ERGMs for Rank-Order Relational Data

References

Suppose that we reprsent ranking (or ordinal rating) of j by i by the value of y_i, j. What reference can we use for ranks?

help("ergm-references", "ergm.rank")

Terms

For details, see Krivitsky and Butts (2017). It’s not meaningful to

• compare ranks across different egos.
• take rank difference within an ego.

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 Δ_i, j^↗g(y) ≡ g(y^{i: j ⇄ j+}) − g(y).

Let N^k≠ 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: g_D(y) = ∑_{(i, j, l) ∈ N^3≠}y_{l: j ≻ i}y_{i: l ≻ j} Δ_i, j^↗g_D(y) = 2(y_{j⁺: i ≻ j} + y_{j: j⁺ ≻ 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: g_A(y; x) = ∑_{(i, j, k) ∈ N^3≠}y_{i: j ≻ k}(x_j − x_k). Δ_i, j^↗g_A(y; x) = 2(x_j − 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. g_I(y; y′) = ∑_{(i, j, k) ∈ Ns^3≠}[y_{i: j ≻ k}(1 − y′_{i: j ≻ k}) + (1 − y_{i: j ≻ k})y′_{i: j ≻ k}], with promotion statistic being simply Δ_i, j^↗g_I(y; y′) = 2(y′_{i: j⁺ ≻ j} − y′_{i: j ≻ j⁺}). Optionally, the count can be weighted by the weights argument, which can be either a 3D n × n × 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. g_A(y; x) = ∑_{(i, j, k) ∈ N^3≠}y_{i: j ≻ k}(x_j − x_k). Δ_i, j^↗g_A(y; x) = 2(x_j − 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: g_GNC(y) = ∑_{(i, j, k, l) ∈ N^4≠}y_{l: j ≻ k}(1 − y_{i: j ≻ k}) Δ_i, j^↗g_GNC(y) = 2∑_{l ∈ N ∖ {i, j, j⁺}}(y_{l: j⁺ ≻ j} − y_{l: j ≻ 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: g_LNC(y) = ∑_{(i, j, k, l) ∈ N^4≠}y_{i: l ≻ j}y_{i: l ≻ k}y_{l: j ≻ k}(1 − y_{i: j ≻ 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.

Example

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  0
as.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  0

Let’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)

## Warning: 'glpk' selected as the solver, but package 'Rglpk' is not available;
## falling back to 'lpSolveAPI'. This should be fine unless the sample size and/or
## the number of parameters is very big.

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          -0.003910   0.003159      0  -1.238    0.216    
## nonconformity.localAND -0.009680   0.009471      0  -1.022    0.307    
## deference              -0.152165   0.035963      0  -4.231   <1e-04 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Check 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           0.0007324  0.0025546      0   0.287    0.774    
## nonconformity.localAND -0.0430711  0.0074902      0  -5.750   <1e-04 ***
## deference              -0.3275382  0.0756276      0  -4.331   <1e-04 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Check diagnostics:

mcmc.diagnostics(newc.fit15)