# Graph Diffusion in Graph Neural Networks¶

This tutorial first briefly introduces the diffusion process on graphs. It then illustrates how Graph Neural Networks can utilize this concept to enhance prediction power.

[ ]:

# Install required packages.
import os
import torch
os.environ['TORCH'] = torch.__version__
os.environ['DGLBACKEND'] = "pytorch"

# Uncomment below to install required packages. If the CUDA version is not 11.8,
# check the https://www.dgl.ai/pages/start.html to find the supported CUDA
# version and corresponding command to install DGL.
#!pip install dgl -f https://data.dgl.ai/wheels/cu118/repo.html > /dev/null
#!pip install --upgrade scipy networkx > /dev/null

try:
import dgl
installed = True
except ImportError:
installed = False
print("DGL installed!" if installed else "Failed to install DGL!")


## Graph Diffusion¶

Diffusion describes the process of substances moving from one region to another. In the context of graph, the diffusing substances (e.g., real-value signals) travel along edges from nodes to nodes.

Mathematically, let $$\vec x$$ be the vector of node signals, then a graph diffusion operation can be defined as:

$\vec{y} = \tilde{A} \vec{x}$

, where $$\tilde{A}$$ is the diffusion matrix that is typically derived from the adjacency matrix of the graph. Although the selection of diffusion matrices may vary, the diffusion matrix is typically sparse and $$\tilde{A} \vec{x}$$ is thus a sparse-dense matrix multiplication.

Let us understand it more with a simple example. First, we obtain the adjacency matrix of the famous Karate Club Network.

[ ]:

import dgl
import dgl.sparse as dglsp
from dgl.data import KarateClubDataset

# Get the graph from DGL's builtin dataset.
dataset = KarateClubDataset()
dgl_g = dataset

indices = torch.stack(dgl_g.edges())
N = dgl_g.num_nodes()
A = dglsp.spmatrix(indices, shape=(N, N))
print(A.to_dense())

tensor([[0., 1., 1.,  ..., 1., 0., 0.],
[1., 0., 1.,  ..., 0., 0., 0.],
[1., 1., 0.,  ..., 0., 1., 0.],
...,
[1., 0., 0.,  ..., 0., 1., 1.],
[0., 0., 1.,  ..., 1., 0., 1.],
[0., 0., 0.,  ..., 1., 1., 0.]])


We use the graph convolution matrix from Graph Convolution Networks as the diffusion matrix in this example. The graph convolution matrix is defined as:

$\tilde{A} = \bar{D}^{-\frac{1}{2}}\bar{A}\bar{D}^{-\frac{1}{2}}$

with $$\bar{A} = A + I$$, where $$A$$ denotes the adjacency matrix and $$I$$ denotes the identity matrix, $$\bar{D}$$ refers to the diagonal node degree matrix of $$\bar{A}$$.

[ ]:

# Compute graph convolution matrix.
I = dglsp.identity(A.shape)
A_hat = A + I
D_hat = dglsp.diag(A_hat.sum(dim=1))
D_hat_invsqrt = D_hat ** -0.5
A_tilde = D_hat_invsqrt @ A_hat @ D_hat_invsqrt
print(A_tilde.to_dense())

tensor([[0.0588, 0.0767, 0.0731,  ..., 0.0917, 0.0000, 0.0000],
[0.0767, 0.1000, 0.0953,  ..., 0.0000, 0.0000, 0.0000],
[0.0731, 0.0953, 0.0909,  ..., 0.0000, 0.0836, 0.0000],
...,
[0.0917, 0.0000, 0.0000,  ..., 0.1429, 0.1048, 0.0891],
[0.0000, 0.0000, 0.0836,  ..., 0.1048, 0.0769, 0.0654],
[0.0000, 0.0000, 0.0000,  ..., 0.0891, 0.0654, 0.0556]])


For node signals, we set all nodes but one to be zero.

[ ]:

# Initial node signals. All nodes except one are set to zero.
X = torch.zeros(N)
X = 5.

# Number of diffusion steps.
r = 8

# Record the signals after each diffusion step.
results = [X]
for _ in range(r):
X = A_tilde @ X
results.append(X)


The program below visualizes the diffusion process with animation. To play the animation, click the “play” icon. You will see how node features converge over time.

[ ]:

import matplotlib.pyplot as plt
import networkx as nx
from IPython.display import HTML
from matplotlib import animation

nx_g = dgl_g.to_networkx().to_undirected()
pos = nx.spring_layout(nx_g)

fig, ax = plt.subplots()
plt.close()

def animate(i):
ax.cla()
# Color nodes based on their features.
nodes = nx.draw_networkx_nodes(nx_g, pos, ax=ax, node_size=200, node_color=results[i].tolist(), cmap=plt.cm.Blues)
# Set boundary color of the nodes.
nodes.set_edgecolor("#000000")
nx.draw_networkx_edges(nx_g, pos, ax=ax)

ani = animation.FuncAnimation(fig, animate, frames=len(results), interval=1000)
HTML(ani.to_jshtml())


## Graph Diffusion in GNNs¶

Scalable Inception Graph Neural Networks (SIGN) leverages multiple diffusion operators simultaneously. Formally, it is defined as:

$\begin{split}Z=\sigma([X\Theta_{0},A_1X\Theta_{1},\cdots,A_rX\Theta_{r}])\\ Y=\xi(Z\Omega)\end{split}$

where: * $$\sigma$$ and $$\xi$$ are nonlinear activation functions. * $$[\cdot,\cdots,\cdot]$$ is the concatenation operation. * $$X\in\mathbb{R}^{n\times d}$$ is the input node feature matrix with $$n$$ nodes and $$d$$-dimensional feature vector per node. * $$\Theta_0,\cdots,\Theta_r\in\mathbb{R}^{d\times d'}$$ are learnable weight matrices. * $$A_1,\cdots, A_r\in\mathbb{R}^{n\times n}$$ are linear diffusion operators. In the example below, we consider $$A^i$$ for $$A_i$$, where $$A$$ is the convolution matrix of the graph. - $$\Omega\in\mathbb{R}^{d'(r+1)\times c}$$ is a learnable weight matrix and $$c$$ is the number of classes.

