R/measure_centrality_between.R
measure_central_between.RdThese functions calculate common betweenness-related centrality measures for one- and two-mode networks:
node_by_betweenness() measures the betweenness centralities of nodes in a network.
node_by_induced() measures the induced betweenness centralities of nodes in a network.
node_by_flow() measures the flow betweenness centralities of nodes in a network,
which uses an electrical current model for information spreading
in contrast to the shortest paths model used by normal betweenness centrality.
node_by_stress() measures the stress centrality of nodes in a network.
These four differ in what they count:
node_by_betweenness() sums the proportion of shortest paths between
each pair that run through a node, so every pair of nodes contributes at
most one unit however many shortest paths connect it;
node_by_stress() instead sums the raw count of those paths, so pairs
joined by many equally short routes count for more;
node_by_flow() abandons shortest paths altogether for maximum flow,
crediting nodes that carry traffic along longer routes as well;
and node_by_induced() asks a different question again — not how much
passes through a node, but how much total betweenness the network would
lose if it were removed.
For ties rather than nodes, see tie_by_betweenness().
All measures attempt to use as much information as they are offered,
including whether the networks are directed, weighted, or multimodal.
If this would produce unintended results,
first transform the salient properties using e.g. to_undirected() functions.
All centrality and centralization measures return normalised or scaled
measures where available, reported when the measure is printed.
node_by_betweenness(.data, normalized = TRUE, cutoff = NULL)
node_by_induced(.data, normalized = TRUE, cutoff = NULL)
node_by_flow(.data, normalized = TRUE)
node_by_stress(.data, normalized = TRUE)A network object of class stocnet, igraph, tbl_graph, network, or similar.
Internally any of these will be coerced to an efficient implementation.
For more information on possible coercions, see e.g. manynet::as_stocnet().
Logical scalar, whether scores are normalized. Different denominators may be used depending on the measure, whether the object is one-mode or two-mode, and other arguments. By default TRUE.
The maximum path length to consider when calculating betweenness. If negative or NULL (the default), there's no limit to the path lengths considered.
A node_measure numeric vector the length of the nodes in the network,
providing the scores for each node.
If the network is labelled,
then the scores will be labelled with the nodes' names.
The object also carries the measure it computed, the range its values
can fall within, and whether and how those values were normalized.
These are shown as a one-line header when the object is printed.
Where a measure offers a choice between several ways of counting the
same thing, it also carries the variant it used.
All can be retrieved with attr().
Betweenness centrality is based on the number of shortest paths between other nodes that a node lies upon: $$C_B(i) = \sum_{j,k:j \neq k, j \neq i, k \neq i} \frac{g_{jik}}{g_{jk}}$$
Setting cutoff counts only those shortest paths no longer than \(k\),
which elsewhere goes by distance-bounded betweenness (Brandes, 2008) or
range-limited betweenness (Ercsey-Ravasz et al., 2012).
Normalization still applies, so a bounded score remains comparable across
networks.
Induced centrality concerns the change in total betweenness centrality
between networks with and without a given node:
$$C_I(i) = C_B(G) - C_B(G\ i)$$
This "remove the node and re-measure" logic is the general
delta centrality framework of Latora and Marchiori (2007);
node_by_induced() is its betweenness instance, and
node_by_vitality() its closeness instance.
Flow betweenness centrality concerns the total maximum flow, \(f\), between other nodes \(j,k\) in a network \(G\) that a given node mediates: $$C_F(i) = \sum_{j,k:j\neq k, j\neq i, k\neq i} f(j,k,G) - f(j,k,G\ i)$$ When normalized (by default) this sum of differences is divided by the sum of flows \(f(i,j,G)\).
Stress centrality is the number of all shortest paths or geodesics, \(g\), between other nodes that a given node mediates: $$C_S(i) = \sum_{j,k:j \neq k, j \neq i, k \neq i} g_{jik}$$ High stress nodes lie on a large number of shortest paths between other nodes, and thus associated with bridging or spanning boundaries.
