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<!doctype html>
<html>
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no">
<title>Symbolic Regression on Network Properties</title>
<link rel="stylesheet" href="css/reveal.css">
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</head>
<body>
<div class="reveal">
<div class="slides">
<section>
<div class="bbox">
<h3>Symbolic Regression on<br/>Network Properties</h3>
<hr class="shadow">
<p class="largefnt" style="text-align: center;">Marcus Märtens, Fernando Kuipers and Piet Van Mieghem</p>
<p class="mediumfnt" style="text-align: center;background-color: lightgrey;margin:10%;">Follow this presentation on your device: <a href="https://coolrunning.github.io/symbreg_pres">https://coolrunning.github.io/symbreg_pres</a></p>
<table style="margin-top:50px">
<tbody class="invis" style="width:100%;">
<tr>
<td>
<img width="200px" height="110px" data-src="./img/NAS_blue_200px.png" alt="NAS logo">
</td>
<td style="padding-right:100px">
Network<br/>Architectures and<br/>Services
</td>
<td>
<img width="394px" height="110px" data-src="./img/TUDLogo.png" alt="TU Delft logo">
</td>
</tr>
</tbody>
</table>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Network representations</h3></div>
<div class="sliderow">
<div class="leftcol" style="width:70%;text-align:left;padding-left:20px">
<ul>
<li>Number of nodes: $N$</li>
<li>Number of links: $L$</li>
</ul>
<p class="smallfnt">
Adjacency matrix $A =
\begin{bmatrix}
0 & 1 & 1 & 1 & 1 & 0 \\
1 & 0 & 0 & 0 & 1 & 1 \\
1 & 0 & 0 & 1 & 1 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 1 & 0 \\
\end{bmatrix}$
</p>
<p class="smallfnt">
Laplacian matrix $Q =
\begin{bmatrix}
4 & -1 & -1 & -1 & -1 & 0 \\
-1 & 3 & 0 & 0 & -1 & -1 \\
-1 & 0 & 3 & 1 & -1 & 0 \\
-1 & 0 & -1 & 2 & 0 & 0 \\
-1 & -1 & -1 & 0 & 4 & -1 \\
0 & -1 & 0 & 0 & -1 & 2 \\
\end{bmatrix}$
</p>
</div>
<div class="rightcol">
<img src="./img/billy.svg" width="400px" alt="a sample network">
<p class="smallfnt">In this example: $N = 6$ and $L = 9$.</p>
</div>
</div>
<div class="slidefooter">
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Triangles in networks</h3></div>
<div class="sliderow">
<div class="leftcol" style="width:55%">
<p class="mediumfnt">How to count the number of <b>node-disjoint</b> triangles?</p>
<pre style="font-size: 10pt"><code class="python" data-trim contenteditable>
def triangles(G, A):
triangles = 0
for n1, n2, n3 in zip(G.nodes(), G.nodes(), G.nodes()):
if (A[n1,n2] and A[n2,n3] and A[n3,n1]):
triangles += 1
return triangles / 6
</code></pre>
<p class="mediumfnt">Algorithms need exhaustive enumeration.</p>
</div>
<div class="rightcol">
<img src="./img/billy2.svg" width="400px" alt="a sample network">
<p class="smallfnt">In this example we have <b>4</b> triangles.</p>
</div>
</div>
<div class="slidefooter" style="background-color:lightgray;">
<p class="mediumfnt">Why are we interested in triangles anyway?</p>
<p class="smallfnt">
$$\text{Clustering coefficient }C = \frac{3 \times \text{number of triangles}}{\text{number of connected triplets of nodes}}$$</p>
<p class="mediumfnt">Important <b>metric</b> to classify networks.</p>
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Triangles in networks</h3></div>
<div class="sliderow">
<div class="leftcol" style="width:55%">
<p class="mediumfnt">Spectral decomposition of the Adjacency Matrix:</p>
<p class="smallfnt">$A = X
\begin{bmatrix}
\lambda_1 & & 0 \\
& \ddots & \\
0 & & \lambda_{N} \\
\end{bmatrix}
X^T
$
</p>
<p class="mediumfnt">$\lambda_1 \geq \ldots \geq \lambda_{N}$ are the <b>eigenvalues</b> of the network.</p>
<p class="smallfnt" style="color:gray">$\mu_1 \geq \ldots \geq \mu_{N}$ are the <b>Laplacian eigenvalues</b><br/>(We will need them later)</p>
</div>
<div class="rightcol">
<img src="./img/billy2.svg" width="400px" alt="a sample network">
<p class="smallfnt">In this example the eigenvalues are:<br/>$3.182, 1.247, -1.802,$<br/>$-1.588,-0.445, -0.594$</p>
</div>
</div>
<div class="slidefooter fragment fade-in" style="bottom:-30px">
<p class="mediumfnt">Computing the <b>number of triangles</b> $\blacktriangle$ becomes simple, once you know the eigenvalues.