The code below implements the diffusion function to compute $$A_1X, A_2X, \cdots, A_rX$$ and the module that combines all the diffused node features.

[ ]:

import torch
import torch.nn as nn
import torch.nn.functional as F

################################################################################
# (HIGHLIGHT) Take the advantage of DGL sparse APIs to implement the feature
# diffusion in SIGN laconically.
################################################################################
def sign_diffusion(A, X, r):
# Perform the r-hop diffusion operation.
X_sign = [X]
for i in range(r):
# A^i X
X = A @ X
X_sign.append(X)
return X_sign

class SIGN(nn.Module):
def __init__(self, in_size, out_size, r, hidden_size=256):
super().__init__()
self.theta = nn.ModuleList(
[nn.Linear(in_size, hidden_size) for _ in range(r + 1)]
)
self.omega = nn.Linear(hidden_size * (r + 1), out_size)

def forward(self, X_sign):
results = []
for i in range(len(X_sign)):
results.append(self.theta[i](X_sign[i]))
Z = F.relu(torch.cat(results, dim=1))
return self.omega(Z)


## Training¶

We train the SIGN model on Cora dataset. The node features are diffused in the pre-processing stage.

[ ]:

from dgl.data import CoraGraphDataset

def evaluate(g, pred):
label = g.ndata["label"]

# Compute accuracy on validation/test set.
return val_acc, test_acc

def train(model, g, X_sign):
label = g.ndata["label"]

for epoch in range(10):
# Switch the model to training mode.
model.train()

# Forward.
logits = model(X_sign)

# Compute loss with nodes in training set.

# Backward.
loss.backward()
optimizer.step()

# Switch the model to evaluating mode.
model.eval()

# Compute prediction.
logits = model(X_sign)
pred = logits.argmax(1)

# Evaluate the prediction.
val_acc, test_acc = evaluate(g, pred)
print(
f"In epoch {epoch}, loss: {loss:.3f}, val acc: {val_acc:.3f}, test"
f" acc: {test_acc:.3f}"
)

# If CUDA is available, use GPU to accelerate the training, use CPU
# otherwise.
dev = torch.device("cuda:0" if torch.cuda.is_available() else "cpu")

# Load graph from the existing dataset.
dataset = CoraGraphDataset()
g = dataset.to(dev)

# Create the sparse adjacency matrix A (note that W was used as the notation
# for adjacency matrix in the original paper).
indices = torch.stack(g.edges())
N = g.num_nodes()
A = dglsp.spmatrix(indices, shape=(N, N))

# Calculate the graph convolution matrix.
I = dglsp.identity(A.shape, device=dev)
A_hat = A + I
D_hat_invsqrt = dglsp.diag(A_hat.sum(dim=1)) ** -0.5
A_hat = D_hat_invsqrt @ A_hat @ D_hat_invsqrt

# 2-hop diffusion.
r = 2
X = g.ndata["feat"]
X_sign = sign_diffusion(A_hat, X, r)

# Create SIGN model.
in_size = X.shape
out_size = dataset.num_classes
model = SIGN(in_size, out_size, r).to(dev)

# Kick off training.
train(model, g, X_sign)

Downloading /root/.dgl/cora_v2.zip from https://data.dgl.ai/dataset/cora_v2.zip...
Extracting file to /root/.dgl/cora_v2
NumNodes: 2708
NumEdges: 10556
NumFeats: 1433
NumClasses: 7
NumTrainingSamples: 140
NumValidationSamples: 500
NumTestSamples: 1000
Done saving data into cached files.
In epoch 0, loss: 1.946, val acc: 0.164, test acc: 0.200
In epoch 1, loss: 1.937, val acc: 0.712, test acc: 0.690
In epoch 2, loss: 1.926, val acc: 0.610, test acc: 0.595
In epoch 3, loss: 1.914, val acc: 0.656, test acc: 0.640
In epoch 4, loss: 1.898, val acc: 0.724, test acc: 0.726
In epoch 5, loss: 1.880, val acc: 0.734, test acc: 0.753
In epoch 6, loss: 1.859, val acc: 0.730, test acc: 0.746
In epoch 7, loss: 1.834, val acc: 0.732, test acc: 0.743
In epoch 8, loss: 1.807, val acc: 0.734, test acc: 0.746
In epoch 9, loss: 1.776, val acc: 0.734, test acc: 0.745


Check out the full example script here. Learn more about how graph diffusion is used in other GNN models:

• Predict then Propagate: Graph Neural Networks meet Personalized PageRank paper code

• Combining Label Propagation and Simple Models Out-performs Graph Neural Networks paper code

• Simplifying Graph Convolutional Networks paper code

• Graph Neural Networks Inspired by Classical Iterative Algorithms paper code