# What is the difference between graph semi-supervised learning and normal semi-supervised learning?

Whenever I look for papers involving semi-supervised learning, I always find some that talk about graph semi-supervised learning (e.g. A Unified Framework for Data Poisoning Attack to Graph-based Semi-supervised Learning).

What is the difference between graph semi-supervised learning and normal semi-supervised learning?

Given their main example, the MNIST dataset, is not graph structured, they detail a method for converting the raw Euclidean data $$X$$ into said form (represented by its adjacency matrix $$S$$), and then compute the Laplacian $$L$$ of this graph:
We consider the graph-based semi-supervised learning (G-SSL) problem. The input include labeled data $$X_{l} ∈ \mathbb{R}^{n_{l}×d}$$ and unlabeled data $$X_{u} \in \mathbb{R}^{n_{u}×d}$$, we define the whole features $$X = [X_{l}; X_{u}]$$. Denoting the labels of $$X_{l}$$ as $$y_{l}$$, our goal is to predict the labels of test data $$y_{u}$$. The learner applies algorithm $$A$$ to predict $$y_{u}$$ from available data $$\{X_{l}, y_{l}, X_{u}\}$$. Here we restrict $$A$$ to label propagation method, where we first generate a graph with adjacency matrix $$S$$ from Gaussian kernel: $$S_{ij} = \exp(−γ\lVert x_i − x_j\rVert ^{2})$$, where the subscripts $$x_{i(j)}$$ represents the $$i(j)$$-th row of $$X$$. Then the graph Laplacian is calculated by $$L = D − S$$, where $$D = \text{diag}\{\sum_{k=1}^{n} S_{ik}\}$$ is the degree matrix.