</p>
<p class="largefnt" style="border: solid 3px red;margin-left:35%;margin-right:35%">
$\blacktriangle = \frac{1}{6} \sum\limits_{i = 1}^{N} \lambda_i^3$
</p>
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Network spectra</h3></div>
<div class="sliderow">
<div class="leftcol" style="border:#333 solid 1px;height:300px">
<p class="mediumfnt">Random network</p>
<p class="minor" style="position:absolute;top:30px;text-align:center;width:100%">Barbasi-Albert ($N$ = 5000)</p>
<img src="./img/g_ba.svg" width="330px" alt="a sample network" style="position:absolute;left:0px;bottom:0px;">
<img src="./img/ba-net.png" height="200px" alt="a sample network" style="position:absolute;left:240px;top:70px">
</div>
<div class="rightcol" style="border:#333 solid 1px;height:300px">
<p class="mediumfnt">Network of AS</p>
<p class="minor" style="position:absolute;top:30px;text-align:center;width:100%">tech-WHOIS, Mahadevan, P., Krioukov, D. et al. ($N$ = 7476)</p>
<img src="./img/tech-whois.svg" width="320px" alt="a sample network" style="position:absolute;left:0px;bottom:0px;">
<img src="./img/tech-whois-net.png" height="200px" alt="a sample network" style="position:absolute;left:220px;top:80px">
</div>
</div>
<div class="slidefooter" style="bottom:50px">
<p class="mediumfnt">The adjacency and the Laplacian spectra might not be as intuitive for humans as the graph representation,
but both contain the <b>complete information</b> about the topology of the network.</p>
</div>
</div>
</section>
<section data-background="./img/questionmark.svg" data-background-size="200px" data-background-position="bottom 20% left 50%" data-background-transition="zoom">
<div class="bbox">
<div style="position:absolute;top:100px">
<h3>Can we extract the hidden relations between the properties of a network and their corresponding spectra?</h3>
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Proof of concept: counting triangles</h3></div>
<div class="sliderow">
<div class="leftcol" style="width:100%">
<ul>
<li>Idea: use <a href="http://www.cartesiangp.co.uk" target="_blank"><b>Cartesian Genetic Programming (CGP)</b></a> to evolve symbolic expression!
<ol>
<li>generate all networks with $N = 6$ nodes</li>
<li>compute $\blacktriangle$ as <i>target</i></li>
<li>compute $\lambda_1, \ldots, \lambda_6$ as <i>features</i></li>
</ol>
<li class="fragment fade-in" style="padding-top:40px">Does it work?</li>
<ol>
<li class="fragment fade-in">allow operations $\{+,\cdot^3\}$, do not bother with the constant $\frac{1}{6}$</li>
<span class="fragment fade-in" style="display:inline-block;width:50%">$6\blacktriangle = \sum\limits_{i = 1}^{6} \lambda_i^3$ in 99%!</span><span class="greencomment fragment fade-in">Okay, that was too easy...</span>
<li class="fragment fade-in">allow operations $\{+,\times,\cdot^3\}$, give constant $\frac{1}{6}$ as input node</li>
<span class="fragment fade-in" style="display:inline-block;width:50%">$\blacktriangle = \frac{1}{6}\sum\limits_{i = 1}^{6} \lambda_i^3$ in 15%!</span><span class="greencomment fragment fade-in">Probably still expected?</span>
<li class="fragment fade-in">allow operations $\{+,\times,\div,\cdot^3\}$, only constants $\{1,2,3\}$ as input nodes</li>
<span class="fragment fade-in" style="display:inline-block;width:50%">$\blacktriangle = \frac{1}{6}\sum\limits_{i = 1}^{6} \lambda_i^3$ in 2%!</span><span class="greencomment fragment fade-in">Maybe this could really work!?</span>
</ol>
</ul>
</div>
</div>
</div>
</section>
