Mutual information is a local centrality measure that evaluates a node's importance based on the information content associated with its edges [2]. Specifically, the mutual information \(I(i)\) of node \(i\) is defined as the sum of the mutual information between node \(i\) and its neighbors:
\[
I(i) = \sum_{j \in \mathcal{N}(i)} I(i,j) = \sum_{j \in \mathcal{N}(i)} (\ln d_i - \ln d_j) = \sum_{j \in \mathcal{N}(i)} \ln \frac{d_i}{d_j},
\]
where \(d_i\) and \(d_j\) are the degrees of nodes \(i\) and \(j\), respectively.
Nodes with higher mutual information are considered more important, as they contain more structural information relative to their neighbors.

References

[1] Shvydun, S. (2025). Zoo of Centralities: Encyclopedia of Node Metrics in Complex Networks. arXiv: 2511.05122 https://doi.org/10.48550/arXiv.2511.05122
[2] Liu, Y., Jin, J., Zhang, Y., & Xu, C. (2014). A new clustering algorithm based on data field in complex networks. The Journal of Supercomputing, 67, 723-737. doi: 10.1007/s11227-013-0984-x.