index.Rmd

---
title: "Small network statistics for the network science of teams\\footnote{Contact: \\url{vegayon@usc.edu}. We thank members of our MURI research team, USC's Center for Applied Network Analysis, Andrew Slaughter, and attendees of the NASN 2018 conference for their comments.}"
author:
  - George G. Vega Yon, MS
  - Kayla de la Haye, PhD
date: "NetSciX 2019, SCL \\linebreak[4]January 3, 2019"
output:
  beamer_presentation:
    slide_level: 2 # revealjs::revealjs_presentation
    highlight: espresso
    latex_engine: xelatex
    includes:
      in_header: notation-def.tex
aspectratio: 169
---

```{r setup, include=FALSE}
knitr::knit_hooks$set(smallsize = function(before, options, envir) {
    if (before) {
        "\\footnotesize\n\n"
    } else {
        "\n\\normalsize\n\n"
    }
})
knitr::opts_chunk$set(echo = TRUE, smallsize=TRUE)
```

## Funding Acknowledgement

\begincols
\begincol{.2\linewidth}

\includegraphics[width=.8\linewidth]{fig/ARO_logo.png}

\endcol

\begincol{.79\linewidth}
This material is based upon work support by, or in part by, the U.S. Army Research
Laboratory and the U.S. Army Research Office under grant number W911NF-15-1-0577
\endcol

\endcols


\begincols
\begincol{.79\linewidth}

Computation for the work described in this paper was supported by the University
of Southern California’s Center for High-Performance Computing (hpc.usc.edu).
\endcol

\begincol{.2\linewidth}
\includegraphics[width = 1\linewidth]{fig/usc.pdf}
\endcol
\endcols


\begin{figure}
\centering
\includegraphics[width = .85\linewidth]{fig/muriteams.png}
\end{figure}

## Context: A tale about social abilities and team performance

Recruited \pause

- 42 mixed gender groups of 3 to 5 participants (unknown)

- Eligibility: (1) 18+ years, (2) Native English speaker \pause

2 hour group session \pause

- Group tasks (2 sets of tasks x 30 minutes each)

- Measure group social networks and individual social intelligence  (SI) \pause

Study motivation \pause

- Overall, a very limited set of SI domains have been tested as predictors of social networks

- Very little research on the emergence of networks in teams.

## Context  (cont'd)

\begin{figure}
\centering
\includegraphics[width = .65\linewidth]{fig/plot-graph-4-1.pdf}
\end{figure}

\pause

How can we go beyond descriptive statistics? 

## Small networks and Exponential Random Graph Models

Exponential Random Graph Models: *What are the structures that give origin to a given observed graph?*

```{r ergm-terms, echo=FALSE, fig.align='center', out.width=".7\\linewidth"}
knitr::include_graphics("fig/ergm-terms.png")
```

(In general, ties are not IID, moreover, the entire graph is a single observation.)

## Small networks and Exponential Random Graph Models (Cont'd)

*   Estimating ERGMs is a complex problem. \pause The likelihood function has
    $2^{n(n-1)}$ terms, which means that for a graph of size 6 we have about
    1 billion terms to compute.\pause

*   Current approaches to estimate ERGMs rely on simulation methods, for example
    MCMC-MLE\pause

When trying to estimate ERGMs in little networks \pause

*   MCMC fails to converge when trying to estimate a block diagonal (structural
    zeros) model,\pause

*   Same happens when trying to estimate an ERGM for a single (little) graph, \pause

*   Even if it converges, model degeneracy, *i.e.* bad fit, arises too often.

    
## Rethinking the problem

\pause

*   1st Step: Forget about MCMC-MLE estimation, take advantage of small
    sample and use exact statistic for MLEs: \pause
    
    $$
    \Pr\left(\mathbf{Y}=\mathbf{y}|\theta, \mathcal{Y}\right) = \frac{\exp{\theta^{\mbox{T}}\mathbf{g}(\mathbf{y})}}{\kappa\left(\theta, \mathcal{Y}\right)},\quad\mathbf{y}\in\mathcal{Y}
    $$
    Where $\mathbf{g}(\mathbf{y})$ is a vector of sufficient statistics, $\theta \in \Theta$ a vector of model parameters, and $\kappa\left(\theta, \mathcal{Y}\right)$ is the normalizing constant (a summation with $2^{n(n-1)}$ terms)
    
*   This solves the problem of been able to estimate a small ergm. \pause

*   For this we started working on the `lergm` R package
    (available at https://github.com/muriteams/lergm):

## Example 1

Let's start by trying to estimate an ERGM for a single graph of size 4

```{r lergm1, echo=TRUE}
library(lergm)
set.seed(12)
x <- sna::rgraph(4)
lergm(x ~ edges + balance + mutual)
```


----------

*   Cool, we are able to estimate ERGMs for little networks! (we actually call
    them ~~lergms~~ ERGMitos[^barnett]),\pause
    
*   Going directly to MLE, we avoid the degeneracy problem.\pause

*   Moreover, due to the size of the networks, we can actually go further and
    estimate pooled ERGMs

[^barnett]: Thanks to George Barnett for suggesting the name!

