In real-world time-series modelling, graph structures are widely adopted because they explicitly encode node topology and capture complex network dynamics. In practice, however, a complete graph is often partitioned across multiple parties; each party can access only its local sub-graph and, owing to privacy regulations, cannot share topology or data, creating pervasive data silos. Federated Graph Learning (FGL) offers a privacy-preserving collaborative-learning paradigm, yet current methods still face two key challenges: (1) the graph topology itself contains sensitive structural information, which can lead to privacy leakage if directly shared during FGL; (2) cross-party edges are crucial for accurate modeling, yet exploiting them without compromising privacy remains a significant challenge. To overcome these obstacles, we propose FedSkeleton, a privacy-preserving framework for time-series prediction that comprises a Skeleton Construction Module and a Dual-stream Forecasting Module, enabling global dependency capture without revealing the topology. Extensive experiments show that FedSkeleton consistently outperforms existing baselines and even surpasses models trained in a centralized setting with full-graph access in certain cases. In addition, we conduct comprehensive security analysis, communication-cost evaluation and scalability experiments, demonstrating that FedSkeleton effectively resists common attacks, keeps communication overhead manageable, and remains robust with respect to key hyper-parameters and the number of participating parties.

Modeling opinion dynamics in social networks has been the focus of multiple disciplines in recent decades. Previous studies have often modeled the opinion dynamics as a discrete and homogeneous process, neglecting its continuous and complex nature. To fill this gap, we propose a Bi-Dynamics Graph Ordinary Differential Equation (BDG-ODE) framework, which models complex opinion dynamics as the result of two dynamical processes: the evolution of positive and negative opinions. The proposed model incorporates a dual opinion encoder that processes positive and negative opinions independently. Furthermore, the temporal opinion evolution is modeled through bidirectional graph ordinary differential equations, which allows the model to capture the changes in opinion in continuous time. We introduce an opinion synthesis decoder that effectively maps the evolved representations from the latent space back to the opinion space. Extensive experiments conducted on six datasets with varying characteristics highlight the superiority of BDG-ODE in forecasting opinion evolution within social networks. It achieved an average accuracy improvement of 23.16%, an average enhancement of 29.46% in the F1 score, and an average mean square error of difference improvement of 90. 30%, and an average correlation coefficient improvement of 45.93%, significantly outperforming eight state-of-the-art models. The code for reproduction is available: https://github.com/tsinghua-fib-lab/Bi-Dynamic-Graph-ODE-for-Opinion-Evolution.