Understanding Convolutions on Graphs

This article is one of two Distill publications about graph neural networks.
Take a look at
A Gentle Introduction to Graph Neural Networks

for a companion view on many things graph and neural network related.

Many systems and interactions – social networks, molecules, organizations, citations, physical models, transactions – can be represented quite naturally as graphs.
How can we reason about and make predictions within these systems?

One idea is to look at tools that have worked well in other domains: neural networks have shown immense predictive power in a variety of learning tasks.
However, neural networks have been traditionally used to operate on fixed-size and/or regular-structured inputs (such as sentences, images and video).
This makes them unable to elegantly process graph-structured data.

Graph neural networks (GNNs) are a family of neural networks that can operate naturally on graph-structured data.
By extracting and utilizing features from the underlying graph,
GNNs can make more informed predictions about entities in these interactions,
as compared to models that consider individual entities in isolation.

GNNs are not the only tools available to model graph-structured data:
graph kernels
and random-walk methods
were some of the most popular ones.
Today, however, GNNs have largely replaced these techniques
because of their inherent flexibility to model the underlying systems
better.

In this article, we will illustrate
the challenges of computing over graphs,
describe the origin and design of graph neural networks,
and explore the most popular GNN variants in recent times.
Particularly, we will see that many of these variants
are composed of similar building blocks.

First, let’s discuss some of the complications that graphs come with.

The Challenges of Computation on Graphs

Lack of Consistent Structure

Graphs are extremely flexible mathematical models; but this means they lack consistent structure across instances.
Consider the task of predicting whether a given chemical molecule is toxic  :

Looking at a few examples, the following issues quickly become apparent:

• Molecules may have different numbers of atoms.
• The atoms in a molecule may be of different types.
• Each of these atoms may have different number of connections.
• These connections can have different strengths.

Representing graphs in a format that can be computed over is non-trivial,
and the final representation chosen often depends significantly on the actual problem.

Node-Order Equivariance

Extending the point above: graphs often have no inherent ordering present amongst the nodes.
Compare this to images, where every pixel is uniquely determined by its absolute position within the image!

As a result, we would like our algorithms to be node-order equivariant:
they should not depend on the ordering of the nodes of the graph.
If we permute the nodes in some way, the resulting representations of
the nodes as computed by our algorithms should also be permuted in the same way.

Scalability

Graphs can be really large! Think about social networks like Facebook and Twitter, which have over a billion users.
Operating on data this large is not easy.

Luckily, most naturally occuring graphs are ‘sparse’:
they tend to have their number of edges linear in their number of vertices.
We will see that this allows the use of clever methods
to efficiently compute representations of nodes within the graph.
Further, the methods that we look at here will have significantly fewer parameters
in comparison to the size of the graphs they operate on.

Problem Setting and Notation

There are many useful problems that can be formulated over graphs:

• Node Classification: Classifying individual nodes.
• Graph Classification: Classifying entire graphs.
• Node Clustering: Grouping together similar nodes based on connectivity.
• Link Prediction: Predicting missing links.
• Influence Maximization: Identifying influential nodes.

A common precursor in solving many of these problems is node representation learning:
learning to map individual nodes to fixed-size real-valued vectors (called ‘representations’ or ‘embeddings’).

In Learning GNN Parameters, we will see how the learnt embeddings can be used for these tasks.

Different GNN variants are distinguished by the way these representations are computed.
Generally, however, GNNs compute node representations in an iterative process.
We will use the notation $h_v^{(k)}$

We will define a graph $G$ as a set of nodes, $V$, with a set of edges $E$ connecting them.
Nodes can have individual features as part of the input: we will denote by $x_v$

For ease of exposition, we will assume $G$ is undirected, and all nodes are of the same type.
These kinds of graphs are called ‘homogeneous’.
Many of the same ideas we will see here
apply to other kinds of graphs:
we will discuss this later in Different Kinds of Graphs.

Sometimes we will need to denote a graph property by a matrix $M$,
where each row $M_v$

Extending Convolutions to Graphs

Convolutional Neural Networks have been seen to be quite powerful in extracting features from images.
However, images themselves can be seen as graphs with a very regular grid-like structure,
where the individual pixels are nodes, and the RGB channel values at each pixel as the node features.

