Fuzzy local dimension (FLD) is a measure designed to identify influential nodes by combining fuzzy set theory with local dimension concepts [2]. In the original local dimension (LD) measure [3], all nodes within a given radius (or box-size) contribute equally to the central node. FLD extends this approach by assigning different contributions to nodes based on their distance from the center node \(i\): the closer a node \(j\) is to \(i\), the greater its contribution.
Specifically, Wen and Jiang [2] use a fuzzy membership function to weight each node within the radius. The weighted number of nodes within radius \(r\) of node \(i\) is given by
\[
B_i(r) = \frac{\sum_{j \in N^{(\leq r)}(i)} e^{-d_{ij}^2 / r^2}}{|N^{(\leq r)}(i)|},
\]
where \(d_{ij}\) is the shortest distance between nodes \(i\) and \(j\), and \(N^{(\leq r)}(i)\) denotes the set of nodes satisfying \(d_{ij} \le r\).
The fuzzy local dimension \(c_{FLD}(i)\) of node \(i\) is then obtained as the slope of the line fitting \(\log_2(B_i(r))\) versus \(\log_2(r)\) on a double logarithmic scale using linear regression. Nodes with higher fuzzy local dimension values are those whose neighborhoods are both dense and close to the node, indicating strong local influence and a greater potential to affect the surrounding network.

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] Wen, T., & Jiang, W. (2019). Identifying influential nodes based on fuzzy local dimension in complex networks. Chaos, Solitons & Fractals, 119, 332-342. doi: 10.1016/j.chaos.2019.01.011.
[3] Silva, F. N., & Costa, L. D. F. (2012). Local dimension of complex networks. arXiv preprint arXiv:1209.2476. doi: 10.48550/arXiv.1209.2476.