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<!DOCTYPE html>
<html>
<head>
<title>Exploring the latent space of LSMs</title>
<meta charset="utf-8">
<meta name="description" content="Exploring the latent space of LSMs">
<meta name="author" content="Kenny Joseph">
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<body style="opacity: 0">
<slides class="layout-widescreen">
<!-- LOGO SLIDE -->
<slide class="title-slide segue nobackground">
<hgroup class="auto-fadein">
<h1>Exploring the latent space of LSMs</h1>
<h2></h2>
<p>Kenny Joseph<br/></p>
</hgroup>
<article></article>
</slide>
<!-- SLIDES -->
<slide class="" id="slide-1" style="background:;">
<hgroup>
<h2>Introduction</h2>
</hgroup>
<article data-timings="">
<p class="centered"> <img src="figure/kriv_2009.png"> </p>
<ul>
<li>With no node-level covariates, the latent space portion of an LSM is just a low-dimensional representation of the network
<ul>
<li>appropriate as a layout algorithm or for uncovering "clusters" (<a href="http://dx.doi.org/10.1016/j.socnet.2009.04.001">Krivitsky et al. 2009</a>).</li>
</ul></li>
</ul>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-2" style="background:;">
<hgroup>
<h2>LSM with node covariates</h2>
</hgroup>
<article data-timings="">
<ul>
<li>When node-covariates are of interest, latent space is a "random effect"
<ul>
<li>influence of node covariates on the network are interpreted as an effect on dyadic formation</li>
</ul></li>
<li>The latent space represents both unobserved node-level covariates and unexplained network effects.</li>
<li>Statistically, LSM are appealing- push out increasingly unexplainable effects to the latent social space</li>
<li>But what is the qualitative meaning of the latent space in this case?
<ul>
<li>Particularly important as LSMs are sold as a way to visualize data</li>
</ul></li>
</ul>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-3" style="background:;">
<hgroup>
<h2>Research questions</h2>
</hgroup>
<article data-timings="">
<ul>
<li><p>how does the latent space inform us of latent dyadic and network-level effects that drove the formation of the network?</p></li>
<li><p>when no effects exist outside observed node-level covariates, what does the latent space look like?</p></li>
<li><p>how do these two things change with different parameterizations of the model?</p></li>
</ul>
<hr>
<h2>Overview of approach</h2>
<ul>
<li><p>Generate some networks </p></li>
<li><p>Run LSM on them with different parameterizations</p></li>
<li><p>Visualize the latent space, try and grasp qualitative meaning</p></li>
<li><p>Check the latent space for (non) uniformity where we expect it (semi-qualitatively)</p></li>
</ul>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-4" style="background:;">
<hgroup>
<h2>LSM formulation in latentnet</h2>
</hgroup>
<article data-timings="">
<p>\[\begin{align*}
\text{logit}\, P[Y_{ij} = 1 ] &= \beta_k^TX_{ijk} - |Z_i-Z_j|, k = 1...\color{red}K \\
\beta_k &\overset{\text{iid}}{\sim} \text{N}(\xi_k,\psi^2_k) \\
Z_i &\overset{\text{iid}}{\sim} \sum_{g=1}^{\color{red}G} \lambda_g\text{MVN}_d(\mu_g,\sigma_g^2I_d) \\
\mu_g &\overset{\text{iid}}{\sim} \text{MVN}_d(0,\omega^2I_d) \\
\sigma_g^2 &\overset{\text{iid}}{\sim} \sigma_0^2\text{Inv}\chi_\alpha^2 \\
(\lambda_1,...,\lambda_g) &\overset{\text{iid}}{\sim} \text{Dirichlet}(\nu_1,...,\nu_g)\end{align*}\]</p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-5" style="background:;">
<hgroup>
<h2>Generating the simulated networks</h2>
</hgroup>
<article data-timings="">
<ul>
<li><p>Overview</p>
<ul>
<li>Calculate a similarity matrix w/ a baseline tie probability</li>
<li>Augment these probabilities with shared group information</li>
<li>Draw a random network from the similarity matrix</li>
</ul></li>
<li><p>Some default parameters:</p></li>
</ul>
<pre><code class="r">N_ACTORS <- 8
OUTGROUP_TIE <- 0.01
INGROUP_TIE <- 1
N_COVARIATES <- 1
N_GROUPS <- 2
</code></pre>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-6" style="background:;">
<hgroup>
<h2>Start with a similarity matrix</h2>
</hgroup>
<article data-timings="">
<pre><code class="r">similarity_matrix <- matrix(0, nrow = N_ACTORS, ncol = N_ACTORS)
similarity_matrix[upper.tri(similarity_matrix)] <- runif(N_ACTORS * (N_ACTORS -
1)/2, 0, OUTGROUP_TIE * 2)
pandoc.table(similarity_matrix, style = "rmarkdown", digits = 2)
</code></pre>
<table><thead>
<tr>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
</tr>
</thead><tbody>
<tr>
<td align="center">0</td>
<td align="center">0.00096</td>
<td align="center">0.0058</td>
<td align="center">0.0099</td>
<td align="center">0.018</td>
<td align="center">0.0097</td>
<td align="center">0.002</td>
