Three measures that score a node from a partition membership.
Community-based centrality (Zhao et al. 2015) is
\(CbC(i) = \sum_w d_{iw} S_w / N\), where
\(d_{iw}\) counts the links of \(i\) into community \(w\) of
size \(S_w\). Comm centrality (Gupta, Singh and Cherifi 2016)
combines intra-community degree \(k^{in}\) and inter-community
degree \(k^{out}\):
$$CC(i) = (1 + \mu_C) \frac{k^{in}_i}{\max_{j \in C} k^{in}_j} R
+ (1 - \mu_C)
\left(\frac{k^{out}_i}{\max_{j \in C} k^{out}_j} R\right)^2,$$
with \(\mu_C\) the mean inter-link fraction in the community of
\(i\). Community-based mediator centrality (Tulu, Hou and Younas 2018)
is \(CbM(i) = H_i\, d_i / \sum_j d_j\),
with \(H_i\) the base-2 entropy of the links of \(i\) over the
communities.
Usage
centrality_community_based(x, membership = NULL, mode = "all", ...)
centrality_comm_centrality(
x,
membership = NULL,
mode = "all",
comm_r = "max_intra",
...
)
centrality_community_mediator(x, membership = NULL, mode = "all", ...)Arguments
- x
Network input accepted by
centrality.- membership
Community labels, one per node, for example from
detect_communities.- mode
For directed networks:
"all"(default),"out"or"in".- ...
Further arguments to
centrality, such asnormalized.- comm_r
Scale \(R\) of Comm centrality:
"max_intra"(default) or a single positive number applied to every community.
Details
Edge weights and self-loops are ignored. Under mode = "out" or
mode = "in" only out-links or in-links count, and the default
ignores direction. Higher values mark more central nodes in all three.
The default comm_r = "max_intra" sets \(R\) to the largest
intra-community degree of each community, the choice the source
recommends. The prose of Gupta et al. writes \(\mu_C\) where their
equation has \(1 + \mu_C\), and the equation is implemented.
Nodes
linked to one community only score 0 on the mediator measure. Without
membership each function raises a warning of classes cograph_bad_membership and
cograph_undefined_measure and
returns NA. A membership that is not one non-missing label per node
raises an error of class cograph_bad_membership, and an invalid
comm_r raises cograph_bad_parameter.
References
Zhao, Z., Wang, X., Zhang, W., & Zhu, Z. (2015). A community-based approach to identifying influential spreaders. Entropy, 17(4), 2228-2252.
Gupta, N., Singh, A., & Cherifi, H. (2016). Centrality measures for networks with community structure. Physica A, 452, 46-59.
Tulu, M. M., Hou, R., & Younas, T. (2018). Identifying influential nodes based on community structure to speed up the dissemination of information in complex network. IEEE Access, 6, 7390-7401.
Examples
centrality_community_based(regulation_net, membership = rep(1:2, each = 5))
#> Explore Plan Monitor Adapt Reflect Discuss Synthesize
#> 2.5 3.0 3.5 3.0 2.5 2.5 2.0
#> Evaluate Create Share
#> 2.5 3.0 2.5
centrality_comm_centrality(regulation_net, membership = rep(1:2, each = 5))
#> Explore Plan Monitor Adapt Reflect Discuss Synthesize
#> 4.428686 4.765714 6.954171 5.299886 4.428686 3.760000 1.980000
#> Evaluate Create Share
#> 3.760000 6.453750 3.760000
centrality_community_mediator(regulation_net,
membership = rep(1:2, each = 5))
#> Explore Plan Monitor Adapt Reflect Discuss Synthesize
#> 0.08990283 0.07222471 0.12771476 0.10203287 0.08990283 0.06684519 0.00000000
#> Evaluate Create Share
#> 0.06684519 0.11111111 0.06684519