A natural idea, then, is to consider generalizing convolutions to arbitrary graphs. Recall, however, the challenges
listed out in the previous section: in particular, ordinary convolutions are not node-order invariant, because
they depend on the absolute positions of pixels.
It is initially unclear as how to generalize convolutions over grids to convolutions over general graphs,
where the neighbourhood structure differs from node to node.

The curious reader may wonder if performing some sort of padding and ordering
could be done to ensure the consistency of neighbourhood structure across nodes.
This has been attempted with some success ,
but the techniques we will look at here are more general and powerful.

We begin by introducing the idea of constructing polynomial filters over node neighbourhoods,
much like how CNNs compute localized filters over neighbouring pixels.
Then, we will see how more recent approaches extend on this idea with more powerful mechanisms.
Finally, we will discuss alternative methods
that can use ‘global’ graph-level information for computing node representations.

Polynomial Filters on Graphs

The Graph Laplacian

Given a graph $G$, let us fix an arbitrary ordering of the $n$ nodes of $G$.
We denote the $0-1$ adjacency matrix of $G$ by $A$, we can construct the diagonal degree matrix $D$ of $G$ as:

where $A_{vu}$

Then, the graph Laplacian $L$ is the square $n \times n$ matrix defined as:
$L = D – A.$

The graph Laplacian gets its name from being the discrete analog of the
Laplacian operator
from calculus.

Although it encodes precisely the same information as the adjacency matrix $A$

In the sense that given either of the matrices $A$ or $L$, you can construct the other.
,
the graph Laplacian has many interesting properties of its own.

The graph Laplacian shows up in many mathematical problems involving graphs:
random walks,
spectral clustering,
and
diffusion, to name a few.

We will see some of these properties
in a later section,
but will instead point readers to
this tutorial
for greater insight into the graph Laplacian.

Polynomials of the Laplacian

Now that we have understood what the graph Laplacian is,
we can build polynomials of the form:
$p_w(L) = w_0 I_n + w_1 L + w_2 L^2 + \ldots + w_d L^d = \sum_{i = 0}^d w_i L^i.$

These polynomials can be thought of as the equivalent of ‘filters’ in CNNs,
and the coefficients $w$ as the weights of the ‘filters’.

For ease of exposition, we will focus on the case where nodes have one-dimensional features:
each of the $x_v$

Using the previously chosen ordering of the nodes,
we can stack all of the node features $x_v$

Once we have constructed the feature vector $x$,
we can define its convolution with a polynomial filter $p_w$

At this point, a natural question to ask is:
How does the degree $d$ of the polynomial influence the behaviour of the convolution?
Indeed, it is not too hard to show that:
This is Lemma 5.2 from .
$\text{dist}_G(v, u) > i \quad \Longrightarrow \quad L_{vu}^i = 0.$

This implies, when we convolve $x$ with $p_w(L)$

Effectively, the convolution at node $v$ occurs only with nodes $u$ which are not more than $d$ hops away.
Thus, these polynomial filters are localized. The degree of the localization is governed completely by $d$.

To help you understand these ‘polynomial-based’ convolutions better, we have created the visualization below.
Vary the polynomial coefficients and the input grid $x$ to see how the result $x’$

ChebNet

ChebNet refines this idea of polynomial filters by looking at polynomial filters of the form:

$p_w(L) = \sum_{i = 1}^d w_i T_i(\tilde{L})$

What is the motivation behind these choices?

• $L$ is actually positive semi-definite: all of the eigenvalues of $L$ are not lesser than $0$.
  If $\lambda_{\max}(L) > 1$
• The Chebyshev polynomials have certain interesting properties that make interpolation more numerically stable.
  We won’t talk about this in more depth here,
  but will advise interested readers to take a look at as a definitive resource.

