ModuLand centrality
ModuLand centrality quantifies a node's importance based on its role within the influence-function-based community landscape constructed by the NodeLand algorithm [2]. For each node \(k\), NodeLand computes an influence function \(f_k\) by iteratively building the set \(A_k\) of nodes strongly influenced by \(k\). Starting from \(A_k = \{k\}\), neighboring nodes are added one at a time only if their inclusion increases the density of the subgraph induced by \(A_k\). This process continues until no further improvement in density is possible.The ModuLand centrality \(c_{\mathrm{ModuLand}}(i)\) of node \(i\) is then defined as\[c_{\mathrm{ModuLand}}(i) = \sum_{j=1}^{N} c(i,j) = \sum_{j=1}^{N} \sum_{k=1}^{N} f_k(i,j),\]where\[f_k(i,j) =\begin{cases}w_{ij}, & \text{if } (i,j) \in A_k,\\0, & \text{otherwise,}\end{cases}\]and \(w_{ij}\) is the weight of edge \((i,j)\). This formulation captures how strongly node \(i\) participates in regions of high influence across the network, reflecting its centrality in the overlapping community landscape.