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# CS224W - Colab 1

In this Colab, we will write a full pipeline for learning node embeddings. We will go through the following 3 steps.

To start, we will load a classic graph in network science, the Karate Club Network. We will explore multiple graph statistics for that graph.

We will then work together to transform the graph structure into a PyTorch tensor, so that we can perform machine learning over the graph.

Finally, we will finish the first learning algorithm on graphs: a node embedding model. For simplicity, our model here is simpler than DeepWalk / node2vec algorithms taught in the lecture. But it's still rewarding and challenging, as we will write it from scratch via PyTorch.

Now let's get started!

Note: Make sure to sequentially run all the cells, so that the intermediate variables / packages will carry over to the next cell

# 1 Graph Basics

To start, we will load a classic graph in network science, the Karate Club Network. We will explore multiple graph statistics for that graph.

## Setup

We will heavily use NetworkX in this Colab.

## Zachary's karate club network

The Karate Club Network is a graph describes a social network of 34 members of a karate club and documents links between members who interacted outside the club.

networkx.classes.graph.Graph
False

## Question 1: What is the average degree of the karate club network? (5 Points)

Average degree of karate club network is 2

## Question 2: What is the average clustering coefficient of the karate club network? (5 Points)

Average clustering coefficient of karate club network is 0.57

## Question 3: What is the PageRank value for node 0 (node with id 0) after one PageRank iteration? (5 Points)

Please complete the code block by implementing the PageRank equation: $$r_j = \sum_{i \rightarrow j} \beta \frac{r_i}{d_i} + (1 - \beta) \frac{1}{N}$$

The PageRank value for node 0 after one iteration is 0.12810457516339868

## Question 4: What is the (raw) closeness centrality for the karate club network node 5? (5 Points)

The equation for closeness centrality is $$c(v) = \frac{1}{\sum_{u \neq v}\text{shortest path length between } u \text{ and } v}$$

The karate club network has closeness centrality 0.38372093023255816

# 2 Graph to Tensor

We will then work together to transform the graph $$G$$ into a PyTorch tensor, so that we can perform machine learning over the graph.

## Setup

Check if PyTorch is properly installed

1.7.1

## PyTorch tensor basics

We can generate PyTorch tensor with all zeros, ones or random values.

tensor([[1., 1., 1., 1.],
[1., 1., 1., 1.],
[1., 1., 1., 1.]])
tensor([[0., 0., 0., 0.],
[0., 0., 0., 0.],
[0., 0., 0., 0.]])
tensor([[0.0280, 0.6091, 0.4008, 0.1651],
[0.1234, 0.8024, 0.6696, 0.8063],
[0.6875, 0.4731, 0.7378, 0.6720]])
torch.Size([3, 4])

PyTorch tensor contains elements for a single data type, the dtype.

torch.float32
torch.int64

## Question 5: Getting the edge list of the karate club network and transform it into torch.LongTensor. What is the torch.sum value of pos_edge_index tensor? (10 Points)

The pos_edge_index tensor has shape torch.Size([2, 78])
The pos_edge_index tensor has sum value 2535

## Question 6: Please implement following function that samples negative edges. Then you will answer which edges (edge_1 to edge_5) can be negative ones in the karate club network? (10 Points)

The neg_edge_index tensor has shape torch.Size([2, 483])

# 3 Node Emebedding Learning

Finally, we will finish the first learning algorithm on graphs: a node embedding model.

## Setup

1.7.1

To write our own node embedding learning methods, we'll heavily use the nn.Embedding module in PyTorch. Let's see how to use nn.Embedding:

Sample embedding layer: Embedding(4, 8)

We can select items from the embedding matrix, by using Tensor indices

tensor([[ 0.1296,  0.3114,  0.9752,  0.1887,  0.7663,  1.1147, -1.2896,  0.4189]],
tensor([[ 0.1296,  0.3114,  0.9752,  0.1887,  0.7663,  1.1147, -1.2896,  0.4189],
[-0.8057, -0.6563, -0.1285,  0.5352, -1.1358,  1.3075, -0.2638, -2.3275]],
torch.Size([4, 8])
tensor([[1., 1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1., 1.]], grad_fn=<EmbeddingBackward>)

Now, it's your time to create node embedding matrix for the graph we have! - We want to have 16 dimensional vector for each node in the karate club network. - We want to initalize the matrix under uniform distribution, in the range of $$[0, 1)$$. We suggest you using torch.rand.

torch.Size([34, 16])
Embedding: Embedding(34, 16)
tensor([[-1.5256, -0.7502, -0.6540, -1.6095, -0.1002, -0.6092, -0.9798, -1.6091,
-0.7121,  0.3037, -0.7773, -0.2515, -0.2223,  1.6871,  0.2284,  0.4676],
[-0.9274,  0.5451,  0.0663, -0.4370,  0.7626,  0.4415,  1.1651,  2.0154,
0.1374,  0.9386, -0.1860, -0.6446,  1.5392, -0.8696, -3.3312, -0.7479]],
grad_fn=<EmbeddingBackward>)

## Visualize the initial node embeddings

One good way to understand an embedding matrix, is to visualize it in a 2D space. Here, we have implemented an embedding visualization function for you. We first do PCA to reduce the dimensionality of embeddings to a 2D space. Then visualize each point, colored by the community it belongs to.

## Question 7: Training the embedding! What is the best performance you can get? Please report both the best loss and accuracy on Gradescope. (20 Points)

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## Visualize the final node embeddings

Visualize your final embedding here! You can visually compare the figure with the previous embedding figure. After training, you should oberserve that the two classes are more evidently separated. This is a great sanitity check for your implementation as well.