Measures of node and tie centrality

Source: R/measure_centrality.R

These functions calculate common centrality measures for one- and two-mode networks. 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 normalized measures by default, including for two-mode networks.

node_degree(object, normalized = TRUE, direction = c("all", "out", "in"))

tie_degree(object, normalized = TRUE)

node_closeness(object, normalized = TRUE, direction = "out", cutoff = NULL)

tie_closeness(object, normalized = TRUE)

node_betweenness(object, normalized = TRUE, cutoff = NULL)

tie_betweenness(object, normalized = TRUE)

node_eigenvector(object, normalized = TRUE, scale = FALSE)

tie_eigenvector(object, normalized = TRUE)

node_reach(object, normalized = TRUE, k = 2)

Arguments

object

An object of a migraph-consistent class:

  • matrix (adjacency or incidence) from {base} R

  • edgelist, a data frame from {base} R or tibble from {tibble}

  • igraph, from the {igraph} package

  • network, from the {network} package

  • tbl_graph, from the {tidygraph} package

normalized

Logical scalar, whether the centrality scores are normalized. Different denominators are used depending on whether the object is one-mode or two-mode, the type of centrality, and other arguments.

direction

Character string, “out” bases the measure on outgoing ties, “in” on incoming ties, and "all" on either/the sum of the two. For two-mode networks, "all" uses as numerator the sum of differences between the maximum centrality score for the mode against all other centrality scores in the network, whereas "in" uses as numerator the sum of differences between the maximum centrality score for the mode against only the centrality scores of the other nodes in that mode.

cutoff

Maximum path length to use during calculations.

scale

Logical scalar, whether to rescale the vector so the maximum score is 1.

k

Integer of steps out to calculate reach

Value

A single centralization score if the object was one-mode, and two centralization scores if the object was two-mode.

Depending on how and what kind of an object is passed to the function, the function will return a tidygraph object where the nodes have been updated

A numeric vector giving the betweenness centrality measure of each node.

A numeric vector giving the eigenvector centrality measure of each node.

Functions

  • node_degree: Calculates the degree centrality of nodes in an unweighted network, or weighted degree/strength of nodes in a weighted network.

  • tie_degree: Calculate the degree centrality of edges in a network

  • node_closeness: Calculate the closeness centrality of nodes in a network

  • tie_closeness: Calculate the closeness of each edge to each other edge in the network.

  • node_betweenness: Calculate the betweenness centralities of nodes in a network

  • tie_betweenness: Calculate number of shortest paths going through an edge

  • node_eigenvector: Calculate the eigenvector centrality of nodes in a network

  • tie_eigenvector: Calculate the eigenvector centrality of edges in a network

  • node_reach: Calculate nodes' reach centrality

References

Faust, Katherine. 1997. "Centrality in affiliation networks." Social Networks 19(2): 157-191. doi:10.1016/S0378-8733(96)00300-0 .

Borgatti, Stephen P., and Martin G. Everett. 1997. "Network analysis of 2-mode data." Social Networks 19(3): 243-270. doi:10.1016/S0378-8733(96)00301-2 .

Borgatti, Stephen P., and Daniel S. Halgin. 2011. "Analyzing affiliation networks." In The SAGE Handbook of Social Network Analysis, edited by John Scott and Peter J. Carrington, 417–33. London, UK: Sage. doi:10.4135/9781446294413.n28 .

Bonacich, Phillip. 1991. “Simultaneous Group and Individual Centralities.” Social Networks 13(2):155–68. doi:10.1016/0378-8733(91)90018-O .

See also

to_undirected() for removing edge directions and to_unweighted() for removing weights from a graph.