Freeman, Linton. 1977. "A set of measures of centrality based on betweenness". Sociometry, 40(1): 35–41. doi:10.2307/3033543
Brandes, Ulrik. 2008. "On variants of shortest-path betweenness centrality and their generic computation". Social Networks 30(2): 136-145. doi:10.1016/j.socnet.2007.11.001
Ercsey-Ravasz, Maria, Ryan N. Lichtenwalter, Nitesh V. Chawla, and Zoltan Toroczkai. 2012. "Range-limited centrality measures in complex networks". Physical Review E 85(6): 066103. doi:10.1103/PhysRevE.85.066103
Everett, Martin and Steve Borgatti. 2010. "Induced, endogenous and exogenous centrality" Social Networks, 32: 339-344. doi:10.1016/j.socnet.2010.06.004
Latora, Vito, and Massimo Marchiori. 2007. "A measure of centrality based on network efficiency". New Journal of Physics 9(6): 188. doi:10.1088/1367-2630/9/6/188
Freeman, Linton C., Stephen P. Borgatti, and Douglas R. White. 1991. "Centrality in Valued Graphs: A Measure of Betweenness Based on Network Flow". Social Networks, 13(2), 141-154. doi:10.1016/0378-8733(91)90017-N
Koschutzki, D., K.A. Lehmann, L. Peeters, S. Richter, D. Tenfelde-Podehl, and O. Zlotowski. 2005. "Centrality Indices". In U. Brandes and T. Erlebach (eds.), Network Analysis: Methodological Foundations. Berlin: Springer.
Shimbel, A. 1953. "Structural Parameters of Communication Networks". Bulletin of Mathematical Biophysics, 15:501-507. doi:10.1007/BF02476438
Other betweenness:
measure_central_tie_between,
measure_centralisation_between
Other centrality:
measure_central_close,
measure_central_degree,
measure_central_eigen,
measure_central_tie_between,
measure_central_tie_close,
measure_central_tie_degree,
measure_central_tie_eigen,
measure_centralisation_between,
measure_centralisation_close,
measure_centralisation_degree,
measure_centralisation_eigen
Other measures:
measure_assort_net,
measure_assort_node,
measure_breadth,
measure_broker_node,
measure_broker_tie,
measure_brokerage,
measure_central_close,
measure_central_degree,
measure_central_eigen,
measure_central_tie_between,
measure_central_tie_close,
measure_central_tie_degree,
measure_central_tie_eigen,
measure_closure,
measure_closure_node,
measure_cohesion,
measure_core,
measure_diffusion_infection,
measure_diffusion_net,
measure_diffusion_node,
measure_diverse_net,
measure_diverse_node,
measure_features,
measure_fit,
measure_fragmentation,
measure_hierarchy,
measure_periods
Other nodal:
mark_core,
mark_degree,
mark_diff,
mark_nodes,
mark_select_node,
measure_assort_node,
measure_broker_node,
measure_brokerage,
measure_central_close,
measure_central_degree,
measure_central_eigen,
measure_closure_node,
measure_core,
measure_diffusion_node,
measure_diverse_node,
member_brokerage,
member_cliques,
member_community,
member_community_hier,
member_community_non,
member_components,
member_core,
member_diffusion,
member_equivalence,
motif_brokerage_node,
motif_clique,
motif_composition,
motif_exposure,
motif_node,
motif_path
node_by_betweenness(ison_southern_women)
#> # Betweenness centrality, normalized [0, 1]
#> ▆▂▁▁▁
#> Evelyn Laura Theresa Brenda Charlotte Frances Eleanor Pearl Ruth Verne Myra
#> 1 0.097 0.052 0.088 0.05 0.011 0.011 0.01 0.007 0.017 0.016 0.016
#> # ... and 7 more values from this nodeset. Use `print_all(...)` to print all values.
#> E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13
#> 1 0.002 0.002 0.018 0.008 0.038 0.065 0.13 0.244 0.226 0.011 0.02 0.018 0.002
#> # ... and 1 more values from this nodeset. Use `print_all(...)` to print all values.
node_by_induced(ison_adolescents)
#> # Induced centrality (-Inf, Inf)
#> ▁▁▄▁▁
#> Betty Sue Alice Jane Dale Pam Carol Tina
#> 1 0.095 0.362 -0.505 0.029 -0.105 1.10 0.895 0.362