<td align="center">0.018</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0.0091</td>
<td align="center">0.016</td>
<td align="center">0.016</td>
<td align="center">0.019</td>
<td align="center">0.011</td>
<td align="center">0.0098</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0.011</td>
<td align="center">0.015</td>
<td align="center">0.0054</td>
<td align="center">0.019</td>
<td align="center">0.018</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0.0069</td>
<td align="center">0.014</td>
<td align="center">0.011</td>
<td align="center">0.0049</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0.00052</td>
<td align="center">0.017</td>
<td align="center">0.008</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0.018</td>
<td align="center">0.018</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0.013</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
</tr>
</tbody></table>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-7" style="background:;">
<hgroup>
<h2>Generate (random) groupings</h2>
</hgroup>
<article data-timings="">
<pre><code class="r">## Get 'random'' groupings
groupings <- data.frame(id = 1:N_ACTORS, Group1 = c(rep(1, N_ACTORS/2), rep(2,
N_ACTORS/2)))
pandoc.table(groupings, style = "rmarkdown", digits = 2)
</code></pre>
<table><thead>
<tr>
<th align="center">id</th>
<th align="center">Group1</th>
</tr>
</thead><tbody>
<tr>
<td align="center">1</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">2</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">3</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">4</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">5</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center">6</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center">7</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center">8</td>
<td align="center">2</td>
</tr>
</tbody></table>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-8" style="background:;">
<hgroup>
<h2>Calculate percentage of group memberships shared</h2>
</hgroup>
<article data-timings="">
<pre><code class="r">## Determine matrix of co-memberships in groups, normalized by number of
## groups
percent_shared_memberships <- ifelse(outer(groupings[, 2], groupings[, 2], FUN = "-") ==
0, 1, 0)
pandoc.table(percent_shared_memberships, style = "rmarkdown", digits = 2)
</code></pre>
<table><thead>
<tr>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
</tr>
</thead><tbody>
<tr>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
</tr>
<tr>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
</tr>
<tr>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
</tr>
<tr>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
</tr>
</tbody></table>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-9" style="background:;">
<hgroup>
<h2>Reconfigure similarity matrix w/ group info</h2>
</hgroup>
<article data-timings="">
<pre><code class="r">## Add to si
similarity_matrix <- similarity_matrix + INGROUP_TIE * percent_shared_memberships
diag(similarity_matrix) <- 0
similarity_matrix[lower.tri(similarity_matrix)] <- 0
similarity_matrix <- ifelse(similarity_matrix > 1, 1, similarity_matrix)
pandoc.table(similarity_matrix, style = "rmarkdown", digits = 2)
</code></pre>
<table><thead>
<tr>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
</tr>
</thead><tbody>
<tr>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">0.018</td>
<td align="center">0.0097</td>
<td align="center">0.002</td>
<td align="center">0.018</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">0.016</td>
<td align="center">0.019</td>
<td align="center">0.011</td>
<td align="center">0.0098</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">0.015</td>
<td align="center">0.0054</td>
<td align="center">0.019</td>
<td align="center">0.018</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0.0069</td>
<td align="center">0.014</td>
<td align="center">0.011</td>
<td align="center">0.0049</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
</tr>
</tbody></table>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-10" style="background:;">
<hgroup>
<h2>Draw a random network from similarity matrix</h2>
</hgroup>
<article data-timings="">
<pre><code class="r">random_draw <- matrix(0, nrow = N_ACTORS, ncol = N_ACTORS)
random_draw[upper.tri(random_draw)] <- rbinom(rep(1, N_ACTORS * (N_ACTORS -
1)/2), 1, as.vector(similarity_matrix[upper.tri(similarity_matrix)]))
lower_indices <- lower.tri(random_draw)
random_draw[lower_indices] <- t(random_draw)[lower_indices]
pandoc.table(random_draw, style = "rmarkdown", digits = 2)
</code></pre>
<table><thead>
<tr>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
<th align="center"></th>
