ERGMs for Valued Networks with Applications to Count Data
Pavel N. Krivitsky, Carter T. Butts, and the Statnet Team
Coverage
This vignette covers valued network modeling in the ergm framework, with an emphasis on count data. Examples pertinent to rank data can be found in the vignette in the ergm.rank package.
Representing valued network data
network (Butts (2008)) objects have three types of attributes:
- network attributes – attributes which pertain to the whole network and include such information as network size, directedness, and multiplicity;
- vertex attributes – attributes which pertain to the individual vertices in the network and include such information as vertex label, as well as group assignment or some other property of the individual being represented;
- edge attributes – attributes which pertain to edges in the network and include such information as edge value.
An edge attribute is only defined for edges that exist in the network. Thus, in a matter of speaking, to set an edge value, one first has to create an edge and then set its attribute.
As with network and vertex attributes, edge attributes that have been set can be listed with list.edge.attributes. Every network has at least one edge attribute: "na", which, if set to TRUE, marks an edge as missing.
Constructing valued networks
There are several ways to create valued networks for use with ergm. Here, we will demonstrate two of the most straightforward approaches.
Sampson’s Monks, pooled
The first dataset that we’ll be using is the (in)famous Sampson’s monks. Dataset samplk in package ergm contains three (binary) networks: samplk1, samplk2, and samplk3, containing the Monks’ top-tree friendship nominations at each of the three survey time points. We are going to construct a valued network that pools these nominations.
Method 1: From a sociomatrix In many cases, a valued sociomatrix is available (or can easily be constructed). In this case, we’ll build one from the Sampson data.
library(ergm.count) # Also loads ergm.
data(samplk)
ls()
## [1] "samplk1" "samplk2" "samplk3"
as.matrix(samplk1)[1:5,1:5]
## John Bosco Gregory Basil Peter Bonaventure
## John Bosco 0 0 1 0 1
## Gregory 1 0 0 0 0
## Basil 1 1 0 0 0
## Peter 0 0 0 0 1
## Bonaventure 0 0 0 1 0
# Create a sociomatrix totaling the nominations.
samplk.tot.m<-as.matrix(samplk1)+as.matrix(samplk2)+as.matrix(samplk3)
samplk.tot.m[1:5,1:5]
## John Bosco Gregory Basil Peter Bonaventure
## John Bosco 0 1 2 0 2
## Gregory 3 0 0 0 0
## Basil 3 1 0 0 0
## Peter 0 0 0 0 3
## Bonaventure 1 0 0 3 0
# Create a network where the number of nominations becomes an attribute of an edge.
samplk.tot <- as.network(samplk.tot.m, directed=TRUE, matrix.type="a",
ignore.eval=FALSE, names.eval="nominations" # Important!
)
# Add vertex attributes. (Note that names were already imported!)
samplk.tot %v% "group" <- samplk1 %v% "group" # Groups identified by Sampson
samplk.tot %v% "group"
## [1] "Turks" "Turks" "Outcasts" "Loyal" "Loyal" "Loyal"
## [7] "Turks" "Waverers" "Loyal" "Waverers" "Loyal" "Turks"
## [13] "Waverers" "Turks" "Turks" "Turks" "Outcasts" "Outcasts"
# We can view the attribute as a sociomatrix.
as.matrix(samplk.tot,attrname="nominations")[1:5,1:5]
