We employ graph neural networks (GNN) to analyse and classify physical gel networks obtained from Brownian dynamics simulations of particles with competing attractive and repulsive interactions. Conventionally such gels are characterized by their position in a state diagram spanned by the packing fraction and the strength of the attraction. Gel networks at different regions of such a state diagram are qualitatively different although structural differences are subtile while dynamical properties are more pronounced. However, using graph classification the GNN is capable of positioning complete or partial snapshots of such gel networks at the correct position in the state diagram based on purely structural input. Furthermore, we demonstrate that not only supervised learning but also unsupervised learning can be used successfully. Therefore, the small structural differences are sufficient to classify the gel networks. Even the trend of data from experiments with different salt concentrations is classified correctly if the GNN was only trained with simulation data. Finally, GNNs are used to compute backbones of gel networks. As the node features used in the GNN are computed in linear time