ERGMs for Rank-Order Relational Data

In ergm.rank: Fit, Simulate and Diagnose Exponential-Family Models for Rank-Order Relational Data

library(knitr)
opts_chunk$set(echo=TRUE,tidy=TRUE,error=FALSE,message=FALSE)

\def\y{\boldsymbol{y}} \def\Y{\boldsymbol{Y}} \def\covariate{\boldsymbol{x}} \def\weight{\boldsymbol{w}} \def\covariates{\mathbb{X}} \def\e{\boldsymbol{e}} \def\actors{{N}} \newcommand{\actorsnot}[1]{\actors\setsub\left{#1\right}} \newcommand{\distuples}[1]{\actors^{#1\ne}} \def\nactors{ n } \def\cnmap{\boldsymbol{\eta}} \def\linpred{\boldsymbol{\eta}} \def\linpar{\boldsymbol{\beta}} \def\sendeff{\boldsymbol{\delta}} \def\recveff{\boldsymbol{\gamma}} \def\Z{\boldsymbol{Z}} \def\nnatpar{ p } \def\latdim{ d } \def\curvpar{\boldsymbol{\theta}} \def\curvpars{\boldsymbol{\Theta}} \def\natcurvpars{\boldsymbol{\Theta}{\text{N}}} \def\designpar{\boldsymbol{\psi}} \def\ncurvpar{ q } \def\meanpar{\boldsymbol{\mu}} \def\genstatsymbol{g} \def\genstats{\boldsymbol{\genstatsymbol}} \newcommand{\genstat}[1]{\boldsymbol{\genstatsymbol}{\text{#1}}} \def\target{\boldsymbol{t}} \def\Design{\boldsymbol{D}} \def\design{\boldsymbol{d}} \def\dyadvals{\mathbb{S}} \def\maxdyadvals{ s } \def\Borel{ \mathfrak{B} }

\def\changeijv{\boldsymbol{\Delta}\sij} \newcommand{\promote}[2]{\Delta_{#1,#2}^\nearrow} \def\setsub{\backslash}

\newcommand{\EN}[3]{\left#1 #3 \right#2} \newcommand{\en}[3]{#1 #3 #2}

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\def\M{P} \def\h{h} \def\Mteg{\M_{\curvpar;\cnmap,\genstats}} \def\Mref{\M_\h} \def\Mtheg{\M_{\curvpar;\Mref,\cnmap,\genstats}}

\def\sigY{\mathsf{Y}}

\def\Pteg{\Prob_{\curvpar;\genstats}} \def\Pheg{\Prob_{\h,\genstats}} \def\Eteg{\E_{\curvpar;\genstats}} \def\fteg{f_{\curvpar;\genstats}} \DeclareMathOperator{\Odds}{Odds} \def\offset{\text{o}} \def\normc{\kappa} \def\ceg{\normc{\genstats}} \def\cegoff{\normc_{\genstats\offset,\cnmap,\genstats}} \def\cheg{\normc_{\h,\genstats}} \def\chegoff{\normc_{\h,\genstat\offset,\genstats}} \DeclareMathOperator{\argmax}{arg\,max} \DeclareMathOperator{\argmin}{arg\,min} \newcommand{\ilogit}{\text{logit}^{-1}} \def\reals{\mathbb{R}} \def\naturals{\mathbb{N}} \def\BB{\mathbb{B}} \def\ij{{i,j}} \def\ji{{j,i}} \def\pij{{(i,j)}} \def\pji{{(j,i)}} \def\ipjp{{i',j'}} \def\pipjp{{(i',j')}} \def\tij{\oplus\pij} \def\ijdysY{{\pij\in\dysY}} \def\ynetsY{{\y\in\netsY}} \def\ypnetsY{{\y'\in\netsY}} \def\sij{{i,j}} \def\sijk{{i,j,k}} \def\sik{{i,k}} \def\l{l} \def\sil{{i,\l}} \def\sji{{j,i}} \def\sli{{\l,i}} \def\slj{{\l,j}} \def\slk{{\l,k}} \def\sipjp{{i',j'}} \def\Yij{\Y!\sij} \def\yij{\y\sij} \def\Yji{\Y!\sji} \def\yji{\y\sji} \def\ytij{\y\tij} \def\Yyij{\Yij=\yij} \def\Yy{\Y=\y} \def\sobs{{\text{obs}}} \def\smis{_{\text{mis}}} \def\Yobs{\Y\sobs} \def\yobs{\y\sobs} \def\Ymis{\Y\smis} \def\ymis{\y\smis} \def\Yyobs{\Yobs=\yobs} \def\Yymis{\Ymis=\ymis} \def\half{\frac{1}{2}} \def\jplus{j^+} \makeatletter \newcommand{\myrel}[3][.3]{\binrel@{#3}% \binrel@@{\mathop{\kern\z@#3}\limits^{\vbox to #1\ex@{\kern-\tw@\ex@ \hbox{\scriptsize #2}\vss}}}} \makeatother \newcommand{\pref}[1][]{\myrel{\ensuremath{\,\,#1}}{\succ}} \newcommand{\npref}[1][]{\myrel[-0.2]{\ensuremath{#1}}{\nsucc}} \newcommand{\indiff}[1][]{\myrel{\ensuremath{#1}}{\cong}}

\newcommand{\yat}[1]{\y^{t#1}} \newcommand{\Yat}[1]{\Y^{t#1}} \newcommand{\Yyat}[1]{\Yat{#1}=\yat{#1}} \newcommand{\Yya}[1]{\Y^{#1}=y^{#1}}

