This vignette covers modelling of rank-order relational data in the
ergm framework. The reader is strongly encouraged to first
work through the vignette on valued ERGMs in the ergm.count
package and read the article by Krivitsky and
Butts (2017).
ergm.ranklibrary(ergm.rank)Note that the implementations so far are very slow, so we will only do a short example.
Suppose that we reprsent ranking (or ordinal rating) of \(j\) by \(i\) by the value of \(\boldsymbol{y}_{i,j}\). What reference can we use for ranks?
help("ergm-references", "ergm.rank")For details, see Krivitsky and Butts (2017). It’s not meaningful to
The only thing we are allowed to do is to ask if \(i\) has ranked \(j\) over \(k\).
Therefore, ordinal relational data call for their own sufficient statistics. These will depend on \[ \begin{equation*} \boldsymbol{y}_{i:\,j\succ k}\equiv\begin{cases} 1 & \text{if $j\stackrel{i}{\succ}k$ i.e., $i$ ranks $j$ above $k$;} \\ 0 & \text{otherwise.} \end{cases} \end{equation*} \] We may interpret them using the promotion statistic \[\boldsymbol{\Delta}_{i,j}^\nearrow\boldsymbol{g}(\boldsymbol{y})\equiv \boldsymbol{g}(\boldsymbol{y}^{i:\,j\rightleftarrows j^+})-\boldsymbol{g}(\boldsymbol{y}).\]
Let \({N}^{k\ne}\) be the set of possible \(k\)-tuples of actor indices where no actors are repeated. Then,
rank.deference: Deference
(aversion): Measures the amount of “deference” in the
network: configurations where an ego \(i\) ranks an alter \(j\) over another alter \(k\), but \(j\), in turn, ranks \(k\) over \(i\): \[
\boldsymbol{g}_{\text{D}}(\boldsymbol{y}) = \sum_{(i,j,l)\in {N}^{3\ne}}
\boldsymbol{y}_{l:\,j\succ i}\boldsymbol{y}_{i:\,l\succ j} \]
\[
\Delta_{i,j}^\nearrow\boldsymbol{g}_{\text{D}}(\boldsymbol{y}) = 2(
\boldsymbol{y}_{j^+:\,i\succ j}+\boldsymbol{y}_{j:\,j^+\succ i} - 1 ).
\] A lower-than-chance value of this statistic and/or a negative
coefficient implies a form of mutuality in the network.
rank.edgecov(x, attrname): Dyadic
covariates: Models the effect of a dyadic covariate on the
propensity of an ego \(i\) to rank
alter \(j\) highly: \[
\boldsymbol{g}_{\text{A}}(\boldsymbol{y};\boldsymbol{x}) =
\sum_{(i,j,k)\in {N}^{3\ne}} \boldsymbol{y}_{i:\,j\succ
k}(\boldsymbol{x}_j-\boldsymbol{x}_k).\] \[
\Delta_{i,j}^\nearrow\boldsymbol{g}_{\text{A}}(\boldsymbol{y};\boldsymbol{x})=
2(\boldsymbol{x}_{j}-\boldsymbol{x}_{j^+}),\] See the
?rank.edgecov ERGM term documentation for
arguments.
rank.inconsistency(x, attrname, weights, wtname, wtcenter):
(Weighted) Inconsistency: Measures the amount of
disagreement between rankings of the focus network and a fixed covariate
network x, by couting the number of pairwise comparisons
for which the two networks disagree. x can be a
network with an edge attribute attrname
containing the ranks or a matrix of appropriate dimension containing the
ranks. If x is not given, it defaults to the LHS network,
and if attrname is not given, it defaults to the
response edge attribute. \[\boldsymbol{g}_{\text{I}}(\boldsymbol{y};\boldsymbol{y}')
= \sum_{(i,j,k)\in{N}^{3\ne}_s} \left[ \boldsymbol{y}_{i:\,j\succ k}(
1-\boldsymbol{y}'_{i:\,j\succ k} ) +
\left(1-\boldsymbol{y}_{i:\,j\succ k}\right)
\boldsymbol{y}'_{i:\,j\succ k} \right],\] with promotion
statistic being simply \[
\Delta_{i,j}^\nearrow\boldsymbol{g}_{\text{I}}(\boldsymbol{y};\boldsymbol{y}')
= 2(\boldsymbol{y}'_{i:\,j^+\succ j}-\boldsymbol{y}'_{i:\,j\succ
j^+}).\] Optionally, the count can be weighted by the
weights argument, which can be either a 3D \(n\times n\times n\)-array whose \((i,j,k)\)th element gives the weight for
the comparison by \(i\) of \(j\) and \(k\) or a function taking three arguments,
\(i\), \(j\), and \(k\), and returning the weight of this
comparison. If wtcenter=TRUE, the calculated weights will
be centered around their mean. wtname can be used to label
this term.