Other measures: centralisation, closure, cohesion(), diversity, features, holes

Examples

node_degree(mpn_elite_mex)
#>   Trevino Madero Carranza Aguilar Obregon Calles `Aleman Gonzalez` `Portes Gil`
#> 1  0.0882  0.176    0.235   0.176   0.176  0.176             0.147        0.235
#> # ... with 27 more in the vector.
node_degree(ison_southern_women)
#>   EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL  RUTH VERNE  MYRA
#> 1  0.571   0.5   0.571    0.5     0.286   0.286   0.286 0.214 0.286 0.286 0.286
#> # ... with 7 more in the vector.
#>      E1    E2    E3    E4    E5    E6    E7    E8    E9   E10   E11   E12   E13
#> 1 0.167 0.167 0.333 0.222 0.444 0.444 0.556 0.778 0.667 0.333 0.222 0.389 0.222
#> # ... with 1 more in the vector.
tie_degree(ison_adolescents)
#>   `Betty-Sue` `Sue-Alice` `Alice-Jane` `Sue-Dale` `Alice-Dale` `Jane-Dale`
#> 1       0.333       0.667        0.444      0.556        0.556       0.333
#> # ... with 4 more in the vector.
node_closeness(mpn_elite_mex)
#>   Trevino Madero Carranza Aguilar Obregon Calles `Aleman Gonzalez` `Portes Gil`
#> 1     0.4  0.405    0.466   0.493   0.436  0.459             0.466        0.493
#> # ... with 27 more in the vector.
node_closeness(ison_southern_women)
#>   EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL  RUTH VERNE  MYRA
#> 1    0.8 0.727     0.8  0.727       0.6   0.667   0.667 0.667 0.706 0.706 0.686
#> # ... with 7 more in the vector.
#>      E1    E2    E3    E4    E5    E6    E7    E8    E9   E10   E11   E12   E13
#> 1 0.524 0.524 0.564 0.537 0.595 0.688 0.733 0.846 0.786 0.564 0.537 0.579 0.537
#> # ... with 1 more in the vector.
(ec <- tie_closeness(ison_adolescents))
#>   `Betty-Sue` `Sue-Alice` `Alice-Jane` `Sue-Dale` `Alice-Dale` `Jane-Dale`
#> 1       0.562       0.692          0.6      0.643        0.643         0.5
#> # ... with 4 more in the vector.
plot(ec)

ison_adolescents %>% 
  activate(edges) %>% mutate(weight = ec) %>% 
  autographr()

node_betweenness(mpn_elite_mex)
#>   Trevino  Madero Carranza Aguilar Obregon Calles `Aleman Gonzalez` `Portes Gil`
#> 1 0.00505 0.00819   0.0309  0.0206 0.00806 0.0249           0.00944       0.0389
#> # ... with 27 more in the vector.
node_betweenness(ison_southern_women)
#>       V1     V2     V3     V4     V5     V6      V7      V8     V9    V10    V11
#> 1 0.0967 0.0517 0.0876 0.0498 0.0107 0.0108 0.00936 0.00673 0.0167 0.0144 0.0134
#> # ... with 7 more in the vector.
#>        V1      V2     V3      V4     V5     V6    V7    V8    V9    V10    V11
#> 1 0.00215 0.00209 0.0181 0.00764 0.0376 0.0620 0.129 0.240 0.213 0.0151 0.0200
#> # ... with 3 more in the vector.
(tb <- tie_betweenness(ison_adolescents))
#>   `Betty-Sue` `Sue-Alice` `Alice-Jane` `Sue-Dale` `Alice-Dale` `Jane-Dale`
#> 1           7           3            5        4.5          2.5           2
#> # ... with 4 more in the vector.
plot(tb)

ison_adolescents %>% 
  activate(edges) %>% mutate(weight = tb) %>% 
  autographr()

node_eigenvector(mpn_elite_mex)
#>   Trevino Madero Carranza Aguilar Obregon Calles `Aleman Gonzalez` `Portes Gil`
#> 1  0.0808  0.109    0.166   0.170   0.130  0.138             0.140        0.208
#> # ... with 27 more in the vector.
node_eigenvector(ison_southern_women)
#>   EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL  RUTH VERNE  MYRA
#> 1  0.423 0.397   0.472  0.402     0.227   0.287   0.319 0.264 0.337 0.327 0.292
#> # ... with 7 more in the vector.
#>      E1    E2    E3    E4    E5    E6    E7    E8    E9   E10   E11   E12   E13
#> 1 0.215 0.228 0.356 0.261 0.431 0.447 0.522 0.639 0.505 0.323 0.159 0.361 0.251
#> # ... with 1 more in the vector.
tie_eigenvector(ison_adolescents)
#>   `Betty-Sue` `Sue-Alice` `Alice-Jane` `Sue-Dale` `Alice-Dale` `Jane-Dale`
#> 1       0.366       0.638        0.447      0.524        0.541       0.333
#> # ... with 4 more in the vector.
node_reach(ison_adolescents)
#>   Betty   Sue Alice  Jane  Dale   Pam Carol  Tina
#> 1 0.429 0.286 0.286 0.286 0.286 0.571 0.286 0.143