## Solution

*   When estimating a block diagonal ERGM we were essentially assuming
    independence across networks.\pause

*   This means that we can actually do the same with exact statistics approach
    to calculate a joint likelihood:\pause
    
    $$
    \Pr\left(\mathbf{Y}=\{\mathbf{y}_{\color{cyan} i}\}|\theta, \left\{\mathcal{Y}_{\color{cyan}i}\right\}\right) = {\color{cyan} \prod_i} \frac{\exp{\theta^{\mbox{T}}\mathbf{g}(\mathbf{y}_{\color{cyan} i})}}{\kappa_{\color{cyan} i}\left(\theta, \mathcal{Y}_{\color{cyan}i}\right)}
    $$
    \pause
    
*   By estimating a pooled version of the ERGM we can increase the power of our
    MLEs.\pause
    
*   We implemented this in the `lergm` package
    
## Example 2

Suppose that we have 3 little graphs of sizes 4, 5, and 5:

```{r lergm2, echo=TRUE}
library(lergm)
set.seed(12)
x1 <- sna::rgraph(4)
x2 <- sna::rgraph(5)
x3 <- sna::rgraph(5)

lergm(list(x1, x2, x3) ~ edges + balance + mutual)
```

<!-- ## Convergence diagnostics -->

<!-- ```{r diagnostics2} -->
<!-- plot(ans) -->
<!-- ``` -->

## Simulation study

### Scenario A

1.  Draw parameters for _edges_ and _mutual_ from a uniform(-3, 3).

2.  Using those parameters, sampled $n\sim\mbox{Poisson}(30)$ networks of size 4

3.  Estimated the pooled ERGMs using both the MLE and the bootstrap version.

\pause

### Scenario B

1.  Idem.

2.  Using those parameters, sampled $n_1\sim\mbox{Poisson}(15), n_2\sim\mbox{Poisson}(15)$
    networks of size 3 and 4 respectively.

3.  Idem.

\pause

(If anyone asks, I just ran about 3 million ERGMs... :))

## Simulation study: Scenario A

### Empirical Bias

\begin{figure}
\centering
\includegraphics[width=.65\linewidth]{fig/bias-01-fixed-sizes-4.pdf}
\end{figure}

## Simulation study: Scenario A

### Power

\begin{figure}
\centering
\includegraphics[width=.65\linewidth]{fig/power-01-fixed-sizes-4.pdf}
\end{figure}


## Simulation study: Scenario B

### Empirical Bias

\begin{figure}
\centering
\includegraphics[width=.65\linewidth]{fig/bias-02-various-sizes-3-4.pdf}
\end{figure}

## Simulation study: Scenario B

### Power

\begin{figure}
\centering
\includegraphics[width=.65\linewidth]{fig/power-02-various-sizes-3-4.pdf}
\end{figure}

## Other approaches

\begin{figure}
\centering
\includegraphics[height=.7\textheight]{fig/similr.png}
\end{figure}

## Discussion

*   First set of results from the simulation study are encouraging\pause

*   Need to conduct more simulations using _nodal_ attributes and networks of size 5
    (right now having problems when building the DGP).\pause

*   Small structures imply a smaller pool of parameters (which is OK), but 
    can be more useful when including nodal attributes.\pause
    
*   When estimating the pooled version, we are essentially hand-waving the fact
    that parameter estimates implicitly encode size of the graph, i.e.
    
    > Does a the estimate of `edge = 0.1` has the same meaning for a network of
    size 3 to a size 5? (but perhaps is not such a big deal)

    \pause

*   Finally, this work can be extended to other types of small networks,
    including: families, ego-nets, etc. And other methods, such as TERGMs.
    
##

### Thank you!

\maketitle

## What have we got so far?

\footnotesize

```r
lergm(networks ~ mutual + edges + triangle + nodematch("male") + 
    diff("Empathy") + nodematch("nonwhite"))
```


```{r muri-pre, cache=TRUE, echo=FALSE}
ans       <- confint(readRDS("muri_estimates.rds"))
ans_less3 <- confint(readRDS("muri_estimates_less3.rds"))

library(magrittr)
# kableExtra::usepackage_latex("booktabs")
knitr::kable(
  x        = cbind(ans, ans_less3),
  digits   = 2,
  format   = "latex",
  booktabs = TRUE,
  caption  = "Preliminary results with our small teams data. The table shows 95\\% confidence intervals for the parameter estimates using the pooled ERGM model." 
  ) %>%
  kableExtra::add_header_above(c(" " = 1, "All (42)" = 2, "All but size 3 (35)" = 2)) %>%
  kableExtra::row_spec(c(2, 3, 5), background = "lightgray")
```