rank.nodeicov(attrname, transform, transformname):
Attractiveness/Popularity covariates: Models the
effect of a nodal covariate on the propensity of an actor to be ranked
highly by the others. \[
\boldsymbol{g}_{\text{A}}(\boldsymbol{y};\boldsymbol{x}) =
\sum_{(i,j,k)\in {N}^{3\ne}} \boldsymbol{y}_{i:\,j\succ
k}(\boldsymbol{x}_j-\boldsymbol{x}_k).\] \[
\Delta_{i,j}^\nearrow\boldsymbol{g}_{\text{A}}(\boldsymbol{y};\boldsymbol{x})=
2(\boldsymbol{x}_{j}-\boldsymbol{x}_{j^+}), \] See the
?nodeicov ERGM term documentation for arguments.
rank.nonconformity(to, par):
Nonconformity: Measures the amount of
``nonconformity’’ in the network: configurations where an ego \(i\) ranks an alter \(j\) over another alter \(k\), but ego \(l\) ranks \(k\) over \(j\).
This statistic has an argument to, which controls to
whom an ego may conform:
"all" (the default) Nonconformity
to all egos is counted: \[
\boldsymbol{g}_{\text{GNC}}(\boldsymbol{y}) = \sum_{(i,j,k,l)\in
{N}^{4\ne}}\boldsymbol{y}_{l:\,j\succ
k}\left(1-\boldsymbol{y}_{i:\,j\succ k}\right) \] \[
\Delta_{i,j}^\nearrow\boldsymbol{g}_{\text{GNC}}(\boldsymbol{y}) =
2\sum_{l\in {N}\backslash\left\{i,j,j^+\right\}}(
\boldsymbol{y}_{l:\,j^+\succ j}-\boldsymbol{y}_{l:\,j\succ j^+} ).
\] A lower-than-chance value of this statistic and/or a negative
coefficient implies a degree of consensus in the network.
"localAND" (Local
nonconformity) Nonconformity of \(i\) to ego \(l\) regarding the relative ranking of \(j\) and \(k\) is only counted if \(i\) ranks \(l\) over both \(j\) and \(k\): \[\boldsymbol{g}_{\text{LNC}}(\boldsymbol{y}) =
\sum_{(i,j,k,l)\in {N}^{4\ne}} \boldsymbol{y}_{i:\,l\succ j}
\boldsymbol{y}_{i:\,l\succ k} \boldsymbol{y}_{l:\,j\succ k}
(1-\boldsymbol{y}_{i:\,j\succ k})\] \[
\begin{align*}
\Delta_{i,j}^\nearrow\boldsymbol{g}_{\text{LNC}}(\boldsymbol{y})=\sum_{k\in
{N}\backslash\left\{i,j,j^+\right\}}(& \boldsymbol{y}_{i:\,k\succ
j^+}\boldsymbol{y}_{k:\,j^+\succ j}-\boldsymbol{y}_{i:\,k\succ
j^+}\boldsymbol{y}_{k:\,j\succ j^+}\\
\vphantom{\sum_{k\in
{N}\backslash\left\{i,j,j^+\right\}}}&+\boldsymbol{y}_{k:\,i\succ
j^+}\boldsymbol{y}_{k:\,j^+\succ j}-\boldsymbol{y}_{k:\,i\succ
j}\boldsymbol{y}_{k:\,j\succ j^+}\\
\vphantom{\sum_{k\in
{N}\backslash\left\{i,j,j^+\right\}}}&+\boldsymbol{y}_{j:\,k\succ
j^+}\boldsymbol{y}_{i:\,j^+\succ k}-\boldsymbol{y}_{j^+:\,k\succ
j}\boldsymbol{y}_{i:\,j\succ k}).
\end{align*}
\] A lower-than-chance value of this statistic and/or a negative
coefficient implies a form of hierarchical transitivity in the
network.
