Joint embedding of structure and features via graph convolutional networks
Sébastien Lerique, Jacobo LevyAbitbol, Márton Karsai
IXXI, École Normale Supérieure de Lyon
a walk through
What are "node embeddings"
DeepWalk, LINE, node2vec, ...
Preserve different properties:
 pairwise distances
 communities
 structural position
Easily answer questions like:
 who is similar to X
 who is X likely to connect to
Twitter users can...
... be tightly connected
... relate through similar interests
... write in similar styles
graph node2vec: \(d_n(u_i, u_j)\)
average user word2vec: \(d_w(u_i, u_j)\)
Questions

Create a taskindependent representation of network + features

What is the dependency between network structure and feature structure

Can we capture that dependency in a representation
network—feature dependencies
network—feature independence
Use deep learning to create embeddings
Introduction
Network embeddings → Twitter → networkfeature dependencies
Building blocks
Neural networks + Graph convolutions + Autoencoders
Capture dependencies
by arranging the building blocks together
Neural networks
x
y
green
red
\(H^{(l+1)} = \sigma(H^{(l)}W^{(l)})\)
\(H^{(0)} = X\)
\(H^{(L)} = Z\)
Inspired by colah's blog
Graphconvolutional neural networks
\(H^{(l+1)} = \sigma(H^{(l)}W^{(l)})\)
\(H^{(0)} = X\)
\(H^{(L)} = Z\)
\(H^{(l+1)} = \sigma(\color{DarkRed}{\tilde{D}^{\frac{1}{2}}\tilde{A}\tilde{D}^{\frac{1}{2}}}H^{(l)}W^{(l)})\)
\(H^{(0)} = X\)
\(H^{(L)} = Z\)
\(\color{DarkGreen}{\tilde{A} = A + I}\)
\(\color{DarkGreen}{\tilde{D}_{ii} = \sum_j \tilde{A}_{ij}}\)
Kipf & Welling (2016)
Semisupervised graphconvolution learning
Four wellmarked communities of size 10, small noise
More semisupervised GCN netflix
Overlapping communities of size 12, small noise
Two feature communities in a nearclique, small noise
Five wellmarked communities of size 20, moderate noise
(Variational) Autoencoders
From blog.keras.io
 Bottleneck compression → creates embeddings
 Flexible training objectives
 Free encoder/decoder architectures
high dimension
high dimension
low dimension
Example — autoencoding MNIST digits
MNIST Examples by Josef Steppan (CCBYSA 4.0)
60,000 training images
28x28 pixels
784 dims
784 dims
2D
From blog.keras.io
GCN + Variational autoencoders = 🎉💖🎉
Socioeconomic status
Language style
Topics
Socioeconomic status
Language style
Topics
Compressed & combined representation of nodes + network
Kipf & Welling (2016)
GCN+VAE learning
Five wellmarked communities of size 10, moderate label noise
Introduction
Network embeddings → Twitter → networkfeature dependencies
Building blocks
Neural networks + Graph convolutions + Autoencoders
Capture dependencies
by arranging the building blocks together
Example networkfeature dependencies
Features finer than network communities
Features coarser than network communities
Allocating network/feature embedding dims
Advantages of overlapping embeddings
No correlation
Full correlation
Full overlap
Medium overlap
No overlap
Overlapping model
Reference model
Joint embedding of structure and features via graph convolutional networks
By Sébastien Lerique
Joint embedding of structure and features via graph convolutional networks
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