The local dimension (LD) quantifies the dimensionality of nodes in a network by examining how the volume of the neighborhood around each node scales with increasing topological distance [2]. In most spatially embedded real networks, which typically lack the small-world property, the distribution of link lengths follows a power law. Consequently, the number of nodes \(B_i(r)\) located within a topological distance \(r\) from a node \(i\) obeys the relationship\[B_i(r) \sim r^d,\]where the constant \(d\) characterizes the effective dimension of the network. Silva and Costa [2] refined this power-law relationship by allowing the dimensionality to vary locally, proposing that\[B_i(r) = α r^{D_i(r)},\]where \(D_i(r)\) represents the local dimension around node \(i\). The local dimension coefficient \(D_i(r)\) can be estimated from the slope of the \(B_i(r)\) curve on a double-logarithmic scale and is discretized as\[D_i(r) \simeq r \frac{n_i(r)}{B_i(r)},\]where \(n_i(r)\) denotes the number of nodes that are exactly at a topological distance \(r\) from the reference node \(i\). Pu et al. [3] further extended the local dimension measure by allowing the distance parameter \(r\) to vary across different nodes in the 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] Silva, F. N., & Costa, L. D. F. (2012). Local dimension of complex networks. arXiv preprint arXiv:1209.2476. doi: 10.48550/arXiv.1209.2476.
[3] 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.