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,
alpha = 0,
direction = c("all", "out", "in")
)

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

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

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

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

node_power(object, normalized = TRUE, scale = FALSE, exponent = 1)

## 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.

alpha

Numeric scalar, the positive tuning parameter introduced in Opsahl et al (2010) for trading off between degree and strength centrality measures. By default, alpha = 0, which ignores tie weights and the measure is solely based upon degree (the number of ties). alpha = 1 ignores the number of ties and provides the sum of the tie weights as strength centrality. Values between 0 and 1 reflect different trade-offs in the relative contributions of degree and strength to the final outcome, with 0.5 as the middle ground. Values above 1 penalise for the number of ties. Of two nodes with the same sum of tie weights, the node with fewer ties will obtain the higher score. This argument is ignored except in the case of a weighted network.

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

exponent

Decay rate for the Bonacich power centrality score.

## 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.

A numeric vector giving each node's power centrality measure.

## Functions

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

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

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

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

• node_reach(): Calculate nodes' reach centrality or how many nodes they can reach within k steps

• node_power(): Calculate the power centrality of nodes in a network

## 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 .

Opsahl, Tore, Filip Agneessens, and John Skvoretz. 2010. "Node centrality in weighted networks: Generalizing degree and shortest paths." Social Networks 32, 245-251. doi:10.1016/j.socnet.2010.03.006

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

Bonacich, Phillip. 1987. “Power and Centrality: A Family of Measures.” The American Journal of Sociology 92(5): 1170–82. doi:10.1086/228631 .

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

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

## Examples

node_degree(mpn_elite_mex)
#>   Trevino Madero Carranza Aguilar Obregon Calles Alema…¹ Porte…² L. Ca…³ Avila…⁴
#> 1  0.0882  0.176    0.235   0.176   0.176  0.176   0.147   0.235   0.353   0.206
#> # ... with 25 more from this nodeset 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 from this nodeset 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 from this nodeset in the vector.
node_closeness(mpn_elite_mex)
#>   Trevino Madero Carranza Aguilar Obregon Calles Alema…¹ Porte…² L. Ca…³ Avila…⁴
#> 1     0.4  0.405    0.466   0.493   0.436  0.459   0.466   0.493   0.586   0.523
#> # ... with 25 more from this nodeset 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 from this nodeset 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 from this nodeset in the vector.
node_betweenness(mpn_elite_mex)
#>   Trevino  Madero Carra…¹ Aguilar Obregon Calles Alema…² Porte…³ L. Ca…⁴ Avila…⁵
#> 1 0.00505 0.00819  0.0309  0.0206 0.00806 0.0249 0.00944  0.0389   0.157  0.0204
#> # ... with 25 more from this nodeset 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 from this nodeset 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 from this nodeset in the vector.
node_eigenvector(mpn_elite_mex)
#>   Trevino Madero Carranza Aguilar Obregon Calles Alema…¹ Porte…² L. Ca…³ Avila…⁴
#> 1  0.0808  0.109    0.166   0.170   0.130  0.138   0.140   0.208   0.302   0.253
#> # ... with 25 more from this nodeset 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 from this nodeset 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 from this nodeset in the vector.