The goal of concorR is to implement the CONCOR (CONvergence of iterated CORrelations) algorithm for positional analysis. Positional analysis divides a network into blocks based on the similarity of links between actors. CONCOR uses structural equivalence—“same ties to same others”—as its criterion for grouping nodes, and calculates this by correlating columns in the adjacency matrix. For more details on CONCOR, see the original description by Breiger, Boorman, and Arabie (1975), or Chapter 9 in Wasserman and Faust (1994).
You can install the released version of concorR from CRAN with:
install.packages("concorR")And the development version from GitHub with:
# install.packages("devtools")
devtools::install_github("ATraxLab/concorR")This is a basic example which shows a common task: using CONCOR to partition a single adjacency matrix.
library(concorR)
a <- matrix(c(0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1,
1, 0, 1, 0, 1, 1, 0, 0, 0, 0), ncol = 5)
rownames(a) <- letters[1:5]
colnames(a) <- letters[1:5]
concor(list(a))
#> block vertex
#> 1 1 b
#> 2 1 c
#> 3 1 d
#> 4 2 a
#> 5 2 eAdditional helper functions are included for using the igraph package:
library(igraph)
#>
#> Attaching package: 'igraph'
#> The following objects are masked from 'package:stats':
#>
#> decompose, spectrum
#> The following object is masked from 'package:base':
#>
#> union
library(viridis)
#> Loading required package: viridisLite
plot(graph_from_adjacency_matrix(a))
glist <- concor_make_igraph(list(a))
col_pal_a = viridis(2)
plot(glist[[1]], vertex.color = col_pal_a[V(glist[[1]])$csplit1])
The blockmodel shows the permuted adjacency matrix, rearranged to group nodes by CONCOR partition.
bm <- make_blk(list(a), 1)[[1]]
plot_blk(bm, labels = TRUE)
The reduced matrix represents each position as a node, and calculates links by applying a density threshold to the ties between (and within) positions.
(r_mat <- make_reduced(list(a), nsplit = 1))
#> $reduced_mat
#> $reduced_mat[[1]]
#> Block 1 Block 2
#> Block 1 1 0
#> Block 2 1 1
#>
#>
#> $dens
#> [1] 0.6
r_igraph <- make_reduced_igraph(r_mat$reduced_mat[[1]])
plot_reduced(r_igraph)
In the prior example, the reduced network was created using an edge
density threshold. For some applications, it may be preferred to use a
degree-based measure instead. If we define
to be the adjacency
matrix, define the sub-adjacency matrix
as
follows:
We’d like to use a simple criterion to determine whether to draw an edge or not, we will use the scaled degree for this purpose. For our definition, we will divide by the max observed degree.
Note that for this definition, the sub-adjacency matrix will not be
square if there are different numbers of elements in each block. Also,
while it is more common to use normalized degree, it is identical to
edge density, and therefore of no help to us here.
To use this criteria, we have created an argument stat. The default to
this argument is 'density', which does the analysis in the previous
section. To use this criterion instead, use the option 'degree'.
(r_mat_deg <- make_reduced(list(a), nsplit = 1, stat = 'degree'))
#> $reduced_mat
#> $reduced_mat[[1]]
#> Block 1 Block 2
#> Block 1 1 0
#> Block 2 1 1
#>
#>
#> $deg
#> $deg[[1]]
#> [1] 0.6
r_deg_igraph <- make_reduced_igraph(r_mat_deg$reduced_mat[[1]])
plot_reduced(r_deg_igraph)
Sometimes, there are isolated nodes in a network. The CONCOR algorithm
works with this network by creating a block for isolated members, and
then running the standard algorithm on the network with the isolated
members removed. Therefore the number of blocks will be 2^nsplit + 1
blocks in that event. Consider this example:
isoA = matrix(c(0,1,1,1,1,0,
1,0,1,1,1,0,
1,1,0,1,1,0,
0,0,0,0,1,0,
0,0,0,1,0,0,
0,0,0,0,0,0),
nrow=6,byrow=TRUE)
rownames(isoA) = LETTERS[1:6]
colnames(isoA) = LETTERS[1:6]
concor(list(isoA),nsplit=1)
#> block vertex
#> 1 1 D
#> 2 1 E
#> 3 2 A
#> 4 2 B
#> 5 2 C
#> 6 3 FThis network looks like this:
plot(graph_from_adjacency_matrix(isoA))
## With CONCOR block coloring
gISOlist <- concor_make_igraph(list(isoA))
col_pal_iso = viridis(3)
plot(gISOlist[[1]], vertex.color = col_pal_iso[V(gISOlist[[1]])$csplit1])
bm = make_blk(list(isoA),nsplit=1)[[1]]
plot_blk(bm, labels = TRUE)
rmDen = make_reduced(list(isoA),nsplit=1,stat='density')
rmDeg = make_reduced(list(isoA),nsplit=1,stat='degree')
rmDen.g = make_reduced_igraph(rmDen$reduced_mat[[1]])
plot_reduced(rmDen.g)
rmDeg.g = make_reduced_igraph(rmDeg$reduced_mat[[1]])
plot_reduced(rmDeg.g)
CONCOR can use multiple adjacency matrices to partition nodes based on
all relations simultaneously. The package includes igraph data files
for the Krackhardt (1987) high-tech managers study, which gives networks
for advice, friendship, and reporting among 21 managers at a firm.
