Measures of eigenvector-like centrality and centralisation

These functions calculate common eigenvector-related centrality measures, or walk-based eigenmeasures, for one- and two-mode networks:

• node_eigenvector() measures the eigenvector centrality of nodes in a network.

• node_power() measures the Bonacich, beta, or power centrality of nodes in a network.

• node_alpha() measures the alpha or Katz centrality of nodes in a network.

• node_pagerank() measures the pagerank centrality of nodes in a network.

• node_hub() measures how well nodes in a network serve as hubs pointing to many authorities.

• node_authority() measures how well nodes in a network serve as authorities from many hubs.

• tie_eigenvector() measures the eigenvector centrality of ties in a network.

• net_eigenvector() measures the eigenvector centralization for a network.

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_eigenvector(.data, normalized = TRUE, scale = FALSE)

node_power(.data, normalized = TRUE, scale = FALSE, exponent = 1)

node_alpha(.data, alpha = 0.85)

node_pagerank(.data)

node_authority(.data)

node_hub(.data)

tie_eigenvector(.data, normalized = TRUE)

net_eigenvector(.data, normalized = TRUE)

Arguments

.data

An object of a manynet-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.

scale

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

exponent

Decay rate or attentuation factor for the Bonacich power centrality score. Can be positive or negative.

alpha

A constant that trades off the importance of external influence against the importance of connection. When \(\alpha = 0\), only the external influence matters. As \(\alpha\) gets larger, only the connectivity matters and we reduce to eigenvector centrality. By default \(\alpha = 0.85\).

Value

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

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

Details

We use {igraph} routines behind the scenes here for consistency and because they are often faster. For example, igraph::eigencentrality() is approximately 25% faster than sna::evcent().

Eigenvector centrality

Eigenvector centrality operates as a measure of a node's influence in a network. The idea is that being connected to well-connected others results in a higher score. Each node's eigenvector centrality can be defined as: $$x_i = \frac{1}{\lambda} \sum_{j \in N} a_{i,j} x_j$$ where \(a_{i,j} = 1\) if \(i\) is linked to \(j\) and 0 otherwise, and \(\lambda\) is a constant representing the principal eigenvalue. Rather than performing this iteration, most routines solve the eigenvector equation \(Ax = \lambda x\).

Power or beta (or Bonacich) centrality

Power centrality includes an exponent that weights contributions to a node's centrality based on how far away those other nodes are. $$c_b(i) = \sum A(i,j) (\alpha = \beta c(j))$$ Where \(\beta\) is positive, this means being connected to central people increases centrality. Where \(\beta\) is negative, this means being connected to central people decreases centrality (and being connected to more peripheral actors increases centrality). When \(\beta = 0\), this is the outdegree. \(\alpha\) is calculated to make sure the root mean square equals the network size.

Alpha centrality

Alpha or Katz (or Katz-Bonacich) centrality operates better than eigenvector centrality for directed networks. Eigenvector centrality will return 0s for all nodes not in the main strongly-connected component. Each node's alpha centrality can be defined as: $$x_i = \frac{1}{\lambda} \sum_{j \in N} a_{i,j} x_j + e_i$$ where \(a_{i,j} = 1\) if \(i\) is linked to \(j\) and 0 otherwise, \(\lambda\) is a constant representing the principal eigenvalue, and \(e_i\) is some external influence used to ensure that even nodes beyond the main strongly connected component begin with some basic influence. Note that many equations replace \(\frac{1}{\lambda}\) with \(\alpha\), hence the name.

For example, if \(\alpha = 0.5\), then each direct connection (or alter) would be worth \((0.5)^1 = 0.5\), each secondary connection (or tertius) would be worth \((0.5)^2 = 0.25\), each tertiary connection would be worth \((0.5)^3 = 0.125\), and so on.

Rather than performing this iteration though, most routines solve the equation \(x = (I - \frac{1}{\lambda} A^T)^{-1} e\).

References

On eigenvector centrality

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

On power centrality

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

On alpha centrality

Katz, Leo 1953. "A new status index derived from sociometric analysis". Psychometrika. 18(1): 39–43.

Bonacich, P. and Lloyd, P. 2001. “Eigenvector-like measures of centrality for asymmetric relations” Social Networks. 23(3):191-201.

On pagerank centrality

Brin, Sergey and Page, Larry. 1998. "The anatomy of a large-scale hypertextual web search engine". Proceedings of the 7th World-Wide Web Conference. Brisbane, Australia.

On hub and authority centrality

Kleinberg, Jon. 1999. "Authoritative sources in a hyperlinked environment". Journal of the ACM 46(5): 604–632. doi:10.1145/324133.324140

See also

Other centrality: measure_central_between, measure_central_close, measure_central_degree

Other measures: measure_attributes, measure_central_between, measure_central_close, measure_central_degree, measure_closure, measure_cohesion, measure_diffusion_infection, measure_diffusion_net, measure_diffusion_node, measure_features, measure_heterogeneity, measure_hierarchy, measure_holes, measure_periods, measure_properties, member_diffusion

Examples

node_eigenvector(ison_southern_women)
#>   Evelyn Laura Theresa Brenda Charlotte Frances Eleanor Pearl  Ruth Verne  Myra
#> 1   0.45 0.425     0.5   0.43     0.246   0.308   0.338 0.274 0.351 0.326  0.28
#> # ... with 7 more values from this nodeset unprinted. Use `print(..., n = Inf)` to print all values.
#>      E1    E2    E3    E4    E5    E6    E7    E8    E9   E10   E11   E12   E13
#> 1 0.237  0.25 0.392 0.287  0.47 0.479 0.533  0.64 0.497 0.264 0.144 0.305 0.186
#> # ... with 1 more values from this nodeset unprinted. Use `print(..., n = Inf)` to print all values.
node_power(ison_southern_women, exponent = 0.5)
#>   Evelyn Laura Theresa Brenda Charlotte Frances Eleanor Pearl  Ruth Verne  Myra
#> 1  -1.44 -1.38   -1.44  -1.38     -1.00   -1.38   -1.38 -1.52 -1.44 -1.44 -1.52
#> # ... with 7 more values from this nodeset unprinted. Use `print(..., n = Inf)` to print all values.
#>      E1    E2    E3    E4    E5    E6    E7    E8    E9   E10   E11   E12   E13
#> 1 -1.32 -1.32 -1.32 -1.32 -1.32 -1.62 -1.62 -1.62 -1.62 -1.32 -1.32 -1.32 -1.32
#> # ... with 1 more values from this nodeset unprinted. Use `print(..., n = Inf)` to print all values.
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 values from this nodeset unprinted. Use `print(..., n = Inf)` to print all values.
net_eigenvector(ison_southern_women)
#> Mode 1 Mode 2 
#> 0.0849 0.2630