<section data-background="./img/greencheck.svg" data-background-size="160px" data-background-position="bottom 30% left 50%" data-background-transition="zoom">
<div class="bbox">
<div style="position:absolute;top:100px">
<h3>Rediscover a known formula for a network property</h3>
</div>
</div>
</section>
<section data-background="./img/questionmark.svg" data-background-size="160px" data-background-position="bottom 30% left 50%" data-background-transition="zoom">
<div class="bbox">
<div style="position:absolute;top:100px">
<h3>Discover <b>an unknown formula</b> for a network property</h3>
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Network diameter</h3></div>
<div class="sliderow">
<div class="leftcol" style="width:55%">
<p class="mediumfnt">Length of the <b>longest of all shortest</b> paths in a network.</p>
<!--<p class="minor">The diameter of a disconnected network is $\infty$ but we will only consider connected networks here.</p>-->
<pre style="font-size: 10pt"><code class="python" data-trim contenteditable>
def diameter(G):
diam = 0
for src, dest in zip(G.nodes(), G.nodes()):
path = shortest_path(src, dest)
diam = max(len(path), diam)
return diam
</code></pre>
<p class="smallfnt">The exact formula for the <b>network diameter</b> $\rho$ implies this exhaustive enumeration:</p>
<p class="smallfnt" style="border: solid 3px blue;margin-left:5%;margin-right:5%;padding-top:0px;">$$\rho = \max\limits_{src, dest} \min dist(src, dest)$$</p>
</div>
<div class="rightcol">
<img src="./img/billy3.svg" width="400px" alt="a sample network">
<p class="smallfnt">In this example the diameter is <b>3</b>.</p>
</div>
</div>
<div class="slidefooter" style="bottom:0px;background-color: lightgray">
<p class="mediumfnt">Why are we interested in the diameter anyway?</p>
<p class="smallfnt">Diameter = <b>worst-case lower bound</b> on number of hops<br/>
i.e. limits the speed of spreading/flooding in a network</p>
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Symbolic regression for the diameter</h3></div>
<div class="sliderow">
<div class="leftcol" style="width:100%">
<ul>
<li>Idea: Try (systematically) different configurations with <b>CGP</b>!
<ol>
<li>generate all networks with $N = 6$ nodes</li>
<li>compute $\rho$ as the <i>target</i></li>
<li>compute $\lambda_1, \ldots, \lambda_6$ as <i>features</i></li>
<li>include $N, L$ and some basic constants $\{1,2,3\}$ as features as well</li>
<li>use different sets of operators like $\{+,-,\times,\cdot^2\}$ or $\{+,-,\times,\div,\sqrt{\cdot}\}$</li>
</ol>
</ul>
<p class="mediumfnt">Pick the expression that minimizes the <b>sum of absolute errors</b>:</p>
<p class="largefnt" style="border: solid 3px blue;margin-left:25%;margin-right:25%;padding-top:10px">$f(\hat{\rho}) = \sum\limits_{G} \mid \rho(G) - \widehat{\rho(G)} \mid$</p>
<ul>
<li class="fragment fade-in" style="padding-top:40px">Does it work?</li>
<span class="fragment fade-in" style="display:inline-block;width:50%">$\hat{\rho} = 2$</span><span class="greencomment fragment fade-in">What happened?</span>
</ul>
</div>
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Target distributions</h3></div>
<div class="sliderow">
<div class="leftcol" style="border:#333 solid 1px;height:370px">
<p class="largefnt">Triangles $\blacktriangle$</p>
<img src="./img/barchart_triangle.svg" width="100%" alt="">
</div>
<div class="rightcol" style="border:#333 solid 1px;height:370px">
<p class="largefnt">Diameter $\rho$</p>
<img src="./img/barchart_diameter.svg" width="100%" alt="">
</div>
</div>
<div class="slidefooter" style="bottom:0px">
