DMS-EPSRC The Dynamics and Structure of Multiway Networks

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.