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<!DOCTYPE html>
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Sebastian Mohr
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Spectral graph embedding
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Basics
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Main objective
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Solution
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Example
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<h1 id="Spectral-graph-embedding">
Spectral graph embedding
</h1>
<p>
In the following I want to take a short look into how to perform laplacian spectral graph embedding. We will start with the mathematical basics on graphs. If you don't care about the math and just want to see the code you can jump right into the example.
</p>
<h2 id="Basics">
Basics
</h2>
<p>
We want to consider a directed graph $G = (V,E)$ with in total $|V|=N$ vertexes. Edges are defined by an adjacency matrix $A$, with the matrix values $a_{i,j}>0$. In the most simple case the adjacency matrix only contains ones and zeros but you can also weight connections by increasing the corresponding matrix values.
</p>
<p>
A usefull property for graphs is the degree matrix $D$ which contains information about the degree of each node. In our directed graph this corresponds to the incomming connections.
$$
D :=
\begin{cases}
deg(v_{i}) & \text{if } i=j\\
0 & \text{else}
\end{cases}
$$
</p>
<p>
Whereby the $deg(v_i)$ is the number of times a edge terminates at vertex $v_i$. In our directed graph that is the number of incomming connections.
</p>
<p>
The Laplacian matrix is defined for directed graphs as
</p>
$$
L :=
\begin{cases}
deg(v_{i}) & \text{if } i=j\\
-1 & \text{if } i \neq j \text{ and } v_i \text{ is adjacent to } v_j \\
0 & \text{else}
\end{cases}
$$
<p>
In most cases one would use the symmetric normalized Laplacian matrix. It is normalized using the degree matrix.
</p>
$$
L_{\text{sym}} = D^{-1/2}LD^{-1/2} = I- D^{-1/2}AD^{-1/2}
$$
<h2 id="Main-objective">
Main objective
</h2>
<p>
Given a directed graph we want to compute a low dimensions representation of this graph. Thus we want want to create a set of vectors $ X := \{ x_1,...x_N\} \in \!R^2$ such that the larger the values of the adjacent vectors the smaller the distance $||x_i-x_j||^2$ between the resulting vertices. We consider the two dimensional case, but in theory the method should also work for a higher number of dimensions.
</p>
$$
\text{min}_{X \in \!R^2} \sum_{i,j}a_{i,j}||x_i-x_j||^2\\
=\text{min}_{X \in \!R^2} \text{ trace}(XLX^T)
$$
<p>
Trivially this is solved by $X=0$ but we are not searching for the singular solution, we avoid this by introducting further constrains to our problem. All solution vectors should be orthorgonal and the set should be normalized:
$$
XX^T = I \\
X\cdot\!1 = 0
$$
</p>
<h2 id="Solution">
Solution
</h2>
<p>
We can solve the above defined problem by the graph visualization spectral algorithm.
</p>
<ul>
<li>
Compute Laplacian matrix $L$
</li>
<li>
Compute lowest three eigenpairs $(e_1,\lambda_1),(e_2,\lambda_2) ,(e_3,\lambda_3)$ i.e. $Le_k=\lambda_ke_k$
</li>
<li>
Construct the embedding $X$ using the eigenvectors $e_2,e_3$
$$
X=\begin{bmatrix}e_2^T \\ e_3^T\end{bmatrix}
$$
</li>
</ul>
<p>
I feel like deriving these formulas would be beyond the scope of this short blog entry. If you want to read up on the math I recommend this
<a href="http://www.cs.yale.edu/homes/spielman/PAPERS/SGTChapter.pdf">
book
</a>
by Daniel Spielman.
</p>
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<h2 id="Example">
Example
</h2>
<p>
Next let us code up an example using the proposed algorithm.
</p>
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<pre><code class="language-python"># Normally you would create this adjacency matrix from your data
import numpy as np
A = np.zeros((14,14))
for i in range(13):
A[i,i+1] = 1
A[i+1,i] = 1
A[13,0] = 1
A[0,13] = 1
print(f"{A=}")
</code></pre>
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<pre><code class="language-python">A=array([[0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1.],
[1., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 1., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 1., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 1., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 1., 0., 1., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 1., 0., 1., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 1., 0., 1., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 1., 0., 1., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 1., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 1., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 1., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 1.],
[1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 0.]])
</code></pre>
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<pre><code class="language-python"># Compute the degree
deg = np.sum(A, axis=1)
D = np.diag(deg)
print(f"{D=}")
</code></pre>
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<pre><code class="language-python">D=array([[2., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 2., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 2., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 2., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 2., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 2., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 2., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 2., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 2., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 2., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 2., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 2., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 2., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 2.]])
</code></pre>
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<pre><code class="language-python"># Compute Laplacian matrix
L = D-A
print(f"L=array({np.round(L,1)})")
</code></pre>
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<pre><code class="language-python">L=array([[ 2. -1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -1.],
[-1. 2. -1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.],
[ 0. -1. 2. -1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.],
[ 0. 0. -1. 2. -1. 0. 0. 0. 0. 0. 0. 0. 0. 0.],
[ 0. 0. 0. -1. 2. -1. 0. 0. 0. 0. 0. 0. 0. 0.],
[ 0. 0. 0. 0. -1. 2. -1. 0. 0. 0. 0. 0. 0. 0.],
[ 0. 0. 0. 0. 0. -1. 2. -1. 0. 0. 0. 0. 0. 0.],
[ 0. 0. 0. 0. 0. 0. -1. 2. -1. 0. 0. 0. 0. 0.],
[ 0. 0. 0. 0. 0. 0. 0. -1. 2. -1. 0. 0. 0. 0.],
[ 0. 0. 0. 0. 0. 0. 0. 0. -1. 2. -1. 0. 0. 0.],
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. -1. 2. -1. 0. 0.],
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -1. 2. -1. 0.],
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -1. 2. -1.],
[-1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -1. 2.]])
</code></pre>
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<pre><code class="language-python"># Compute eigenpairs
eigenvals, eigenvectors = np.linalg.eigh(L)
# sort by eigenvalues
i = eigenvals.argsort()[::-1]
eigenvals = eigenvals[i]
eigenvectors = eigenvectors[i]
</code></pre>
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<pre><code class="language-python"># Plotting
import matplotlib.pyplot as plt
def plot(x,y,A):
# Remove axis and set figuresize
plt.subplots(figsize=(10, 10))
plt.axis('off')
# Plot edges
for i, b in enumerate(A):
for j, a in enumerate(b):
if a > 0:
x_s = [x[i],x[j]]
y_s = [y[i],y[j]]
plt.plot(x_s,y_s,color="tab:gray")
# Plot points
plt.scatter(x,y,marker="o",s=150,zorder=10,color="tab:green")
# Plot the coordinates for each point
plot(eigenvectors[:,1],eigenvectors[:,2],A)
</code></pre>
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