<p class="mediumfnt">Exhaustive learning on all networks is no longer feasible.</p>
<p class="mediumfnt">Networks need to be <b>sampled</b> to create a good (diverse) learning set.</p>
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Sampling networks</h3></div>
<div class="sliderow">
<div class="leftcol" style="border:#333 solid 1px;height:370px">
<p class="largefnt">
Augmented path graphs</p>
<table>
<tbody class="invis" style="width:100%">
<tr>
<td style="padding-top:10px;padding-bottom:10px;text-align: center;">
<img src="./img/apath1.svg" alt="aug path" style="width:300px">
</td>
<td>$\rho = 11$</td>
</tr>
<tr>
<td style="padding-top:10px;padding-bottom:10px;text-align: center;">
<img src="./img/apath2.svg" alt="aug path" style="width:300px">
</td>
<td>$\rho = 10$</td>
</tr>
<tr>
<td style="padding-top:10px;padding-bottom:10px;text-align: center;">
<img src="./img/apath3.svg" alt="aug path" style="width:300px">
</td>
<td>$\rho = 8$</td>
</tr>
</tbody>
</table>
<ul style="position:absolute;bottom:0px;padding:20px">
<li>sparse networks</li>
<li>varies $\rho$ by keeping $N$ constant</li>
</ul>
</div>
<div class="rightcol" style="border:#333 solid 1px;height:370px">
<p class="largefnt">
Barbell graphs</p>
<table>
<tbody class="invis" style="width:100%">
<tr>
<td style="padding-top:3px;padding-bottom:3px;text-align: center;">
<img src="./img/barbell_network1.svg" alt="aug path" style="height:50px"></td>
<td style="padding-bottom:10px">$\rho = 6$</td>
</tr>
<tr>
<td style="padding-top:3px;padding-bottom:3px;text-align: center;">
<img src="./img/barbell_network2.svg" alt="aug path" style="height:50px"></td>
<td style="padding-bottom:10px">$\rho = 8$</td>
</tr>
<tr>
<td style="padding-top:3px;padding-bottom:3px;text-align: center;">
<img src="./img/barbell_network3.svg" alt="aug path" style="height:75px"></td>
<td style="padding-bottom:17px">$\rho = 8$</td>
</tr>
</tbody>
</table>
<ul style="position:absolute;bottom:0px;padding:20px">
<li>dense networks</li>
<li>varies $N$ by keeping $\rho$ constant</li>
</ul>
</div>
</div>
<div class="slidefooter" style="border:#333 solid 1px; bottom:-10px">
<p class="largefnt">Mixed graphs</p>
<p class="smallfnt">50% augmented path + 50% barbell</p>
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Computation pipeline</h3></div>
<div class="sliderow" style="height:500px">
<img src="./img/pipeline.svg" alt="computation pipeline">
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>CGP parametrization</h3></div>
<div class="sliderow">
<div class="leftcol" style="width:100%">
<table style="width:70%">
<thead>
<tr>
<th>parameter</th>
<th>value</th>
</tr>
</thead>
<tbody>
<tr>
<td>fitness function</td>
<td>sum of absolute errors</td>
</tr>
<tr>
<td>evolutionary strategy</td>
<td>1+4</td>
</tr>
<tr>
<td>mutation type and rate</td>
<td>probabilistic (0.1)</td>
</tr>
<tr>
<td>node layout</td>
<td>1 row<br/>
200 columns</td>
</tr>
<tr>
<td>levels-back</td>
<td>unrestricted</td>
</tr>
<tr>
<td>number of generations</td>
<td>200000</td>
</tr>
<tr>
<td>operators</td>
<td>$+,-,\times,\div,\cdot^2,\cdot^3,\sqrt{\cdot},\log$</td>
</tr>
<tr>
<td>constants</td>
<td>$1,2,3,4,5,6,7,8,9$</td>
</tr>
<tr>
<td>featuresets</td>
<td>A) $N,L,\lambda_1,\lambda_2,\lambda_3,\lambda_N$<br/>
B) $N,L,\mu_1,\mu_{N-1},\mu_{N-2},\mu_{N-3}$</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Automatically generated formulas</h3></div>
<div class="sliderow">
<div class="leftcol" style="width:100%">
<p class="mediumfnt" style="background-color:darkseagreen;padding:10px">
$\begin{equation}
