DMS-EPSRC The Dynamics and Structure of Multiway Networks

网络科学是为交互系统和连接数据建模的强大框架。网络科学的优势来自于它将连接提炼成核心元素（节点和边缘）的通用性，这些核心元素可以组合起来形成间接连接。许多社会、自然和工程系统可以表示为网络，例如国际关系、基因调控、机场网络和互联网。动态系统的建模，如信息或病毒在网络上的传播，揭示了结构和动力学之间的相互作用。尽管取得了很大的成功，但网络科学的节点和边缘范式在建模方面存在根本性的局限性。这些限制，再加上详细网络数据的可用性，导致了几个具有更丰富交互的高阶网络模型的早期发展。这一建议的中心是多路网络的数学发展，它模拟的相互作用不能分解成两两边，因为原子相互作用涉及两个以上的节点。例如，化学反应网络模拟了几种化合物之间的相互作用，人们在学校和企业的项目中一起工作的小团队，大脑活动是由神经元群调节的。多个实体的联合协调不是通过组合两两交互来捕获的，而是可以用多路网络（如超图和简单复合体）的模型进行分析。作为起点，我们将考虑在多路网络上定义动态过程的问题。我们将考虑各种方法，从简单的线性马尔可夫随机漫步及其对偶共识模型开始，旨在理解某些超图结构如何转化为相关算子的谱性质。下一步，我们将考虑不能在标准图中编码的非线性和非马尔可夫过程，以充分揭示节点之间非二进制相互作用的重要性。在Hodge Laplacian扩散的基础上，对简单复合体上的随机游走动力学进行类似的探索。由这些动态模型生成的概率流将被用于构建有效的排序和聚类算法，这些算法利用了丰富的多路网络结构。

Network science is a powerful framework for modelling interacting systems and connected data. The strength of network science comes from its generality in distilling connectivity into core elements --- nodes and edges --- that can combine to form indirect connections. Many social, natural and engineered systems can be represented as networks, such as international relationships, gene regulation, airport networks and the Internet. Modelling dynamical systems such as information or virus spreading on networks reveals the interplay between structure and dynamics. Despite much success, the node-and-edge paradigm of network science has fundamental modelling limitations. These limitations, combined with the availability of detailed network data, have led to the early development of several higher-order network models of richer interactions. This proposal centres on the mathematical development of multiway networks, which model interactions that cannot be decomposed into pairwise edges simply because the atomic interactions involve more than two nodes. For example, chemical reaction networks model interactions between several compounds, small teams of people work together on projects in schools and businesses, and brain activity is mediated by groups of neurones. The joint coordination of multiple entities is not captured by combining pairwise interactions, but can be analyzed with models for multiway networks, such as hypergraphs and simplicial complexes. As a starting point, we will consider the problem of defining dynamical processes on multiway networks. We will consider a variety of approaches, starting with simple, linear Markov random walks, and their dual consensus model, aiming to understand how certain hypergraph structures translate into spectral properties of associated operators. As a next step, we will consider non-linear and non-Markovian processes that cannot be encoded in a standard graph, in order to reveal in full the importance of non-binary interactions between the nodes. A similar exploration will be conducted for random walk dynamics on simplicial complexes, building on the diffusion based on Hodge Laplacian. The flows of probability generated by these dynamical models will then be used to construct efficient ranking and clustering algorithms that take advantage of the rich multiway network structure.