(These networks were used in the examples of Wasserman and Faust
(1994).)
First, take a look at the CONCOR partitions for two splits (four positions), considering only the advice or only the friendship networks.
par(mfrow = c(1, 2))
plot_socio(krack_advice) # plot_socio imposes some often-useful plot parameters
plot_socio(krack_friend)
m1 <- igraph::as_adjacency_matrix(krack_advice, sparse = FALSE)
m2 <- igraph::as_adjacency_matrix(krack_friend, sparse = FALSE)
g1 <- concor_make_igraph(list(m1), nsplit = 2)
g2 <- concor_make_igraph(list(m2), nsplit = 2)
gadv <- set_vertex_attr(krack_advice, "csplit2", value = V(g1[[1]])$csplit2)
gfrn <- set_vertex_attr(krack_friend, "csplit2", value = V(g2[[1]])$csplit2)
par(mfrow = c(1, 2))
plot_socio(gadv, nsplit = 2)
plot_socio(gfrn, nsplit = 2)
Next, compare with the multi-relation blocking:
gboth <- concor_make_igraph(list(m1, m2), nsplit = 2)
gadv2 <- set_vertex_attr(krack_advice, "csplit2", value = V(gboth[[1]])$csplit2)
gfrn2 <- set_vertex_attr(krack_friend, "csplit2", value = V(gboth[[2]])$csplit2)
par(mfrow = c(1, 2))
plot_socio(gadv2, nsplit = 2)
plot_socio(gfrn2, nsplit = 2)
Including information from both relations changes the block membership of several nodes.
It also affects the reduced networks, as can be seen from comparing the single-relation version:
red1 <- make_reduced(list(m1), nsplit = 2)
red2 <- make_reduced(list(m2), nsplit = 2)
gred1 <- make_reduced_igraph(red1$reduced_mat[[1]])
gred2 <- make_reduced_igraph(red2$reduced_mat[[1]])
par(mfrow = c(1, 2))
plot_reduced(gred1)
plot_reduced(gred2)
with the multi-relation version:
redboth <- make_reduced(list(m1, m2), nsplit = 2)
gboth <- lapply(redboth$reduced_mat, make_reduced_igraph)
par(mfrow = c(1, 2))
plot_reduced(gboth[[1]])
plot_reduced(gboth[[2]])
red1d <- make_reduced(list(m1), nsplit = 2, stat='degree')
red2d <- make_reduced(list(m2), nsplit = 2, stat='degree')
gred1d <- make_reduced_igraph(red1d$reduced_mat[[1]])
gred2d <- make_reduced_igraph(red2d$reduced_mat[[1]])
par(mfrow = c(1, 2))
plot_reduced(gred1d)
plot_reduced(gred2d)
with the multi-relation version:
redbothd <- make_reduced(list(m1, m2), nsplit = 2, stat='degree')
gbothd <- lapply(redbothd$reduced_mat, make_reduced_igraph)
par(mfrow = c(1, 2))
plot_reduced(gbothd[[1]])
plot_reduced(gbothd[[2]])
This work was supported by National Science Foundation awards DUE-1712341 and DUE-1711017.
R. L. Breiger, S. A. Boorman, P. Arabie, An algorithm for clustering relational data with applications to social network analysis and comparison with multidimensional scaling. J. of Mathematical Psychology. 12, 328 (1975). http://doi.org/10.1016/0022-2496(75)90028-0
D. Krackhardt, Cognitive social structures. Social Networks. 9, 104 (1987). http://doi.org/10.1016/0378-8733(87)90009-8
S. Wasserman and K. Faust, Social Network Analysis: Methods and Applications (Cambridge University Press, 1994).