\hat{\rho} = \frac{\log{\left (2 L \mu_{N-3} + 6 \right )} + 6}{\log{\left (L \mu_{N-3} + \sqrt{5} \sqrt{\frac{1}{\mu_{N-1}}} \right )}} + \sqrt{5} \sqrt{\frac{1}{\mu_{N-1}}} + 3 \sqrt{82} \sqrt{\frac{1}{729 L \mu_{N-2} \mu_{N-3} - 5}}
\end{equation}$</p>
<p class="mediumfnt" style="background-color: thistle;padding:10px">
$\begin{equation}
\hat{\rho} = N - \frac{1 - \frac{1}{\left(L - N\right)^{\frac{3}{2}}}}{\frac{6 - \frac{6}{\left(L - N\right)^{\frac{3}{2}}}}{\sqrt{L - N}} + 4 \sqrt{L - N}} - 2 \sqrt{L - N} - \frac{1}{\sqrt{L - N}}
\end{equation}$</p>
<p class="mediumfnt" style="background-color: lightsteelblue;padding:10px">
$\begin{equation}
\hat{\rho} = \sqrt{\sqrt{N} + \frac{45 \mu_{N-3}}{\left(\mu_{N-1} + \mu_{N-3}\right)^{2}} + \log{\left (\frac{216}{\left(\mu_{N-1} + \mu_{N-3}\right)^{2}} \right )} - \frac{16}{9 \mu_{N-3}} + \frac{8 \sqrt[4]{\mu_{N-3}}}{L \mu_{N-1} \mu_{N-2}}}
\end{equation}$</p>
</div>
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Bounds on the diameter</h3></div>
<div class="sliderow">
<p class="mediumfnt" style="background-color:darkseagreen;padding:10px">
$\begin{equation}
\hat{\rho} = \frac{\log{\left (2 L \mu_{N-3} + 6 \right )} + 6}{\log{\left (L \mu_{N-3} + \sqrt{5} \sqrt{\frac{1}{\mu_{N-1}}} \right )}} + \sqrt{5} \sqrt{\frac{1}{\mu_{N-1}}} + 3 \sqrt{82} \sqrt{\frac{1}{729 L \mu_{N-2} \mu_{N-3} - 5}}
\end{equation}$</p>
<div class="leftcol" style="width:69%;top:120px">
<p class="mediumfnt">Comparison to known analytical results from graph theory:</p>
<p class="smallfnt">Upper bound by Chung, F.R., Faber, V. and Manteuffel, T.A.: "An Upper bound on the diameter of a graph from eigenvalues associated with its Laplacian."</p>
<p class="minor">(See also Van Dam, E.R., Haemers, W.H.: "Eigenvalues and the diameter of graphs.")</p>
<p class="mediumfnt" style="border: solid 3px blue;margin-left:15%;margin-right:15%;padding:10px">
$\rho \leq \left\lfloor \frac{\cosh^{-1}(N-1)}{\cosh^{-1}\left(\frac{\mu_1 + \mu_{N-1}}{\mu_1 - \mu_{N-1}}\right)}\right\rfloor + 1$
</p>
</div>
<div class="rightcol" style="width:29%;top:120px">
<div style="color:#F00;font-size:18pt;position:absolute;right:85px;top:45px">$\hat{\rho}/\rho$</div>
<p class="mediumfnt">
<img src="./img/diameter-approx.svg" alt="This is how it looks">
</p>
</div>
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Diameter on real networks</h3></div>
<div class="sliderow">
<div class="leftcol" style="height:500px">
<table style="width:60%">
<thead>
<tr>
<th>parameter</th>
<th>value</th>
</tr>
</thead>
<tbody>
<tr>
<td>$N$</td>
<td>379</td>
</tr>
<tr>
<td>$L$</td>
<td>914</td>
</tr>
<tr>
<td>true diameter $\rho$</td>
<td>17</td>
</tr>
<tr style="background-color:lightcoral">
<td>approximated diameter $\hat{\rho}$</td>
<td>21.48</td>
</tr>
<tr style="background-color:lightsteelblue">
<td>upper bound</td>
<td>160.09</td>
</tr>
</tbody>
</table>
<img src="./img/ca-netscience_net.svg" style="height:220px;position:absolute;left:120px;bottom:40px;" alt="ca-netscience"/>
</div>
<div class="rightcol" style="height:500px">
<p class="mediumfnt">ca-netscience
<img src="./img/diamplot_ca-netscience.svg" style="height:400px;" alt="ca-netscience"/>
</p>
<p class="smallfnt" style="position:absolute;bottom:0px;right:40px;">source: <a href='http://www.networkrepository.com'>http://www.networkrepository.com</a></p>
</div>
<div class="slidefooter">
</div>
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Diameter on real networks</h3></div>
<div class="sliderow">
<div class="leftcol" style="height:500px">
<table style="width:60%">
<thead>
<tr>
