Fixed-Radius, Fuzzy and Volume Local Dimensions

Three members of the local-dimension family, with the center node counted in its own ball. The fixed-radius local dimension (Silva and Costa 2013) is \(D_i(r) = r\, n_i(r) / B_i(r)\) at one radius \(r\), where \(n_i(r)\) counts the nodes at distance \(r\) and \(B_i(r)\) those within it. The fuzzy local dimension (Wen and Jiang 2019) is the slope of \(\log N_i(r)\) on \(\log r\) for the fuzzy ball $$N_i(r) = \frac{\sum_{d_{ij} \le r} e^{-d_{ij}^2 / r^2}} {|\{j : d_{ij} \le r\}|}.$$ The local volume dimension (Li and Deng 2021) is the slope of \(\ln V_i(l)\) on \(\ln l\) for the volume \(V_i(l) = \sum_{d_{ij} \le l} k_j\).

Usage

centrality_local_dimension_fixed(x, mode = "all", ld_radius = 2, ...)

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

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

Arguments

x

Network input accepted by centrality.

mode

For directed networks: "all" (default), "out" or "in".

ld_radius

Radius \(r\) for local_dimension_fixed, in hops (default 2).

...

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. The fixed-radius form is a structural descriptor, and nodes whose eccentricity is below the radius score 0. Larger fuzzy dimensions and smaller volume dimensions mark more influential nodes. The two regression forms return NaN for a node with fewer than two radii. The volume dimension follows a later preprint of the authors and the Centrality Zoo entry. A ld_radius that is not a single number of at least 1 raises an error of class cograph_bad_parameter.

References

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

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

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(5), 2150069.

See also

centrality_local_dimension, centrality_local_information_dimension, centrality.

Examples

centrality_local_dimension_fixed(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>        0.8        0.6        0.4        0.6        0.8        0.8        1.0 
#>   Evaluate     Create      Share 
#>        0.8        0.6        0.8 
centrality_fuzzy_local_dimension(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>  0.4277285  0.5646062  0.6855285  0.5646062  0.4277285  0.4277285  0.2686074 
#>   Evaluate     Create      Share 
#>  0.4277285  0.5646062  0.4277285 
centrality_local_volume_dimension(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>  0.7548875  0.5069600  0.2954559  0.5454341  0.8006912  0.7104934  0.9475326 
#>   Evaluate     Create      Share 
#>  0.6256045  0.4694853  0.6256045