Local Dimension

The local dimension (Silva and Costa 2013; Pu et al. 2014) is the growth exponent of the ball around a node. With $B_i(r)$ the number of nodes within $r$ hops of $i$, the node itself included, it is the least-squares slope of $\ln B_i(r)$ on $\ln r$ over $r = 1, \ldots, d_{\max}(i)$: $$D_i = \frac{d \ln B_i(r)}{d \ln r}.$$

Usage

centrality_local_dimension(x, mode = "all", ...)

Arguments

x: Network input accepted by centrality.
mode: For directed networks: "all" (default), "out" or "in".
...: Further arguments to centrality, such as normalized.

Value

A named numeric vector with one score per node, in input node order.

Details

Distances are hop counts, so edge weights are ignored. On a directed network mode sets the direction of the paths. A node that reaches most of the network in a few hops has a small exponent, so lower values mark more influential nodes. A node with a single radius returns the discretized derivative $r\, n_i(r) / B_i(r)$ at $r = 1$, where $n_i(r)$ counts the nodes at distance exactly $r$. A node that reaches no other node returns NaN.

References

Silva, F. N., & Costa, L. da F. (2013). Local dimension of complex networks. arXiv:1209.2476.

Pu, J., Chen, X., Wei, D., Liu, Q., & Deng, Y. (2014). Identifying influential nodes based on local dimension. EPL, 107(1), 10010.

Wen, T., & Jiang, W. (2019). Identifying influential nodes based on fuzzy local dimension in complex networks. Chaos, Solitons & Fractals, 119, 332-342.

Examples

centrality_local_dimension(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>  0.7369656  0.5145732  0.3219281  0.5145732  0.7369656  0.7369656  1.0000000 
#>   Evaluate     Create      Share 
#>  0.7369656  0.5145732  0.7369656