Polynomial Filters are Node-Order Equivariant

The polynomial filters we considered here are actually independent of the ordering of the nodes.
This is particularly easy to see when the degree of the polynomial $p_w$

Embedding Computation

We now describe how we can build a graph neural network
by stacking ChebNet (or any polynomial filter) layers
one after the other with non-linearities,
much like a standard CNN.
In particular, if we have $K$ different polynomial filter layers,
the $k^{\text{th}}$

Note that these networks
reuse the same filter weights across different nodes,
exactly mimicking weight-sharing in Convolutional Neural Networks (CNNs)
which reuse weights for convolutional filters across a grid.

Modern Graph Neural Networks

ChebNet was a breakthrough in learning localized filters over graphs,
and it motivated many to think of graph convolutions from a different perspective.

We return back to the result of convolving $x$ by the polynomial kernel $p_w(L) = L$

As we noted before, this is a $1$-hop localized convolution.
But more importantly, we can think of this convolution as arising of two steps:

• Aggregating over immediate neighbour features $x_u$
• Combining with the node’s own feature $x_v$

Key Idea:
What if we consider different kinds of ‘aggregation’ and ‘combination’ steps,
beyond what are possible using polynomial filters?

By ensuring that the aggregation is node-order equivariant,
the overall convolution becomes node-order equivariant.

These convolutions can be thought of as ‘message-passing’ between adjacent nodes:
after each step, every node receives some ‘information’ from its neighbours.

By iteratively repeating the $1$-hop localized convolutions $K$ times (i.e., repeatedly ‘passing messages’),
the receptive field of the convolution effectively includes all nodes upto $K$ hops away.

Embedding Computation

Message-passing forms the backbone of many GNN architectures today.
We describe the most popular ones in depth below:

• Graph Convolutional Networks (GCN)
• Graph Attention Networks (GAT)
• Graph Sample and Aggregate (GraphSAGE)
• Graph Isomorphism Network (GIN)

Thoughts

An interesting point is to assess different aggregation functions: are some better and others worse?
demonstrates that aggregation functions indeed can be compared on how well
they can uniquely preserve node neighbourhood features;
we recommend the interested reader take a look at the detailed theoretical analysis there.

Here, we’ve talk about GNNs where the computation only occurs at the nodes.
More recent GNN models
such as Message-Passing Neural Networks
and Graph Networks
perform computation over the edges as well;
they compute edge embeddings together with node embeddings.
This is an even more general framework –
but the same ‘message passing’ ideas from this section apply.

Interactive Graph Neural Networks

Below is an interactive visualization of these GNN models on small graphs.
For clarity, the node features are just real numbers here, shown inside the squares next to each node,
but the same equations hold when the node features are vectors.

In practice, each iteration above is generally thought of as a single ‘neural network layer’.
This ideology is followed by many popular Graph Neural Network libraries,

For example: PyTorch Geometric
and StellarGraph.

allowing one to compose different types of graph convolutions in the same model.

From Local to Global Convolutions

The methods we’ve seen so far perform ‘local’ convolutions:
every node’s feature is updated using a function of its local neighbours’ features.

While performing enough steps of message-passing will eventually ensure that
information from all nodes in the graph is passed,
one may wonder if there are more direct ways to perform ‘global’ convolutions.

The answer is yes; we will now describe an approach that was actually first put forward
in the context of neural networks by ,
much before any of the GNN models we looked at above.

Spectral Convolutions

As before, we will focus on the case where nodes have one-dimensional features.
After choosing an arbitrary node-order, we can stack all of the node features to get a
‘feature vector’ $x \in \mathbb{R}^n$

Key Idea:
Given a feature vector $x$,
the Laplacian $L$ allows us to quantify how smooth $x$ is, with respect to $G$.

How?

After normalizing $x$ such that $\sum_{i = 1}^n x_i^2 = 1$

$L$ is a real, symmetric matrix, which means it has all real eigenvalues $\lambda_1 \leq \ldots \leq \lambda_{n}$

Spectral Representations of Natural Images

As discussed before, we can consider any image as a grid graph, where each pixel is a node,
connected by edges to adjacent pixels.
Thus, a pixel can have either $3, 5,$ or $8$ neighbours, depending on its location within the image grid.
Each pixel gets a value as part of the image. If the image is grayscale, each value will be a single
real number indicating how dark the pixel is. If the image is colored, each value will be a $3$-dimensional
vector, indicating the values for the red, green and blue (RGB) channels.
We use the alpha channel as well in the visualization below, so this is actually RGBA.