<th>parameter</th>
<th>value</th>
</tr>
</thead>
<tbody>
<tr>
<td>$N$</td>
<td>4941</td>
</tr>
<tr>
<td>$L$</td>
<td>6594</td>
</tr>
<tr>
<td>true diameter $\rho$</td>
<td>46</td>
</tr>
<tr style="background-color:lightcoral">
<td>approximated diameter $\hat{\rho}$</td>
<td>97.52</td>
</tr>
<tr style="background-color:lightsteelblue">
<td>upper bound</td>
<td>749.49</td>
</tr>
</tbody>
</table>
<img src="./img/inf-power_net.svg" style="height:220px;position:absolute;left:120px;bottom:40px;" alt="ca-netscience"/>
</div>
<div class="rightcol" style="height:500px">
<p class="mediumfnt">inf-power
<img src="./img/diamplot_inf-power.svg" style="height:400px;" alt="ca-netscience"/>
</p>
<p class="smallfnt" style="position:absolute;bottom:0px;right:40px;">source: <a href='http://www.networkrepository.com'>http://www.networkrepository.com</a></p>
</div>
<div class="slidefooter">
</div>
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Diameter on real networks</h3></div>
<div class="sliderow">
<div class="leftcol" style="height:500px">
<table style="width:60%">
<thead>
<tr>
<th>parameter</th>
<th>value</th>
</tr>
</thead>
<tbody>
<tr>
<td>$N$</td>
<td>1458</td>
</tr>
<tr>
<td>$L$</td>
<td>1948</td>
</tr>
<tr>
<td>true diameter $\rho$</td>
<td>19</td>
</tr>
<tr style="background-color:lightcoral">
<td>approximated diameter $\hat{\rho}$</td>
<td>18.59</td>
</tr>
<tr style="background-color:lightsteelblue">
<td>upper bound</td>
<td>207.52</td>
</tr>
</tbody>
</table>
<img src="./img/bio-yeast_net.svg" style="height:220px;position:absolute;left:120px;bottom:40px;" alt="bio-yeast network"/>
</div>
<div class="rightcol" style="height:500px">
<p class="mediumfnt">bio-yeast
<img src="./img/diamplot_bio-yeast.svg" style="height:400px;" alt="bio-yeast"/>
</p>
<p class="smallfnt" style="position:absolute;bottom:0px;right:40px;">source: <a href='http://www.networkrepository.com'>http://www.networkrepository.com</a></p>
</div>
<div class="slidefooter">
</div>
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Diameter on real networks</h3></div>
<div class="sliderow">
<img src="./img/diamplotoverview.svg" style="height:520px" alt="Diameterplot overview"/>
</div>
</div>
</section>
<section data-background="./img/questionmark.svg" data-background-size="200px" data-background-position="bottom 20% left 50%" data-background-transition="zoom">
<div class="bbox">
<div style="position:absolute;top:100px">
<h3>Can we extract the hidden relations between the properties of a network and their corresponding spectra?</h3>
</div>
</div>
</section>
<section>
<div class="bbox">
<div class="slideheader"><h3>Closing thoughts</h3></div>
<div class="sliderow">
<div class="leftcol" style="width:100%">
<ul>
<li>Automatically evolving approximate equations for network properties is possible.</li>
<li>Equations are unbiased by human preconceptions.</li>
<li>Unexpected results may aid and stimulate research in spectral graph theory and network science in general.</li>
</ul>
<h4 style="padding-top:40px">Challenges and possible (?) improvements</h4>
<ul>
<li>Sampling network space for stronger diversity.</li>
<li>Make complexity of formula part of the objective function.</li>
<li>Automatically evolve constants rather than using them as inputs.</li>
<li>Apply more advanced feature-selection methods to increase quality of formulas.</li>
</ul>
</div>
<div class="rightcol">
</div>
</div>
<div class="slidefooter" style="background-color:lightgray">
</div>
</div>
</section>
</div>
</div>
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