Maximal Clique Centrality (MCC)
The
maximal clique centrality
(MCC) is based on the observation that essential proteins in a yeast protein-protein interaction network tend to be highly clustered [2]. A
maximal clique
is a fully connected subgraph that is not contained in any larger fully connected subgraph. Let \(S_i\) denote the set of maximal cliques containing node \(i\). The MCC of node \(i\) is then defined as
\begin{equation*}
c_{\mathrm{MCC}}(i) = \sum_{C \in S_i} (|C|-1)!,
\end{equation*}
where \(|C|\) is the size of clique \(C\). Under this definition, the MCC of an isolated node is \(1\). For a node \(i\) whose neighbors are all disconnected (i.e., there is no edge between any two neighbors of node \(i\)), the MCC reduces to the degree of node \(i\):
\begin{equation*}
c_{\mathrm{MCC}}(i) = \sum_{j=1}^{N} a_{ij} = d_i.
\end{equation*}