The multi-local dimension (MLD) centrality is a variant of the local dimension measure for identifying influential spreaders in complex networks [2]. MLD evaluates the structural information around a node by considering concentric boxes of increasing radius. For a given node \(i\), the box radius \(l\) ranges from 1 to the maximum shortest-path distance from \(i\).The proportion of nodes within a box of radius \(l\) is\begin{equation*}μ_i(l) = \frac{N_i(l)}{N},\end{equation*}where \(N_i(l)\) is the number of nodes covered by the box, and \(N\) is the total number of nodes in the network. Based on \(μ_i(l)\), the generalized partition function \(Z_i(q,l)\) is defined as\begin{equation*}Z_i(q,l) = \begin{cases}μ_i(l)^q, & q \notin \{0,1\},\\[1mm]1/μ_i(l), & q = 0,\\[1mm]μ_i(l)\, \log_2 μ_i(l), & q = 1,\end{cases}\end{equation*}where \(q \in \mathbb{R}\) is a tunable parameter controlling the emphasis on different structural scales.The multi-local dimension \(c_{\textsc{MLD}}(i,q)\) of node \(i\) is then defined as\begin{equation*}c_{\textsc{MLD}}(i,q) = \begin{cases}\displaystyle \lim_{l \to 0} \frac{\log_2 Z_i(q,l)}{(q-1)\, \log_2 l}, & q \neq 1,\\[1mm]\displaystyle \lim_{l \to 0} \frac{Z_i(q,l)}{\log_2 l}, & q = 1.\end{cases}\end{equation*}In practice, \(c_{\textsc{MLD}}(i,q)\) is estimated numerically as the slope of a linear regression: if \(q \neq 1\), the regression is of \(\frac{\log_2 Z_i(q,l)}{q-1}\) versus \(\log_2 l\); if \(q = 1\), it is of \(Z_i(q,l)\) versus \(\log_2 l\). Wen et al. [2] show that \textsc{MLD} reduces to the local information dimensionality (LID) [3] when \(q=1\), and to the local dimension measure [4] when \(q=0\).

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., Pelusi, D., & Deng, Y. (2020). Vital spreaders identification in complex networks with multi-local dimension. Knowledge-Based Systems, 195, 105717. doi: 10.1016/j.knosys.2020.105717.
[3] Wen, T., & Deng, Y. (2020). Identification of influencers in complex networks by local information dimensionality. Information Sciences, 512, 549-562. doi: 10.1016/j.ins.2019.10.003.
[4] Pu, J., Chen, X., Wei, D., Liu, Q., & Deng, Y. (2014). Identifying influential nodes based on local dimension. Europhysics Letters, 107(1), 10010. doi: 10.1209/0295-5075/107/10010.