复杂网络传播动力学被广泛用于疾病传播和信息扩散等问题的建模。网络传播过程的 预测、分析与理解至关重要，需要开发既考虑传播动力学随机性又兼顾真实传播网络结构 复杂性的理论近似方法。动态消息传递(Dynamic Message Passing, DMP)和循环动态消 息传递(recurrent Dynamic Message Passing, rDMP)是一类用于复杂网络传播推断与预测 的重要理论近似方法。尽管 DMP/rDMP 在稀疏网络上能较准确地预测传播规模和识别传 播阈值，但其在真实传播网络中面临以下挑战:在成对交互网络中，原始 DMP/rDMP 难以 处理局部复杂环结构带来的节点相关性;在高阶交互传播网络中，原始 DMP/rDMP 不能 直接用来预测和分析高阶传播过程。本文分别基于平均场近似、高阶信念传播和可学习图 神经网络，开展了考虑局部环结构的 DMP/rDMP 改进和面向高阶传播网络的 DMP/rDMP 扩展的研究。主要研究内容及贡献如下:

1)本文使用平均场近似和配对近似严格推导了成对交互网络上的 rDMP 方法，并考 虑了 rDMP 方法中局部环结构的问题。本文的理论推导提供了扩展 rDMP 方法的原则性框 架，修正后的 rDMP 形式与近期基于空腔主方程理论得到的形式相同，并能直观地扩展到 更高阶的环结构。同时，本文发现 rDMP 在局部环中仍会存在回声室效应，导致传播过程 被高估。利用上述原则性框架，本文将原始 rDMP 方法扩展到考虑局部环的情况。实验表 明，新方法能抑制局部环的回声室效应，改善 rDMP 的预测效果。此外，新方法的稳定性 分析也给出了网络传播阈值的全新估计。

2)针对非循环传播动力学的 DMP 方法在具有复杂局部结构的网络上失效的问题，本 文使用可学习的图神经网络(Graph Neural Networks, GNN)模型作为 DMP 的优化模块而 组成混合模型，通过监督学习训练获得更好的推断精度。首先将 GNN 扩展到线图，以完 成与 DMP 的结构对齐，因为 DMP 是线图上的消息传递过程。本文设计了一个全新的模 型，用于融合 GNN 的可学习能力与 DMP 中的物理先验，使其既能通过学习捕获复杂局 部结构导致的节点间的动力学相关性，又能在训练集之外具有良好的泛化性能。在人造和 真实网络上的数值实验表明，本文提出的模型在不同的网络结构和动力学参数上具有最佳 的拟合效果和泛化效果，可作为 DMP 的替代推断模型，用于非循环传播的预测。

3)本文从高阶平均场理论和高阶信念传播方法分别推导出高阶循环和非循环传播动 力学的消息传递理论，用于高阶传播过程的预测和高阶传播阈值的识别。具体而言，本文 为二阶单纯复形网络上的单纯形传播模型提出了高阶 rDMP，数值实验表明高阶 rDMP 能 很好地预测高阶传播过程，尤其是传播早期。通过线性稳定性分析，还可以得到单纯形传 播模型传播阈值的预测值。本文为三阶超图上的易感-感染传播模型提出了高阶 DMP 框 架，由超图上的动态信念传播和易感-感染传播模型的非循环性质严格推导而来，因此理 论上该方法在稀疏超图上是渐进精确的。数值实验表明高阶 DMP 在稀疏超图上能近乎精 确地预测高阶传播，在稠密的真实超图上也比简单的平均场近似效果好。本文首次将动态 消息传递方法扩展到高阶传播过程，为高阶传播动力学的推断/预测和理论分析提供了全 新的工具和视角。

Complex network spreading dynamics is widely used for modeling problems such as dis- ease transmission and information spreading, where prediction, analysis and understanding of the spreading process are crucial. Message passing (MP) is an effective class of methods for network analysis and approximate inference of marginal probabilities, and the MP variants for spreading dynamics prediction and analysis are known as dynamic message passing (DMP) methods and re- current dynamic message passing (rDMP) methods. Although DMP/rDMP can accurately predict the spreading size and transmission threshold on networks with local tree structures, its applica- tion to real propagation networks faces the following problems: in pairwise interaction networks, the original DMP/rDMP has difficulty in handling node correlations due to local complex loop structures, resulting in overestimation of the spreading size; in higher-order interaction spread- ing networks, the original DMP/rDMP is not applicable and cannot be directly used to predict and analyze higher-order spreading processes. In this paper, based on mean-field approximation, higher-order belief propagation and learnable graph neural network, respectively, we focus on two aspects of DMP/rDMP improvement considering local loop structure and DMP/rDMP extension for higher-order propagation networks, and the main research contents and contributions are as follows.

(1) In this paper, the rDMP method on pairwise interaction networks is rigorously derived, and the local loop structure is considered in the rDMP method. The modified form of rDMP is first derived using mean-field approximation and pairwise approximation methods, and then the local ring structure is extended in rDMP. The theoretical derivation in this paper gives a principled framework for extending the rDMP method, and the obtained modified form of rDMP is the same as the one obtained recently based on the theory of cavity master equations and can be extended intuitively to higher order loop structures. At the same time, this paper finds that the rDMP still suffers from echo chamber effects in the local ring, making it overestimate the propagation process.

Using the above principled framework, this paper extends the original rDMP method to consider only the local loop case. Experiments show that the new method can suppress the echo chamber effect of the local ring and improve the prediction of rDMP. In addition, the stability analysis of the new method gives a new estimate of the transmission threshold.

(2) To address the problem that the DMP method with non-recurrent spreading dynamics will fail on networks with complex local structures, this paper uses the learnable GNN model as the rrefining module of DMP, and composes a hybrid model to obtain better inference accuracy through supervised learning. The GNN is first extended to the line graph to complete the alignment with DMP, which is a message passing process on the line graph. A completely new model was designed for fusing the learnable capability of GNN with the physical prior in DMP to enable both learning to capture dynamical correlations between nodes due to complex local structures and good generalization performance outside the training set. Numerical experiments on artificial and real networks show that the model proposed in this paper has the best fitting and generalization effects on different network structures and dynamics parameters, and can be used as an alternative inferential model to DMP for non-recurrent spreading prediction.

(3) In this paper, the message passing theory of higher-order recurrent and non-recurrent spreading dynamics is derived from higher-order mean field theory and higher-order belief propa- gation methods, respectively, for the prediction of higher-order spreading processes and the iden- tification of higher-order spreading thresholds. Specifically, in this paper, a higher-order rDMP is proposed for the SCM model on second-order simplicial complex networks. Numerical ex- periments show that the higher-order rDMP can predict the higher-order spreading process well, especially in the early stage of propagation. The predicted value of the SCM propagation threshold can also be obtained by linear stability analysis. In this paper, a higher-order DMP is proposed for the SI model on the third-order hypergraph, which is strictly derived from the dynamic belief prop- agation on the hypergraph and the non-recurrent nature of SI, so that the method is theoretically asymptotically accurate on the sparse hypergraph. Numerical experiments show that the higher- order DMP can predict the higher-order spreading nearly accurately on sparse hypergraphs and also better than the simple mean-field approximation on dense real hypergraphs.