Centrality

Centrality indices for signed networks

There exist dozens of indices for networks with positive ties, but for signed networks they are rather scarce. The package implements three indices so far. Versions of degree and eigenvector centrality, and PN centrality by Everett & Borgatti.

Degree centrality can be calculated in four different ways with degree_signed(), specified by the type parameter:

• type="pos" count only positive neighbors
• type="neg" count only negative neighbors
• type="ratio" positive neighbors/(positive neighbors+negative neighbors)
• type="net" positive neighbors-negative neighbors

The mode parameter can be used to get “in” and “out” versions for directed networks.

The PN index is very similar to Katz status and Hubbell’s measure for networks with only positive ties. The technical details can be found in the paper by Everett & Borgatti.

The below example illustrates all indices with a network where signed degree can not distinguish vertices.

A <- matrix(c(0,  1,  0,  1,  0,  0,  0, -1, -1,  0,  
               1,  0,  1, -1,  1, -1, -1,  0,  0,  0,  
               0,  1,  0,  1, -1,  0,  0,  0, -1,  0,  
               1, -1,  1,  0,  1, -1, -1,  0,  0,  0,  
               0,  1, -1,  1,  0,  1,  0, -1,  0, -1,  
               0, -1,  0, -1,  1,  0,  1,  0,  1, -1,  
               0, -1,  0, -1,  0,  1,  0,  1, -1,  1,  
              -1,  0,  0,  0, -1,  0,  1,  0,  1,  0,  
              -1,  0, -1,  0,  0,  1, -1,  1,  0,  1,  
               0,  0,  0,  0, -1, -1,  1,  0,  1,  0),10,10)

g <- graph_from_adjacency_matrix(A,"undirected",weighted = "sign")

degree_signed(g,type="ratio")
#>  [1] 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
eigen_centrality_signed(g)
#>  [1] 0.62214960 1.00000000 0.74518850 1.00000000 0.89990041 0.64289592
#>  [7] 0.35828159 0.37471921 0.28087411 0.07834568
pn_index(g)
#>  [1] 0.9009747 0.8613482 0.9076997 0.8613482 0.8410658 0.8496558 0.8617321
#>  [8] 0.9015909 0.8509848 0.9072930

Note that PN centrality and eigenvector centrality differ significantly for this network.

cor(eigen_centrality_signed(g),pn_index(g),method = "kendall")
#> [1] -0.2

A note on eigenvector centrality

The adjacency matrix of a signed network may not have a dominant eigenvalue. This means it is not clear which eigenvector should be used. In addition it is possible for the adjacency matrix to have repeated eigenvalues and hence multiple linearly independent eigenvectors. In this case certain centralities can be arbitrarily assigned. The eigen_centrality_signed() function returns an error if this is the case.

A <- matrix(c( 0,  1,  1, -1,  0,  0, -1,  0,  0, 
               1,  0,  1,  0, -1,  0,  0, -1,  0, 
               1,  1,  0,  0,  0, -1,  0,  0, -1, 
              -1,  0,  0,  0,  1,  1, -1,  0,  0, 
               0, -1,  0,  1,  0,  1,  0, -1,  0, 
               0,  0, -1,  1,  1,  0,  0,  0, -1, 
              -1,  0,  0, -1,  0,  0,  0,  1,  1, 
               0, -1,  0,  0, -1,  0,  1,  0,  1, 
               0,  0, -1,  0,  0, -1,  1,  1, 0), 9, 9)

g <- igraph::graph_from_adjacency_matrix(A,"undirected",weighted = "sign")
eigen_centrality_signed(g)
#> Error in eigen_centrality_signed(g): no dominant eigenvalue exists

References

Everett, Martin G., and Stephen P. Borgatti. 2014. “Networks Containing Negative Ties.” Social Networks 38: 111–20.

Bonacich, Phillip, and Paulette Lloyd. 2004. “Calculating Status with Negative Relations.” Social Networks 26 (4): 331–38.