Segregation measures

Measures of how much the ties of a network stay within the groups given by an attribute of the nodes, reviewed by Bojanowski and Corten (2014).

Usage

segregation(
  A,
  att,
  method = c("assortativity", "gam", "orwg", "coleman", "freeman"),
  digraph = FALSE,
  loops = FALSE
)

Arguments

A

A square matrix

att

A vector with the group of each node

method

The measure: assortativity (default), gam, orwg, coleman or freeman

digraph

Whether the matrix is directed or undirected. The coleman index is defined for directed networks

loops

Whether to consider the loops of the matrix

Value

This function returns the value of the measure, which is a value per group for the coleman index.

Details

All the measures are computed from the mixing matrix, which counts the ties within and between groups:

assortativity: the proportion of ties that are within groups, compared with the proportion expected if the ties were distributed at random keeping how active each group is (Newman, 2003). It is one when every tie is within a group, and zero under random mixing.

gam: the index of Gupta, Anderson and May (1989), the trace of the matrix of the proportion of the ties of each group that go to every other group, rescaled to go from \(-1/(K-1)\) to one. It is defined for undirected networks, and every group should have at least one tie.

orwg: the odds of a tie within a group divided by the odds of a tie between groups (Moody, 2001). Unlike the other measures, it takes into account the pairs of nodes that are not tied, so it is not affected by the density of the network.

coleman: the homophily index of Coleman (1958), computed for each group: how many ties the group sends to itself compared with the ties it would send if it chose the other nodes at random. It is one when the group only relates to itself.

freeman: the segregation index of Freeman (1978) for two groups: how many fewer ties between the groups there are than the ones expected in a random network with the same density and group sizes. It is zero when there are as many as expected, or more.

References

Bojanowski, M. and Corten, R. (2014). Measuring segregation in social networks. Social Networks, 39, 14–32. doi:10.1016/j.socnet.2014.04.001

Coleman, J. (1958). Relational analysis: The study of social organizations with survey methods. Human Organization, 17(4), 28–36. doi:10.17730/humo.17.4.q5604m676260q8n7

Freeman, L. C. (1978). Segregation in social networks. Sociological Methods and Research, 6(4), 411–429. doi:10.1177/004912417800600401

Gupta, S., Anderson, R. M. and May, R. M. (1989). Networks of sexual contacts: implications for the pattern of spread of HIV. AIDS, 3(12), 807–817. doi:10.1097/00002030-198912000-00005

Moody, J. (2001). Race, school integration, and friendship segregation in America. American Journal of Sociology, 107(3), 679–716. doi:10.1086/338954

Newman, M. E. J. (2003). Mixing patterns in networks. Physical Review E, 67(2), 026126. doi:10.1103/PhysRevE.67.026126

Author

Alejandro Espinosa-Rada

Examples

A <- matrix(c(
  0, 1, 1, 0, 0, 0,
  1, 0, 1, 0, 0, 0,
  1, 1, 0, 1, 0, 0,
  0, 0, 1, 0, 1, 1,
  0, 0, 0, 1, 0, 1,
  0, 0, 0, 1, 1, 0
), byrow = TRUE, ncol = 6)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)
att <- c("a", "a", "a", "b", "b", "b")

segregation(A, att)
#> [1] 0.7142857
segregation(A, att, method = "orwg")
#> [1] Inf
segregation(A, att, method = "coleman", digraph = FALSE)
#>         a         b 
#> 0.7619048 0.7619048