## John Bosco Gregory Basil Peter Bonaventure
## John Bosco 0 1 2 0 2
## Gregory 3 0 0 0 0
## Basil 3 1 0 0 0
## Peter 0 0 0 0 3
## Bonaventure 1 0 0 3 0
# Also, note that samplk.tot now has an edge if i nominated j *at least once*.
as.matrix(samplk.tot)[1:5,1:5]
## John Bosco Gregory Basil Peter Bonaventure
## John Bosco 0 1 1 0 1
## Gregory 1 0 0 0 0
## Basil 1 1 0 0 0
## Peter 0 0 0 0 1
## Bonaventure 1 0 0 1 0Method 2: Form an edgelist Sociomatrices are simple to work with, but not very convenient for large, sparse networks. In the latter case, edgelists are often preferred. For our present case, suppose that instead of a sociomatrix we have an edgelist with values:
samplk.tot.el <- as.matrix(samplk.tot, attrname="nominations",
matrix.type="edgelist")
samplk.tot.el[1:5,]
## [,1] [,2] [,3]
## [1,] 2 1 3
## [2,] 3 1 3
## [3,] 5 1 1
## [4,] 6 1 2
## [5,] 7 1 1
# and an initial empty network.
samplk.tot2 <- samplk1 # Copy samplk1
delete.edges(samplk.tot2, seq_along(samplk.tot2$mel)) # Empty it out
samplk.tot2 #We could also have used network.initialize(18)
## Network attributes:
## vertices = 18
## directed = TRUE
## hyper = FALSE
## loops = FALSE
## multiple = FALSE
## bipartite = FALSE
## total edges= 0
## missing edges= 0
## non-missing edges= 0
##
## Vertex attribute names:
## cloisterville group vertex.names
##
## No edge attributes
samplk.tot2[samplk.tot.el[,1:2], names.eval="nominations", add.edges=TRUE] <-
samplk.tot.el[,3]
as.matrix(samplk.tot2,attrname="nominations")[1:5,1:5]
## John Bosco Gregory Basil Peter Bonaventure
## John Bosco 0 1 2 0 2
## Gregory 3 0 0 0 0
## Basil 3 1 0 0 0
## Peter 0 0 0 0 3
## Bonaventure 1 0 0 3 0In general, the construction net[i,j, names.eval="attrname", add.edges=TRUE] <- value can be used to modify individual edge values for attribute "attrname". This way, we can also add more than one edge attribute to a network. Note that network objects can support an almost unlimited number of vertex, edge, or network attributes, and that these attributes can contain any data type. (Not all data types are compatible with all interface methods; see ?network and related documentation for more information.)
Zachary’s Karate club
The other dataset we’ll be using is almost as (in)famous Zachary’s Karate Club dataset. We will be employing here a collapsed multiplex network that counts the number of social contexts in which each pair of individuals associated with the Karate Club in question interacted. A total of 8 contexts were considered, but as the contexts themselves were determined by the network process, this limit itself can be argued to be endogenous.
Over the course of the study, the club split into two factions, one led by the instructor (“Mr. Hi”) and the other led by the Club President (“John A.”). Zachary also recorded the faction alignment of every regular attendee in the club. This dataset is included in the ergm.count package, as zach.
Visualizing a valued network
The network’s plot method for networks can be used to plot a sociogram of a network. When plotting a valued network, we it is often useful to color the ties depending on their value. Function gray can be used to generate a gradient of colors, with gray(0) generating black and gray(1) generating white. This can then be passed to the edge.col argument of plot.network.
Sampson’s Monks For the monks, let’s pass value data using a matrix.
par(mar=rep(0,4))
samplk.ecol <-
matrix(gray(1 - (as.matrix(samplk.tot, attrname="nominations")/3)),
nrow=network.size(samplk.tot))
plot(samplk.tot, edge.col=samplk.ecol, usecurve=TRUE, edge.curve=0.0001,
displaylabels=TRUE, vertex.col=as.factor(samplk.tot%v%"group"))
Edge color can also be passed as a vector of colors corresponding to edges. It’s more efficient, but the ordering in the vector must correspond to network object’s internal ordering, so it should be used with care. Note that we can also vary line width and/or transparency in the same manner:
par(mar=rep(0,4))
valmat<-as.matrix(samplk.tot,attrname="nominations") #Pull the edge values
samplk.ecol <-
matrix(rgb(0,0,0,valmat/3),
nrow=network.size(samplk.tot))
plot(samplk.tot, edge.col=samplk.ecol, usecurve=TRUE, edge.curve=0.0001,
displaylabels=TRUE, vertex.col=as.factor(samplk.tot%v%"group"),
edge.lwd=valmat^2)