</tr>
</thead><tbody>
<tr>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
</tr>
<tr>
<td align="center">1</td>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
</tr>
<tr>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
</tr>
<tr>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">0</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">0</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">1</td>
<td align="center">0</td>
</tr>
</tbody></table>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-11" style="background:;">
<hgroup>
<h2>Finally, get the network</h2>
</hgroup>
<article data-timings="">
<p><img src="figure/network.png" title="plot of chunk network" alt="plot of chunk network" style="display: block; margin: auto;" /></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-12" style="background:;">
<hgroup>
<h2>Results</h2>
</hgroup>
<article data-timings="">
<ul>
<li>CIDnetworks vs latentnet packages</li>
<li>Visual comparison of latent spaces</li>
<li>Check Ripley's L, a common spatial statistic, for uniformity in the latent space where we (don't) expect it
\[\begin{align*}
\hat{K}(t) &= \lambda^{-1}\sum_{i \neq j}\frac{I(d_{ij} \, < \, t )}{n} \\
\hat{L}(t) &= (\frac{\hat{K}}{\pi}) \end{align*}\]
<ul>
<li>Do these latter two for different \(NG\), \(K\) and \(G\)</li>
</ul></li>
</ul>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-13" style="background:;">
<hgroup>
<h2>Initial Network Settings</h2>
</hgroup>
<article data-timings="">
<pre><code class="r">N_ACTORS <- 50
OUTGROUP_TIE <- 0.03
N_COVARIATES <- 1
N_GROUPS <- 2
INGROUP_TIE <- 0.9
N_GAUSSIANS <- 1
</code></pre>
<p><img src="figure/unnamed-chunk-9.png" title="plot of chunk unnamed-chunk-9" alt="plot of chunk unnamed-chunk-9" style="display: block; margin: auto;" /></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-14" style="background:;">
<h2>CIDnetworks vs ERGMM</h2>
<p><img src="figure/unnamed-chunk-11.png" title="plot of chunk unnamed-chunk-11" alt="plot of chunk unnamed-chunk-11" style="display: block; margin: auto;" /></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-15" style="background:;">
<hgroup>
<h2>Partialling out node covariates</h2>
</hgroup>
<article data-timings="">
<pre><code class="r">N_COVARIATES <- 3
</code></pre>
<p><img src="figure/unnamed-chunk-14.png" title="plot of chunk unnamed-chunk-14" alt="plot of chunk unnamed-chunk-14" style="display: block; margin: auto;" /></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-16" style="background:;">
<hgroup>
<h2>Modifying the Number of Gaussians</h2>
</hgroup>
<article data-timings="">
<pre><code class="r">N_GAUSSIANS <- 3
</code></pre>
<p><img src="figure/unnamed-chunk-17.png" title="plot of chunk unnamed-chunk-17" alt="plot of chunk unnamed-chunk-17" style="display: block; margin: auto;" /></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-17" style="background:;">
<hgroup>
<h2>With more groups</h2>
</hgroup>
<article data-timings="">
<pre><code class="r">N_COVARIATES <- 2
N_GROUPS <- 4
N_GAUSSIANS <- 1
</code></pre>
<p><img src="figure/unnamed-chunk-20.png" title="plot of chunk unnamed-chunk-20" alt="plot of chunk unnamed-chunk-20" style="display: block; margin: auto;" /></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-18" style="background:;">
<hgroup>
<h2>Modifying the Number of Gaussians (again)</h2>
</hgroup>
<article data-timings="">
<pre><code class="r">N_GAUSSIANS <- 2
</code></pre>
<p><img src="figure/unnamed-chunk-23.png" title="plot of chunk unnamed-chunk-23" alt="plot of chunk unnamed-chunk-23" style="display: block; margin: auto;" /></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-19" style="background:;">
<hgroup>
<h2>Ripley's L - Uniform data, 2D-Gaussian data</h2>
</hgroup>
<article data-timings="">
<p><img src="figure/unnamed-chunk-25.png" title="plot of chunk unnamed-chunk-25" alt="plot of chunk unnamed-chunk-25" style="display: block; margin: auto;" /></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-20" style="background:;">
<hgroup>
<h2>Ripley's L - Partialling data</h2>
</hgroup>
<article data-timings="">
<p><img src="figure/unnamed-chunk-26.png" title="plot of chunk unnamed-chunk-26" alt="plot of chunk unnamed-chunk-26" style="display: block; margin: auto;" /></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-21" style="background:;">
<hgroup>
<h2>Ripley's L - W/ ++ Gaussians</h2>
</hgroup>
<article data-timings="">
<p><img src="figure/unnamed-chunk-27.png" title="plot of chunk unnamed-chunk-27" alt="plot of chunk unnamed-chunk-27" style="display: block; margin: auto;" /></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-22" style="background:;">
<hgroup>
<h2>Ripley's L - More Groups</h2>
