Higher-order structure in networks: construction and its impact on dynamics

Networks are often characterised in terms of their degree distribution and global clustering coefficient. It is assumed that these provide a sufficient parametrisation of networks. However, since the global clustering coefficient is only sensitive to the total number of triangles found in the network, it is evident that two networks could have the same number of triangles but significantly different higher-order structure, i.e., the topologies that result from the placement of closed subgraphs around nodes. The two main objectives of my work are: (1) developing network generating algorithms and network based epidemic models with controllable higher-order structure and (2) investigating the impact of higher-order structure on dynamics on networks. This thesis is based on three papers, corresponding to Chapters. 3, 4 and 5. Chapter 3 presents a novel higher-order structure based network generating algorithm and subgraph counting algorithm. Chapter. 4, generalises a previously proposed ODE model that accurately captures the time evolution of the susceptible-infected-recovered (SIR) dynamics on networks constructed using arbitrary subgraphs. Chapter. 5, improves, extends and generalises the network generating algorithms proposed in the previous two papers. All three chapters demonstrate that for a fixed degree distribution and global clustering, diverse higher-order structure is still possible and that this structure will impact significantly on dynamics unfolding on networks. Hence, we suggest that higher-order structure should receive more attention when analysing network-based systems and dynamics.