The local volume dimension (LVD) is a centrality measure for identifying influential nodes based on the local dimension concept [2]. For each node \(i\), the LVD centrality evaluates the total degree within a box \(V_i(l)\), defined as the sum of the degrees of all nodes at distance \(l\) from node \(i\). It is assumed that the total degree within each box follows a power-law distribution.
The LVD centrality \(c_{\textsc{LVD}}(i)\) of node \(i\) is given by
\[
c_{\textsc{LVD}}(i) = \frac{d \ln V_i(l)}{d \ln l}.
\]
The LVD centrality of node \(i\) is estimated numerically as the slope of the linear regression of \(\ln V_i(l)\) against \(\ln l\).

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] Li, H., & Deng, Y. (2021). Local volume dimension: A novel approach for important nodes identification in complex networks. International Journal of Modern Physics B, 35(05), 2150069. doi: 10.1142/S0217979221500697.