</hgroup>
<article data-timings="">
<p><img src="figure/unnamed-chunk-28.png" title="plot of chunk unnamed-chunk-28" alt="plot of chunk unnamed-chunk-28" style="display: block; margin: auto;" /></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-23" style="background:;">
<hgroup>
<h2>Ripley's L - More Groups, ++ Gaussians</h2>
</hgroup>
<article data-timings="">
<p><img src="figure/unnamed-chunk-29.png" title="plot of chunk unnamed-chunk-29" alt="plot of chunk unnamed-chunk-29" style="display: block; margin: auto;" /></p>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-25" style="background:;">
<hgroup>
<h2>Ripley's L - Low tie density </h2>
</hgroup>
<article data-timings="">
<p> <img src="figure/weird_low_tie_prob.png" style="display: block; margin: auto;" /></p>
</article>
</slide>
<slide class="" id="slide-25" style="background:;">
<hgroup>
<h2>Ripley's L - Low tie density - cont.</h2>
</hgroup>
<article data-timings="">
<p> <img src="figure/weird_low_tie_prob_2.png" style="display: block; margin: auto;" /></p>
</article>
</slide>
<slide class="" id="slide-24" style="background:;">
<hgroup>
<h2>Next Steps - Experimental Design</h2>
</hgroup>
<article data-timings="">
<table><thead>
<tr>
<th align="center">Parameter Name</th>
<th align="center">Description</th>
<th align="center">Conditions</th>
</tr>
</thead><tbody>
<tr>
<td align="center">\(N\)</td>
<td align="center">Number of actors</td>
<td align="center">50</td>
</tr>
<tr>
<td align="center">\(d_{out}\)</td>
<td align="center">Baseline tie density</td>
<td align="center">.01, .1</td>
</tr>
<tr>
<td align="center">\(d_{in}\)</td>
<td align="center">Within group tie probability</td>
<td align="center">.3, .6</td>
</tr>
<tr>
<td align="center">\(K\)</td>
<td align="center">Number of covariates in model</td>
<td align="center">0, 1, 3</td>
</tr>
<tr>
<td align="center">\(NG\)</td>
<td align="center">Number of groups per covariate</td>
<td align="center">2, 4</td>
</tr>
<tr>
<td align="center">\(G\)</td>
<td align="center">Number of Gaussians</td>
<td align="center">1, 2, 3</td>
</tr>
<tr>
<td align="center">\(\sigma_0\)</td>
<td align="center">Covariance of Gaussians</td>
<td align="center">.125, ?</td>
</tr>
<tr>
<td align="center">...</td>
<td align="center">Graph structure</td>
<td align="center">Core-periphary</td>
</tr>
</tbody></table>
<ul>
<li>Run an LSM on each combination of these parameters</li>
<li>Get some kind of point estimate of \(\hat{L}(t)\)</li>
</ul>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-25" style="background:;">
<hgroup>
<h2>Conclusion</h2>
</hgroup>
<article data-timings="">
<ul>
<li>"Findings"
<ul>
<li>The latent space does give some inclination as to dyadic covariates missing from the model</li>
<li>When the latent space is representing only noise, it appears close to a random process along Ripley's L</li>
<li>Neither of these things appear to change much with a "mis-parameterized" LSM</li>
</ul></li>
<li>None of this is particularly surprising. But...
<ul>
<li>In playing around, I noticed that as OUTGROUP_TIE decreased, these results deteriorate</li>
<li>Thus, it would be interesting to work with more realistic networks and see if these results hold</li>
</ul></li>
</ul>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="" id="slide-26" style="background:;">
<hgroup>
<h2>Bibliography</h2>
</hgroup>
<article data-timings="">
<ul>
<li>Pavel Krivitsky, Mark Handcock, Adrian Raftery, Peter Hoff, (2009) Representing degree distributions, clustering, and homophily in social networks with latent cluster random effects models. <em>Social Networks</em> <strong>31</strong> (3) 204-213 <a href="http://dx.doi.org/10.1016/j.socnet.2009.04.001">10.1016/j.socnet.2009.04.001</a></li>
</ul>
</article>
<!-- Presenter Notes -->
</slide>
<slide class="backdrop"></slide>
</slides>
<div class="pagination pagination-small" id='io2012-ptoc' style="display:none;">
<ul>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=1 title='Introduction'>
1
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=2 title='LSM with node covariates'>
2
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=3 title='Research questions'>
3
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=4 title='LSM formulation in latentnet'>
4
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=5 title='Generating the simulated networks'>
5
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=6 title='Start with a similarity matrix'>
6
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=7 title='Generate (random) groupings'>
7
</a>
</li>
<li>
<a href="#" target="_self" rel='tooltip'
data-slide=8 title='Calculate percentage of group memberships shared'>
8