1. Introduction

2. Related Work

3. Methodology

3.3. Graph-SAGE

Graph-SAGE is a method for inductive node embedding. Unlike traditional methods that learn a unique embedding for each node, Graph-SAGE generates embedding by sampling and aggregating features from a node’s local neighborhood. This enables the model to generalize to unseen nodes, making it scalable and versatile.

Figure 2. Graph-SAGE Pipeline.

Figure 2. Graph-SAGE Pipeline.

3.3.1. Neighbor Sampling

For each node v, a fixed-size set of neighbors $\mathcal{N}\left(v\right)$ is sampled. This reduces the computational complexity by considering a limited neighborhood size.

$${\mathcal{N}}_k\left(v\right)=\{u_1,u_2,\dots ,u_k\}$$

(8)

3.3.2. Feature Aggregation

The sampled neighbors’ features are aggregated to form the node’s embedding. The aggregation function can vary, with common choices being mean, sum, or max pooling. The node embedding at layer k, ${\mathbf{h}}_v^{\left(k\right)}$, is computed as follows:

$${\mathbf{h}}_v^{\left(k\right)}=\sigma \left({\mathbf{W}}^{\left(k\right)}·{\mathrm{AGG}}^{\left(k\right)}\left(\left\{{\mathbf{h}}_u^{(k-1)},\forall u\in \mathcal{N}\left(v\right)\right\}\right)\right)$$

(9)

where ${\mathbf{W}}^{\left(k\right)}$ is a trainable weight matrix, $\sigma$ is a nonlinear activation function (e.g. ReLU), and AGG is a function that combines the features of the neighbors.

3.3.3. Multi-Layer Structure

Graph-SAGE employs multiple layers to capture higher-order dependencies. The embedding of a node at the final layer K is influenced by its K-hop neighborhood. For instance, a two-layer Graph-SAGE model would capture information from a node’s neighbors and the neighbors of those neighbors.

$${\mathbf{h}}_v^{\left(K\right)}={\mathrm{AGG}}^{\left(K\right)}\left({\mathbf{h}}_v^{(K-1)},\left\{{\mathbf{h}}_u^{(K-1)},\forall u\in \mathcal{N}\left(v\right)\right\}\right)$$

(10)

3.3.4. Unsupervised Learning

For unsupervised node representation learning, Graph-SAGE uses a loss function that maximizes the similarity between the embedding of adjacent nodes while minimizing the similarity between the embedding of randomly sampled nodes. The loss function is defined as:

$$L=-\sum _{(v,u)\in E}log\sigma ({\mathbf{h}}_v·{\mathbf{h}}_u)-Q·{\mathbb{E}}_{u_n\sim P_n\left(v\right)}log\sigma (-{\mathbf{h}}_v·{\mathbf{h}}_{u_n})$$

(11)

where E is the set of edges, Q is the number of negative samples, and $P_n\left(v\right)$ is a negative sampling distribution.

4. Experiments and Results

5. Conclusion

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