\newcommand{\natpar}[1][]{\curvpar#1} \newcommand{\natparS}[1]{\curvpar^{#1}}

\newcommand{\myexp}[1]{\exp\mathchoice{\left(#1\right)}{(#1)}{(#1)}{(#1)}} \newcommand{\I}[1]{\mathbb{I}\left(#1\right)} \newcommand{\egopref}[4]{#1_{#2:\,#3\succ #4}} \newcommand{\ypref}[3]{\egopref{\y}{#1}{#2}{#3}} \newcommand{\yvpref}[3]{\egopref{\yv}{#1}{#2}{#3}} \newcommand{\egoswapr}[4]{#1^{#2:\,#3\rightleftarrows#4}}

\def\ipromotej{\promote{i}{j}} \newcommand{\promotev}[2]{\boldsymbol{\Delta}_{#1,#2}^\nearrow} \def\ipromotejv{\promotev{i}{j}}

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\providecommand{\abs}[1]{\left\lvert#1\right\rvert}

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\newcommand{\centercol}[1]{\multicolumn{1}{c}{#1}} \newcommand{\coef}[2]{$#1$ $(#2)$} \newcommand{\scoef}[2]{$\mathbf{#1}$ $(#2)$}

\def\Mct{\mu} \def\Mlbg{\lambda} \def\drefdct{\frac{d\Mref}{d\Mct}} \def\drefdlbg{\frac{d\Mref}{d\Mlbg}} \def\dtegdref{\frac{d\Mteg}{d\Mref}} \def\dtegdct{\frac{d\Mteg}{d\Mct}}

---

Coverage

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 @KrBu12e.

Modeling ordinal relational data using ergm.rank

library(ergm.rank)

Note that the implementations so far are very slow, so we will only do a short example.

References

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

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

Terms

For details, see @KrBu12e. 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 $$ \begin{equation} \ypref{i}{j}{k}\equiv\begin{cases} 1 & \text{if $j\stackrel{i}{\succ}k$ i.e., $i$ ranks $j$ above $k$;} \ 0 & \text{otherwise.} \end{cases} \end{equation} $$ We may interpret them using the promotion statistic $$\ipromotejv \genstats(\y)\equiv \genstats(\egoswapr{\y}{i}{j}{\jplus})-\genstats(\y).$$

Let $\distuples{k}$ 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 $i$: $$ \genstat{D}(\y) = \sum_{(i,j,\l)\in \distuples{3}} \ypref{\l}{j}{i}\ypref{i}{\l}{j} $$ $$ \ipromotej \genstat{D}(\y) = 2\en(){\ypref{\jplus}{i}{j}+\ypref{j}{\jplus}{i} - 1}. $$ 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: $$ \genstat{A}(\y;\covariate) = \sum_{(i,j,k)\in \distuples{3}} \ypref{i}{j}{k}(\covariate_j-\covariate_k).$$ $$ \ipromotej \genstat{A}(\y;\covariate)= 2(\covariate_{j}-\covariate_{\jplus}),$$ 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. $$\genstat{I}(\y;\y') = \sum_{(i,j,k)\in\distuples{3}_s} \left[ \ypref{i}{j}{k}\en(){1-\egopref{\y'}{i}{j}{k}} + \left(1-\ypref{i}{j}{k}\right) \egopref{\y'}{i}{j}{k} \right],$$ with promotion statistic being simply $$ \ipromotej \genstat{I}(\y;\y') = 2(\egopref{\y'}{i}{\jplus}{j}-\egopref{\y'}{i}{j}{\jplus}).$$ 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. $$ \genstat{A}(\y;\covariate) = \sum_{(i,j,k)\in \distuples{3}} \ypref{i}{j}{k}(\covariate_j-\covariate_k).$$ $$ \ipromotej \genstat{A}(\y;\covariate)= 2(\covariate_{j}-\covariate_{\jplus}), $$ 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: $$ \genstat{GNC}(\y) = \sum_{(i,j,k,\l)\in \distuples{4}}\ypref{\l}{j}{k}\left(1-\ypref{i}{j}{k}\right) $$ $$ \ipromotej \genstat{GNC}(\y) = 2\sum_{\l \in \actorsnot{i,j,\jplus}}\en(){\ypref{\l}{\jplus}{j}-\ypref{\l}{j}{\jplus}}. $$ 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 $k$: $$\genstat{LNC}(\y) = \sum_{(i,j,k,\l)\in \distuples{4}} \ypref{i}{\l}{j} \ypref{i}{\l}{k} \ypref{\l}{j}{k} (1-\ypref{i}{j}{k})$$ $$ \begin{align} \ipromotej \genstat{LNC}(\y)=\sum_{k\in \actorsnot{i,j,\jplus}}(& \ypref{i}{k}{\jplus}\ypref{k}{\jplus}{j}-\ypref{i}{k}{\jplus}\ypref{k}{j}{\jplus}\ \vphantom{\sum_{k\in \actorsnot{i,j,\jplus}}}&+\ypref{k}{i}{\jplus}\ypref{k}{\jplus}{j}-\ypref{k}{i}{j}\ypref{k}{j}{\jplus}\ \vphantom{\sum_{k\in \actorsnot{i,j,\jplus}}}&+\ypref{j}{k}{\jplus}\ypref{i}{\jplus}{k}-\ypref{\jplus}{k}{j}\ypref{i}{j}{k}). \end{align} $$ 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")
as.matrix(newcomb[[1]], attrname="descrank")

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)

summary(newc.fit1)

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)

Check diagnostics:

mcmc.diagnostics(newc.fit15)

References

ergm.rank documentation built on June 2, 2022, 1:06 a.m.