This paper examines the theoretical performance of distributed diffusion models in environments where computational resources and data availability vary significantly among workers. Traditional models centered on single-worker scenarios fall short in such distributed settings, particularly when some workers are resource-constrained. This discrepancy in resources and data diversity challenges the assumption of accurate score function estimation foundational to single-worker models. We establish the inaugural generation error bound for distributed diffusion models in resource-limited settings, establishing a linear relationship with the data dimension $d$ and consistency with established single-worker results. Our analysis highlights the critical role of hyperparameter selection in influencing the training dynamics, which are key to the performance of model generation. This study provides a streamlined theoretical approach to optimizing distributed diffusion models, paving the way for future research in this area.

This paper investigates decentralized unlearning, aiming to eliminate the impact of a specific client on the whole decentralized system. However, decentralized communication characterizations pose new challenges for effective unlearning: the indirect connections make it difficult to trace the specific client's impact, while the dynamic topology limits the scalability of retraining-based unlearning methods. In this paper, we propose the first **P**rovable **D**ecentralized **U**nlearning algorithm under **D**ynamic **T**opologies called PDUDT. It allows clients to eliminate the influence of a specific client without additional communication or retraining. We provide rigorous theoretical guarantees for PDUDT, showing it is statistically indistinguishable from perturbed retraining. Additionally, it achieves an efficient convergence rate of $\mathcal{O}(\frac{1}{T})$ in subsequent learning, where $T$ is the total communication rounds. This rate matches state-of-the-art results. Experimental results show that compared with the Retrain method, PDUDT saves more than 99\% of unlearning time while achieving comparable unlearning performance.

Federated Learning (FL) is a new learning framework in which$s$clients collaboratively train a model under the guidance of a central server. Meanwhile, with the advent of the era of large models, the parameters of models are facing explosive growth. Therefore, it is important to design federated learning algorithms for edge environment. However, the edge environment is severely limited in computing, storage, and network bandwidth resources. Concurrently, adaptive gradient methods show better performance than constant learning rate in non-distributed settings. In this paper, we propose a pruning-based distributed Adam (PD-Adam) algorithm, which combines model pruning and adaptive learning steps to achieve asymptotically optimal convergence rate of$O(1/\sqrt[4]{K})$. At the same time, the algorithm can achieve convergence consistent with the centralized model. Finally, extensive experiments have confirmed the convergence of our algorithm, demonstrating its reliability and effectiveness across various scenarios. Specially, our proposed algorithm is$2$% and$18$% more accurate than the current state-of-the-art FedAvg algorithm on the ResNet and CIFAR datasets.

Hypergraphs are applicable to various domains such as social contagion, online groups, and protein structures due to their effective modeling of multivariate relationships. However, the increasing size of hypergraphs has led to high computation costs, necessitating efficient acceleration strategies. Existing approaches often require consideration of algorithm-specific issues, making them difficult to directly apply to arbitrary hypergraph processing tasks. In this paper, we propose a compression-array acceleration strategy involving hypergraph reordering to improve memory access efficiency, which can be applied to various hypergraph processing tasks without considering the algorithm itself. We introduce a new metric called closeness to optimize the ordering of vertices and hyperedges in the one-dimensional array representation. Moreover, we present an$\frac{1}{2w}$-approximation algorithm to obtain the optimal ordering of vertices and hyperedges. We also develop an efficient update mechanism for dynamic hypergraphs. Our extensive experiments demonstrate significant improvements in hypergraph processing performance, reduced cache misses, and reduced memory footprint. Furthermore, our method can be integrated into existing hypergraph processing frameworks, such as Hygra, to enhance their performance.

Vertex engagement has extraordinary significance for social resilience and network stability. There have been lots of existing work studying this fundamental problem in pairwise graphs, but in the more generalized hypergraphs, it has not been well explored, due to the great challenges of sparsity, complex connectivity and dynamicity of hypergraphs. In this work, we initialize the study of the vertex engagement problem in hypergraphs. Based on the observation that the engagement of vertices in hypergraphs needs to consider two critical parameters, group engagement and neighbor engagement, we propose a vertex engagement model integrating the merits of these two measures, called constrained core, to address the ineffectiveness and incomprehensiveness caused by just using a single engagement factor. By giving an algorithm for the constrained core decomposition, we show that the constrained core number of vertices can be computed in linear time. Furthermore, by showing a localized property of contained core, efficient maintenance algorithms for updating the constrained core number of vertices in dynamic hypergraphs are proposed, to avoid the large amount of redundant computations caused by the decomposition from scratch. Extensive experiments conducted on real-world hypergraphs well exhibit the effectiveness of our model and the efficiency of the proposed algorithms.