centrality score for each node.