Map equation centrality
Map equation centrality is a vitality-based centrality measure derived from the information-theoretic community detection framework known as the map equation [2]. It quantifies the importance of a node based on how much it contributes to the overall description length of flow on the network. Assume that the network \(G\) has a non-overlapping community structure represented by the partition \(M\). The map equation centrality \(c_{ME}(i)\) of node \(i\) measures the reduction in the average codeword length if node \(i\) is silenced, meaning that when a random walker visits \(i\), no codeword for \(i\) is transmitted. Formally,\begin{equation*}c_{ME}(i) = L(G, M) - L^{*}_i(G, M),\end{equation*}where \(L(G, M)\) denotes the optimal description length of flow for the original network under partition \(M\), and \(L^{*}_i(G, M)\) is the optimal description length when node \(i\) is silenced. Hence, \(c_{ME}(i)\) represents the marginal contribution of node \(i\) to the total information cost of describing flow on the network. Nodes with higher \(c_{ME}(i)\) values are more important for maintaining the network’s information flow structure, while nodes with lower or negative values have less or even disruptive influence on modular information dynamics.