## 3 Machine Learning Models for Studying Dynamics on Networks  
Modeling the dynamics of processes on networks often requires capturing both temporal patterns and network structure. In the past decade, researchers have increasingly turned to machine learning--and particularly deep learning models--to model and predict network dynamics. Recurrent neural networks (RNNs), especially Long Short-Term Memory (LSTM) networks [25], have shown notable success in forecasting traffic flow and travel speeds by effectively learning long-range temporal dependencies [26]. Similarly, LSTMs have been applied to social networks to predict the evolution of user engagement and information diffusion over time, leveraging their strength in sequence modeling to anticipate bursts of activity or link formations. Complementary methods such as eXtreme Gradient Boosting (e.g., XGBoost [27]), consisting of the use of ensembles of decision trees to model nonlinear relationships, have been applied either as standalone models or in hybrid schemes with LSTMs to refine predictions.  
More recently, the Transformer architecture [28] has revolutionized sequence modeling by leveraging self-attention to capture both local and global interactions--yielding improved performance in traffic forecasting and social-interaction modeling [29]. Transformers rely on self-attention mechanisms to capture relationships in sequential data, either temporal patterns or correlations between locations, and they are highly parallelizable, enabling very large models. In social-network analysis, Transformers (and their variant, graph Transformers) have been employed to model sequences of interactions or events (like sequences of posts or the evolution of relationships), benefiting from attention to identify critical earlier interactions that influence later outcomes [30].  
Hybrid approaches combine spatial feature extraction with temporal dynamics, including convolutional Transformers [31, 32] and convolutional LSTM networks [33, 34, 35]. The idea behind them is to use convolutional layers to capture local spatial structure (for example, nearby roads on a map or local clusters in a network) and Transformer/LSTM layers to capture broader contextual or temporal patterns. Variants like graph-structured LSTMs (applying ConvLSTM, also known as CNN-LSTM, ideas to graph adjacency instead of image grids) have been developed to handle network data more directly [36]. Similarly, for social networks, one could imagine using CNNs on graph-structured data (via graph convolution operations) combined with Transformers for temporal sequences to model phenomena like epidemic spreading or information cascades with high fidelity [37].  
Finally, emerging models such as Extended LSTM (xLSTM) [38] push the limits of recurrent architectures by introducing modifications such as exponential memory gating or parallelizable memory units. Early experiments in sequence modeling (primarily language modeling benchmarks) suggest that, with proper scaling and gating enhancements, xLSTMs can achieve competitive performance to Transformers [38] without the training difficulties that vanilla LSTMs face. Robust synthetic data generation becomes indispensable to effectively evaluate and validate these machine learning methods, as detailed in the following section.