  Clustering is typically measured by the ratio of triangles to all triples,
open or closed. Generating clustered networks, and how clustering affects
dynamics on networks, is reasonably well understood for certain classes of
networks \cite{vmclust, karrerclust2010}, e.g., networks composed of lines and
non-overlapping triangles. In this paper we show that it is possible to
generate networks which, despite having the same degree distribution and equal
clustering, exhibit different higher-order structure, specifically, overlapping
triangles and other order-four (a closed network motif composed of four nodes)
structures. To distinguish and quantify these additional structural features,
we develop a new network metric capable of measuring order-four structure
which, when used alongside traditional network metrics, allows us to more
accurately describe a network's topology. Three network generation algorithms
are considered: a modified configuration model and two rewiring algorithms. By
generating homogeneous networks with equal clustering we study and quantify
their structural differences, and using SIS (Susceptible-Infected-Susceptible)
and SIR (Susceptible-Infected-Recovered) dynamics we investigate
computationally how differences in higher-order structure impact on epidemic
threshold, final epidemic or prevalence levels and time evolution of epidemics.
Our results suggest that characterising and measuring higher-order network
structure is needed to advance our understanding of the impact of network
topology on dynamics unfolding on the networks.