Consider the Newcomb’s fraternity data:
data(newcomb)
as.matrix(newcomb[[1]], attrname = "rank")
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
## 1 0 7 12 11 10 4 13 14 15 16 3 9 1 5 8 6 2
## 2 8 0 16 1 11 12 2 14 10 13 15 6 7 9 5 3 4
## 3 13 10 0 7 8 11 9 15 6 5 2 1 16 12 4 14 3
## 4 13 1 15 0 14 4 3 16 12 7 6 9 8 11 10 5 2
## 5 14 10 11 7 0 16 12 4 5 6 2 3 13 15 8 9 1
## 6 7 13 11 3 15 0 10 2 4 16 14 5 1 12 9 8 6
## 7 15 4 11 3 16 8 0 6 9 10 5 2 14 12 13 7 1
## 8 9 8 16 7 10 1 14 0 11 3 2 5 4 15 12 13 6
## 9 6 16 8 14 13 11 4 15 0 7 1 2 9 5 12 10 3
## 10 2 16 9 14 11 4 3 10 7 0 15 8 12 13 1 6 5
## 11 12 7 4 8 6 14 9 16 3 13 0 2 10 15 11 5 1
## 12 15 11 2 6 5 14 7 13 10 4 3 0 16 8 9 12 1
## 13 1 15 16 7 4 2 12 14 13 8 6 11 0 10 3 9 5
## 14 14 5 8 6 13 9 2 16 1 3 12 7 15 0 4 11 10
## 15 16 9 4 8 1 13 11 12 6 2 3 5 10 15 0 14 7
## 16 8 11 15 3 13 16 14 12 1 9 2 6 10 7 5 0 4
## 17 9 15 10 2 4 11 5 12 3 7 8 1 6 16 14 13 0as.matrix(newcomb[[1]], attrname = "descrank")
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
## 1 0 10 5 6 7 13 4 3 2 1 14 8 16 12 9 11 15
## 2 9 0 1 16 6 5 15 3 7 4 2 11 10 8 12 14 13
## 3 4 7 0 10 9 6 8 2 11 12 15 16 1 5 13 3 14
## 4 4 16 2 0 3 13 14 1 5 10 11 8 9 6 7 12 15
## 5 3 7 6 10 0 1 5 13 12 11 15 14 4 2 9 8 16
## 6 10 4 6 14 2 0 7 15 13 1 3 12 16 5 8 9 11
## 7 2 13 6 14 1 9 0 11 8 7 12 15 3 5 4 10 16
## 8 8 9 1 10 7 16 3 0 6 14 15 12 13 2 5 4 11
## 9 11 1 9 3 4 6 13 2 0 10 16 15 8 12 5 7 14
## 10 15 1 8 3 6 13 14 7 10 0 2 9 5 4 16 11 12
## 11 5 10 13 9 11 3 8 1 14 4 0 15 7 2 6 12 16
## 12 2 6 15 11 12 3 10 4 7 13 14 0 1 9 8 5 16
## 13 16 2 1 10 13 15 5 3 4 9 11 6 0 7 14 8 12
## 14 3 12 9 11 4 8 15 1 16 14 5 10 2 0 13 6 7
## 15 1 8 13 9 16 4 6 5 11 15 14 12 7 2 0 3 10
## 16 9 6 2 14 4 1 3 5 16 8 15 11 7 10 12 0 13
## 17 8 2 7 15 13 6 12 5 14 10 9 16 11 1 3 4 0Let’s fit a model for the two types of nonconformity and deference at the first time point:
newc.fit1 <- ergm(newcomb[[1]] ~ rank.nonconformity + rank.nonconformity("localAND") +
rank.deference, response = "descrank", reference = ~CompleteOrder, control = control.ergm(MCMC.burnin = 4096,
MCMC.interval = 32, CD.conv.min.pval = 0.05), eval.loglik = FALSE)summary(newc.fit1)
## Call:
## ergm(formula = newcomb[[1]] ~ rank.nonconformity + rank.nonconformity("localAND") +
## rank.deference, response = "descrank", reference = ~CompleteOrder,
## eval.loglik = FALSE, control = control.ergm(MCMC.burnin = 4096,
## MCMC.interval = 32, CD.conv.min.pval = 0.05))
##
## Monte Carlo Maximum Likelihood Results:
##
## Estimate Std. Error MCMC % z value Pr(>|z|)
## nonconformity -0.003820 0.003360 0 -1.137 0.255611
## nonconformity.localAND -0.010306 0.009583 0 -1.075 0.282189
## deference -0.150513 0.039677 0 -3.793 0.000149 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Check diagnostics:
mcmc.diagnostics(newc.fit1)newc.fit15 <- ergm(newcomb[[15]] ~ rank.nonconformity + rank.nonconformity("localAND") +
rank.deference, response = "descrank", reference = ~CompleteOrder, control = control.ergm(MCMC.burnin = 4096,
MCMC.interval = 32, CD.conv.min.pval = 0.05), eval.loglik = FALSE)summary(newc.fit15)
## Call:
## ergm(formula = newcomb[[15]] ~ rank.nonconformity + rank.nonconformity("localAND") +
## rank.deference, response = "descrank", reference = ~CompleteOrder,
## eval.loglik = FALSE, control = control.ergm(MCMC.burnin = 4096,
## MCMC.interval = 32, CD.conv.min.pval = 0.05))
##
## Monte Carlo Maximum Likelihood Results:
##
## Estimate Std. Error MCMC % z value Pr(>|z|)
## nonconformity 0.0007951 0.0025908 0 0.307 0.759
## nonconformity.localAND -0.0429164 0.0080184 0 -5.352 <1e-04 ***
## deference -0.3438363 0.0747571 0 -4.599 <1e-04 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Check diagnostics:
mcmc.diagnostics(newc.fit15)