Density of the Maximum Neighborhood Component (DMNC)
The
Density of the Maximum Neighborhood Component
(DMNC) centrality extends the maximum neighborhood component (MNC) measure by incorporating the internal link density of the largest connected component within a node’s neighborhood [2].
While MNC centrality considers only the size of the largest connected subgraph among the neighbors of a node, DMNC additionally evaluates how densely those neighbors are connected to each other.
Formally, for a given node \( i \), let \( C_{\max}(G_{\mathcal{N}(i)}) \) denote the largest connected component of the induced subgraph \( G_{\mathcal{N}(i)} \).
Then the DMNC centrality \( c_{\mathrm{dmnc}}(i) \) is defined as
\begin{equation*}
c_{dmnc}(i) = \frac{|E(MNC(G_{\mathcal{N}(i)}))|}{|V(MNC(G_{\mathcal{N}(i)}))|^ε},
\end{equation*}
where \(|E(MNC(G_{\mathcal{N}(i)}))|\) and \(|V(MNC(G_{\mathcal{N}(i)}))|\) denote the number of edges and vertices, respectively, within the largest connected component, and
\( ε \) is a tunable scaling parameter such that \( 1 \leq ε \leq 2 \) (typically \( ε = 1.67 \)). Thus, the DMNC centrality measures how large and how tightly connected a node’s neighbourhood is, giving higher scores to nodes whose neighbours form large, well-connected groups.