Edge clustering coefficient (NC)
Edge clustering coefficient centrality
(NC), also known as the sum of ECC (SoECC) [2], is used to identify essential proteins in networks based on the clustering of edges [3]. The NC centrality of a node \(i\), denoted \(c_{\mathrm{NC}}(i)\), is defined as
\[
c_{\mathrm{NC}}(i) = \sum_{j \in \mathcal{N}(i)} \mathrm{ECC}(i,j),
\]
where \(\mathcal{N}(i)\) is the set of neighbors of node \(i\), and \(\mathrm{ECC}(i,j)\) is the edge clustering coefficient of edge \((i,j)\), given by
\[
\mathrm{ECC}(i,j) = \frac{z_{i,j}}{\min(d_i-1, d_j-1)}.
\]
Here, \(z_{i,j}\) denotes the number of triangles that include the edge \((i,j)\) and \(d_i\) is the degree of node \(i\). The denominator \(\min(d_i - 1, d_j - 1)\) represents the
maximum number of triangles
in which the edge \((i,j)\) can potentially participate.
Thus, the NC centrality accounts for both the
degree of the node
\(d_i\) (i.e., the number of edges incident to node \(i\)) and the
clustering coefficients
of its edges, capturing the node’s involvement in tightly connected regions of the network.