This construction allows us to compute the graph Laplacian and the eigenvector matrix $U$.
Given an image, we can then investigate what its spectral representation looks like.

To shed some light on what the spectral representation actually encodes,
we perform the following experiment over each channel of the image independently:

• We first collect all pixel values across a channel into a feature vector $x$.
• Then, we obtain its spectral representation $\hat{x}$
• We truncate this to the first $m$ components to get $\hat{x}_m$
• Then, we convert this truncated representation $\hat{x}_m$

Finally, we stack the resulting channels back together to get back an image.
We can now see how the resulting image changes with choices of $m$.
Note that when $m = n$, the resulting image is identical to the original image,
as we can reconstruct each channel exactly.

As $m$ decreases, we see that the output image $x_m$

We can relate this to the Fourier decomposition of images:
the more eigenvectors we use, the higher frequencies we can represent on the grid.

To complement the visualization above,
we additionally visualize the first few eigenvectors on a smaller $8 \times 8$ grid below.
We change the coefficients of the first $10$ out of $64$ eigenvectors
in the spectral representation
and see how the resulting image changes:

These visualizations should convince you that the first eigenvectors are indeed smooth,
and the smoothness correspondingly decreases as we consider later eigenvectors.

For any image $x$, we can think of
the initial entries of the spectral representation $\hat{x}$

Embedding Computation

We now have the background to understand spectral convolutions
and how they can be used to compute embeddings/feature representations of nodes.

As before, the model we describe below has $K$ layers:
each layer $k$ has learnable parameters $\hat{w}^{(k)}$

Thus, convolution in the spectral domain enables the use of significantly fewer parameters
than just direct convolution in the natural domain.
Further, by virtue of the smoothness of the Laplacian eigenvectors across the graph,
using spectral representations automatically enforces an inductive bias for
neighbouring nodes to get similar representations.

Assuming one-dimensional node features for now,
the output of each layer is a vector of node representations $h^{(k)}$

We fix an ordering of the nodes in $G$. This gives us the adjacency matrix $A$ and the graph Laplacian $L$,
allowing us to compute $U_m$

The method above generalizes easily to the case where each $h^{(k)} \in \mathbb{R}^{d_k}$

With the insights from the previous section, we see that convolution in the spectral-domain of graphs
can be thought of as the generalization of convolution in the frequency-domain of images.

Spectral Convolutions are Node-Order Equivariant

We can show spectral convolutions are node-order equivariant using a similar approach
as for Laplacian polynomial filters.

The theory of spectral convolutions is mathematically well-grounded;
however, there are some key disadvantages that we must talk about:

• We need to compute the eigenvector matrix $U_m$
• Even if we can compute $U_m$
• The learned filters are specific to the input graphs,
  as they are represented in terms
  of the spectral decomposition of input graph Laplacian $L$.
  This means they do not transfer well to new graphs
  which have significantly different structure (and hence, significantly
  different eigenvalues) .

While spectral convolutions have largely been superseded by
‘local’ convolutions for the reasons discussed above,
there is still much merit to understanding the ideas behind them.
Indeed, a recently proposed GNN model called Directional Graph Networks

actually uses the Laplacian eigenvectors
and their mathematical properties
extensively.

Global Propagation via Graph Embeddings

A simpler way to incorporate graph-level information
is to compute embeddings of the entire graph by pooling node
(and possibly edge) embeddings,
and then using the graph embedding to update node embeddings,
following an iterative scheme similar to what we have looked at here.
This is an approach used by Graph Networks
.
We will briefly discuss how graph-level embeddings
can be constructed in Pooling.
However, such approaches tend to ignore the underlying
topology of the graph that spectral convolutions can capture.

