igraph constraint components (c-size, c-density, c-hierarchy) - r

Question

I would like to compute the components of Burt's constraint discussed here by Burt.
Igraph's constraint command computes Burt's constraint score:
rm(list=ls())
library(igraph)
g <- graph.formula( "A"--------"B":"E":"F":"EGO",
"B"--------"A":"D":"EGO",
"C"--------"EGO",
"D"--------"B":"EGO",
"E"--------"A":"EGO",
"F"--------"A":"EGO",
"EGO"-"A":"B":"C":"D":"E":"F")
coords <- layout_nicely(g)
V(g)$label <- V(g)$name
g$layout <- coords
plot(g)
constraint(g)
Constraint only returns the overall constraint score.
with
and
is the strength (= weight) of connection between two vertices i and j.
is the direct connection between vertices i and j (share of connections from i to j of all of i's connections).
is the sum of indirect connections between vertices i and j (connections to other vertices q that are both connected to i and j).
I want to work with the individual components c-size, c-density, and c-hierarchy.
Burt reshapes the constraint equation like this:
The first term is c-size.
The second term is c-density.
The third term is c-hierarchy.
I want to compute the three components of constraint for each vertex of the network.
I could think of two solutions, both of which are beyond my capabilities.
Maybe there is a way to harness these values directly from igraph's constraint command.
Alternatively, one would have to compute these values manually.
For the example above, I have computed these values by hand using Excel:
node
degrees
constraint
c-size
c-density
c-hierarchy
A
4
0.60
0.25
0.23
0.12
B
3
0.64
0.33
0.24
0.07
C
1
1.00
1.00
0.00
0.00
D
2
0.78
0.50
0.25
0.03
E
2
0.73
0.50
0.21
0.02
F
2
0.73
0.50
0.21
0.02
EGO
6
0.40
0.17
0.16
0.07
strength
direct influence of j
indirect influence of j
combinded influence of j
FROM
TO
A
B
1
0.25
0.25
0.17
0.04
0.09
A
E
1
0.25
0.25
0.17
0.04
0.09
A
F
1
0.25
0.25
0.17
0.04
0.09
A
EGO
1
0.25
0.25
1.33
0.33
0.34
B
A
1
0.33
0.33
0.17
0.06
0.15
B
D
1
0.33
0.33
0.17
0.06
0.15
B
EGO
1
0.33
0.33
0.75
0.25
0.34
C
EGO
1
1.00
1.00
0.00
0.00
1.00
D
B
1
0.50
0.50
0.17
0.08
0.34
D
EGO
1
0.50
0.50
0.33
0.17
0.44
E
A
1
0.50
0.50
0.17
0.08
0.34
E
EGO
1
0.50
0.50
0.25
0.13
0.39
F
A
1
0.50
0.50
0.17
0.08
0.34
F
EGO
1
0.50
0.50
0.25
0.13
0.39
EGO
A
1
0.17
0.17
1.33
0.22
0.15
EGO
B
1
0.17
0.17
0.75
0.13
0.09
EGO
C
1
0.17
0.17
0.00
0.00
0.03
EGO
D
1
0.17
0.17
0.33
0.06
0.05
EGO
E
1
0.17
0.17
0.25
0.04
0.04
EGO
F
1
0.17
0.17
0.25
0.04
0.04