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# NodeFlow and Sampling¶

**Author**: Ziyue Huang, Da Zheng, Quan Gan, Jinjing Zhou, Zheng Zhang

Graph convolutional network ~~~

In an \(L\)-layer graph convolution network (GCN), given a graph \(G=(V, E)\), represented as an adjacency matrix \(A\), with node features \(H^{(0)} = X \in \mathbb{R}^{|V| \times d}\), the hidden feature of a node \(v\) in \((l+1)\)-th layer \(h_v^{(l+1)}\) depends on the features of all its neighbors in the previous layer \(h_u^{(l)}\):

where \(\mathcal{N}(v)\) is the neighborhood of \(v\), \(\tilde{A}\) could be any normalized version of \(A\) such as \(D^{-1} A\) in Kipf et al., \(\sigma(\cdot)\) is an activation function, and \(W^{(l)}\) is a trainable parameter of the \(l\)-th layer.

In the node classification task you minimize the following loss:

where \(y_v\) is the label of \(v\), and \(f(\cdot, \cdot)\) is a loss function, e.g., cross entropy loss.

While training GCN on the full graph, each node aggregates the hidden features of its neighbors to compute its hidden feature in the next layer.

In this tutorial, you run GCN on the Reddit dataset constructed by Hamilton et al., wherein the nodes are posts and edges are established if two nodes are commented by a same user. The task is to predict the category that a post belongs to. This graph has 233,000 nodes, 114.6 million edges and 41 categories. First load the Reddit graph.

```
import numpy as np
import dgl
import dgl.function as fn
from dgl import DGLGraph
from dgl.data import RedditDataset
import mxnet as mx
from mxnet import gluon
# Load MXNet as backend
dgl.load_backend('mxnet')
# load dataset
data = RedditDataset(self_loop=True)
train_nid = mx.nd.array(np.nonzero(data.train_mask)[0]).astype(np.int64)
features = mx.nd.array(data.features)
in_feats = features.shape[1]
labels = mx.nd.array(data.labels)
n_classes = data.num_labels
# construct DGLGraph and prepare related data
g = DGLGraph(data.graph, readonly=True)
g.ndata['features'] = features
```

Out:

```
Finished data loading.
NumNodes: 232965
NumEdges: 114848857
NumFeats: 602
NumClasses: 41
NumTrainingSamples: 153431
NumValidationSamples: 23831
NumTestSamples: 55703
```

Here you define the node UDF, which has a fully-connected layer:

```
class NodeUpdate(gluon.Block):
def __init__(self, in_feats, out_feats, activation=None):
super(NodeUpdate, self).__init__()
self.dense = gluon.nn.Dense(out_feats, in_units=in_feats)
self.activation = activation
def forward(self, node):
h = node.data['h']
h = self.dense(h)
if self.activation:
h = self.activation(h)
return {'activation': h}
```

In DGL, you implement GCN on the full graph with `update_all`

in `DGLGraph`

.
The following code performs two-layer GCN on the Reddit graph.

```
# number of GCN layers
L = 2
# number of hidden units of a fully connected layer
n_hidden = 64
layers = [NodeUpdate(g.ndata['features'].shape[1], n_hidden, mx.nd.relu),
NodeUpdate(n_hidden, n_hidden, mx.nd.relu)]
for layer in layers:
layer.initialize()
h = g.ndata['features']
for i in range(L):
g.ndata['h'] = h
g.update_all(message_func=fn.copy_src(src='h', out='m'),
reduce_func=fn.sum(msg='m', out='h'),
apply_node_func=lambda node: {'h': layers[i](node)['activation']})
h = g.ndata.pop('h')
```

## NodeFlow¶

As the graph scales up to billions of nodes or edges, training on the full graph would no longer be efficient or even feasible.

Mini-batch training allows you to control the computation and memory usage within some budget. The training loss for each iteration is

where \(\tilde{\mathcal{V}}_\mathcal{L}\) is a subset sampled from the total labeled nodes \(\mathcal{V}_\mathcal{L}\) uniformly at random.

Stemming from the labeled nodes \(\tilde{\mathcal{V}}_\mathcal{L}\) in a mini-batch and tracing back to the input forms a computational dependency graph (a directed acyclic graph [DAG]), which captures the computation flow of \(Z^{(L)}\).

In the example below, a mini-batch to compute the hidden features of node D in layer 2 requires hidden features of A, B, E, G in layer 1, which in turn requires hidden features of C, D, F in layer 0.

For that purpose, you define `NodeFlow`

to represent this computation
flow.

`NodeFlow`

is a type of layered graph, where nodes are organized in
\(L + 1\) sequential *layers*, and edges only exist between adjacent
layers, forming *blocks*. You construct `NodeFlow`

backwards, starting
from the last layer with all the nodes whose hidden features are
requested. The set of nodes the next layer depends on forms the previous
layer. An edge connects a node in the previous layer to another in the
next layer if the latter depends on the former. Repeat such process
until all \(L + 1\) layers are constructed. The feature of nodes in
each layer, and that of edges in each block, are stored as separate
tensors.

`NodeFlow`

provides `block_compute`

for per-block computation, which
triggers computation and data propogation from the lower layer to the
next upper layer.

## Neighbor sampling¶

Real-world graphs often have nodes with large degree, meaning that a moderately deep (e.g., three layers) GCN would often depend on input features of the entire graph, even if the computation only depends on outputs of a few nodes, hence its cost-ineffectiveness.

Sampling methods mitigate this computational problem by reducing the receptive field effectively. Fig-c above shows one such example.

Instead of using all the \(L\)-hop neighbors of a node \(v\),
Hamilton et al. propose *neighbor
sampling*, which randomly samples a few neighbors
\(\hat{\mathcal{N}}^{(l)}(v)\) to estimate the aggregation
\(z_v^{(l+1)}\) of its total neighbors \(\mathcal{N}(v)\) in
\(l\)-th GCN layer, by an unbiased estimator
\(\hat{z}_v^{(l+1)}\)

Let \(D^{(l)}\) be the number of neighbors to be sampled for each
node at the \(l\)-th layer, then the receptive field size of each
node can be controlled under \(\prod_{i=0}^{L-1} D^{(l)}\) by
*neighbor sampling*.

You then implement *neighbor sampling* by `NodeFlow`

:

```
class GCNSampling(gluon.Block):
def __init__(self,
in_feats,
n_hidden,
n_classes,
n_layers,
activation,
dropout,
**kwargs):
super(GCNSampling, self).__init__(**kwargs)
self.dropout = dropout
self.n_layers = n_layers
with self.name_scope():
self.layers = gluon.nn.Sequential()
# input layer
self.layers.add(NodeUpdate(in_feats, n_hidden, activation))
# hidden layers
for i in range(1, n_layers-1):
self.layers.add(NodeUpdate(n_hidden, n_hidden, activation))
# output layer
self.layers.add(NodeUpdate(n_hidden, n_classes))
def forward(self, nf):
nf.layers[0].data['activation'] = nf.layers[0].data['features']
for i, layer in enumerate(self.layers):
h = nf.layers[i].data.pop('activation')
if self.dropout:
h = mx.nd.Dropout(h, p=self.dropout)
nf.layers[i].data['h'] = h
# block_compute() computes the feature of layer i given layer
# i-1, with the given message, reduce, and apply functions.
# Here, you essentially aggregate the neighbor node features in
# the previous layer, and update it with the `layer` function.
nf.block_compute(i,
fn.copy_src(src='h', out='m'),
lambda node : {'h': node.mailbox['m'].mean(axis=1)},
layer)
h = nf.layers[-1].data.pop('activation')
return h
```

DGL provides `NeighborSampler`

to construct the `NodeFlow`

for a
mini-batch according to the computation logic of neighbor sampling.
`NeighborSampler`

returns an iterator that generates a `NodeFlow`

each time. This function
has many options to give users opportunities to customize the behavior
of the neighbor sampler, including the number of neighbors to sample or
the number of hops to sample, for example. Please see its API
document for more
details.

```
# dropout probability
dropout = 0.2
# batch size
batch_size = 1000
# number of neighbors to sample
num_neighbors = 4
# number of epochs
num_epochs = 1
# initialize the model and cross entropy loss
model = GCNSampling(in_feats, n_hidden, n_classes, L,
mx.nd.relu, dropout, prefix='GCN')
model.initialize()
loss_fcn = gluon.loss.SoftmaxCELoss()
# use adam optimizer
trainer = gluon.Trainer(model.collect_params(), 'adam',
{'learning_rate': 0.03, 'wd': 0})
for epoch in range(num_epochs):
i = 0
for nf in dgl.contrib.sampling.NeighborSampler(g, batch_size,
num_neighbors,
neighbor_type='in',
shuffle=True,
num_hops=L,
seed_nodes=train_nid):
# When `NodeFlow` is generated from `NeighborSampler`, it only contains
# the topology structure, on which there is no data attached.
# Users need to call `copy_from_parent` to copy specific data,
# such as input node features, from the original graph.
nf.copy_from_parent()
with mx.autograd.record():
# forward
pred = model(nf)
batch_nids = nf.layer_parent_nid(-1).astype('int64')
batch_labels = labels[batch_nids]
# cross entropy loss
loss = loss_fcn(pred, batch_labels)
loss = loss.sum() / len(batch_nids)
# backward
loss.backward()
# optimization
trainer.step(batch_size=1)
print("Epoch[{}]: loss {}".format(epoch, loss.asscalar()))
i += 1
# You only train the model with 32 mini-batches just for demonstration.
if i >= 32:
break
```

Out:

```
Epoch[0]: loss 3.7310597896575928
Epoch[0]: loss 3.0101420879364014
Epoch[0]: loss 2.585841655731201
Epoch[0]: loss 2.159868001937866
Epoch[0]: loss 1.9110859632492065
Epoch[0]: loss 1.7267630100250244
Epoch[0]: loss 1.5326182842254639
Epoch[0]: loss 1.606508731842041
Epoch[0]: loss 1.4062317609786987
Epoch[0]: loss 1.2362780570983887
Epoch[0]: loss 1.3502916097640991
Epoch[0]: loss 1.2101963758468628
Epoch[0]: loss 1.3546274900436401
Epoch[0]: loss 1.0514274835586548
Epoch[0]: loss 1.099158525466919
Epoch[0]: loss 1.0870254039764404
Epoch[0]: loss 1.082731008529663
Epoch[0]: loss 1.0594552755355835
Epoch[0]: loss 1.0993865728378296
Epoch[0]: loss 1.0266661643981934
Epoch[0]: loss 1.0138708353042603
Epoch[0]: loss 0.9079034924507141
Epoch[0]: loss 0.9981871247291565
Epoch[0]: loss 1.066437005996704
Epoch[0]: loss 1.012518048286438
Epoch[0]: loss 1.0037081241607666
Epoch[0]: loss 0.9039970636367798
Epoch[0]: loss 0.9596415162086487
Epoch[0]: loss 0.8718827962875366
Epoch[0]: loss 0.9065074324607849
Epoch[0]: loss 0.8950481414794922
Epoch[0]: loss 0.8208798766136169
```

## Control variate¶

The unbiased estimator \(\hat{Z}^{(\cdot)}\) used in *neighbor
sampling* might suffer from high variance, so it still requires a
relatively large number of neighbors, e.g. \(D^{(0)}=25\) and
\(D^{(1)}=10\) in Hamilton et
al.. With *control variate*, a
standard variance reduction technique widely used in Monte Carlo
methods, 2 neighbors for a node seems sufficient.

*Control variate* method works as follows: Given a random variable
\(X\) and you wish to estimate its expectation
\(\mathbb{E} [X] = \theta\), it finds another random variable
\(Y\) which is highly correlated with \(X\) and whose
expectation \(\mathbb{E} [Y]\) can be easily computed. The *control
variate* estimator \(\tilde{X}\) is

If \(\mathbb{VAR} [Y] - 2\mathbb{COV} [X, Y] < 0\), then \(\mathbb{VAR} [\tilde{X}] < \mathbb{VAR} [X]\).

Chen et al. proposed a *control
variate* based estimator used in GCN training, by using history
\(\bar{H}^{(l)}\) of the nodes which are not sampled, the modified
estimator \(\hat{z}_v^{(l+1)}\) is

This method can also be *conceptually* implemented in DGL as shown
here.

```
have_large_memory = False
# The control-variate sampling code below needs to run on a large-memory
# machine for the Reddit graph.
if have_large_memory:
g.ndata['h_0'] = features
for i in range(L):
g.ndata['h_{}'.format(i+1)] = mx.nd.zeros((features.shape[0], n_hidden))
# With control-variate sampling, you only need to sample two neighbors to train GCN.
for nf in dgl.contrib.sampling.NeighborSampler(g, batch_size, expand_factor=2,
neighbor_type='in', num_hops=L,
seed_nodes=train_nid):
for i in range(nf.num_blocks):
# aggregate history on the original graph
g.pull(nf.layer_parent_nid(i+1),
fn.copy_src(src='h_{}'.format(i), out='m'),
lambda node: {'agg_h_{}'.format(i): node.mailbox['m'].mean(axis=1)})
nf.copy_from_parent()
h = nf.layers[0].data['features']
for i in range(nf.num_blocks):
prev_h = nf.layers[i].data['h_{}'.format(i)]
# compute delta_h, the difference of the current activation and the history
nf.layers[i].data['delta_h'] = h - prev_h
# refresh the old history
nf.layers[i].data['h_{}'.format(i)] = h.detach()
# aggregate the delta_h
nf.block_compute(i,
fn.copy_src(src='delta_h', out='m'),
lambda node: {'delta_h': node.data['m'].mean(axis=1)})
delta_h = nf.layers[i + 1].data['delta_h']
agg_h = nf.layers[i + 1].data['agg_h_{}'.format(i)]
# control variate estimator
nf.layers[i + 1].data['h'] = delta_h + agg_h
nf.apply_layer(i + 1, lambda node : {'h' : layer(node.data['h'])})
h = nf.layers[i + 1].data['h']
# update history
nf.copy_to_parent()
```

You can see full example here, MXNet code and PyTorch code.

Below shows the performance of graph convolution network and GraphSage with neighbor sampling and control variate sampling on the Reddit dataset. Our GraphSage with control variate sampling, when sampling one neighbor, can achieve over 96 percent test accuracy.

## More APIs¶

In fact, `block_compute`

is one of the APIs that comes with
`NodeFlow`

, which provides flexibility to research new ideas. The
computation flow underlying a DAG can be executed in one sweep, by
calling `prop_flows`

.

`prop_flows`

accepts a list of UDFs. The code below defines node update UDFs
for each layer and computes a simplified version of GCN with neighbor sampling.

```
apply_node_funcs = [
lambda node : {'h' : layers[0](node)['activation']},
lambda node : {'h' : layers[1](node)['activation']},
]
for nf in dgl.contrib.sampling.NeighborSampler(g, batch_size, num_neighbors,
neighbor_type='in', num_hops=L,
seed_nodes=train_nid):
nf.copy_from_parent()
nf.layers[0].data['h'] = nf.layers[0].data['features']
nf.prop_flow(fn.copy_src(src='h', out='m'),
fn.sum(msg='m', out='h'), apply_node_funcs)
```

Internally, `prop_flow`

triggers the computation by fusing together
all the block computations, from the input to the top. The main
advantages of this API are 1) simplicity, 2) allowing more system-level
optimization in the future.

**Total running time of the script:** ( 5 minutes 36.983 seconds)