Learning GNN Parameters

All of the embedding computations we’ve described here, whether spectral or spatial, are completely differentiable.
This allows GNNs to be trained in an end-to-end fashion, just like a standard neural network,
once a suitable loss function $\mathcal{L}$ is defined:

• Node Classification: By minimizing any of the standard losses for classification tasks,
  such as categorical cross-entropy when multiple classes are present:
  $\mathcal{L}(y_v, \hat{y_v}) = -\sum_{c} y_{vc} \log{\hat{y_{vc}}}.$
• Graph Classification: By aggregating node representations,
  one can construct a vector representation of the entire graph.
  This graph representation can be used for any graph-level task, even beyond classification.
  See Pooling for how representations of graphs can be constructed.
• Link Prediction: By sampling pairs of adjacent and non-adjacent nodes,
  and use these vector pairs as inputs to predict the presence/absence of an edge.
  For a concrete example, by minimizing the following ‘logistic regression’-like loss:
  $\begin{aligned} \mathcal{L}(y_v, y_u, e_{vu}) &= -e_{vu} \log(p_{vu}) – (1 – e_{vu}) \log(1 – p_{vu}) \\ p_{vu} &= \sigma(y_v^Ty_u) \end{aligned}$
• Node Clustering: By simply clustering the learned node representations.

The broad success of pre-training for natural language processing models
such as ELMo and BERT
has sparked interest in similar techniques for GNNs
.
The key idea in each of these papers is to train GNNs to predict
local (eg. node degrees, clustering coefficient, masked node attributes)
and/or global graph properties (eg. pairwise distances, masked global attributes).

Another self-supervised technique is to enforce that neighbouring nodes get similar embeddings,
mimicking random-walk approaches such as node2vec and DeepWalk :

$L_{G} = \sum_{v} \sum_{u \in N_R(v)} \log\frac{\exp{z_v^T z_u}}{\sum\limits_{u’} \exp{z_{u’}^T z_u}}.$

where $N_R(v)$

Conclusion and Further Reading

While we have looked at many techniques and ideas in this article,
the field of Graph Neural Networks is extremely vast.
We have been forced to restrict our discussion to a small subset of the entire literature,
while still communicating the key ideas and design principles behind GNNs.
We recommend the interested reader take a look at
for a more comprehensive survey.

We end with pointers and references for additional concepts readers might be interested in:

GNNs in Practice

It turns out that accomodating the different structures of graphs is often hard to do efficiently,
but we can still represent many GNN update equations using
as sparse matrix-vector products (since generally, the adjacency matrix is sparse for most real-world graph datasets.)
For example, the GCN variant discussed here can be represented as:
$h^{(k)} = D^{-1} A \cdot h^{(k – 1)} {W^{(k)}}^T + h^{(k – 1)} {B^{(k)}}^T.$

Regularization techniques for standard neural networks,
such as Dropout ,
can be applied in a straightforward manner to the parameters
(for example, zero out entire rows of $W^{(k)}$

Different Kinds of Graphs

Here, we have focused on undirected graphs, to avoid going into too many unnecessary details.
However, there are some simple variants of spatial convolutions for:

• Directed graphs: Aggregate across in-neighbourhood and/or out-neighbourhood features.
• Temporal graphs: Aggregate across previous and/or future node features.
• Heterogeneous graphs: Learn different aggregation functions for each node/edge type.

There do exist more sophisticated techniques that can take advantage of the different structures of these graphs:
see for more discussion.

Pooling

This article discusses how GNNs compute useful representations of nodes.
But what if we wanted to compute representations of graphs for graph-level tasks (for example, predicting the toxicity of a molecule)?

A simple solution is to just aggregate the final node embeddings and pass them through another neural network $\text{PREDICT}_G$

• SortPool: Sort vertices of the graph to get a fixed-size node-order invariant representation of the graph, and then apply any standard neural network architecture.
• DiffPool: Learn to cluster vertices, build a coarser graph over clusters instead of nodes, then apply a GNN over the coarser graph. Repeat until only one cluster is left.
• SAGPool: Apply a GNN to learn node scores, then keep only the nodes with the top scores, throwing away the rest. Repeat until only one node is left.

Supplementary Material

Reproducing Experiments

The experiments from
Spectral Representations of Natural Images
can be reproduced using the following
Colab notebook:
Spectral Representations of Natural Images.

Recreating Visualizations

To aid in the creation of future interactive articles,
we have created ObservableHQ

notebooks for each of the interactive visualizations here: