Modularity vitality
Modularity vitality
is a community-based centrality measure that quantifies the contribution of a node to the overall modularity of a network [2].
Assume that the network \(G\) has a community structure consisting of \(K > 1\) communities (Magelinski et al. [2] employ the Leiden community detection algorithm).
Given a community partition \(C\), the modularity vitality \(c_{MV}(i)\) of node \(i\) is defined as
\begin{equation*}
c_{MV}(i) = Q(G, C) - Q(G_i, C \setminus \{i\}),
\end{equation*}
where \(Q(G, C)\) denotes the modularity of the network \(G\) under partition \(C\), and \(G_i\) is the graph obtained by removing node \(i\) from \(G\). Importantly, the community structure \(C\) is
not recomputed
after the removal of node \(i\); modularity is recalculated using the same partition excluding \(i\).
A positive modularity vitality \(c_{MV}(i)\) indicates that node \(i\) enhances the modular structure of the network, functioning as a community hub.
Conversely, a negative \(c_{MV}(i)\) suggests that the node weakens modularity, typically acting as a bridge between communities.
Thus, modularity vitality not only measures the importance of a node but also distinguishes between hub-like and bridge-like roles within the network.