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Clustering is the propensity of nodes that share a common neighbour to beconnected. It is ubiquitous in many networks but poses many modellingchallenges. Clustering typically manifests itself by a higher than expectedfrequency of triangles, and this has led to the principle of constructingnetworks from such building blocks. This approach has been generalised tonetworks being constructed from a set of more exotic subgraphs. As long asthese are fully connected, it is then possible to derive mean-field models thatapproximate epidemic dynamics well. However, there are virtually no results fornon-fully connected subgraphs. In this paper, we provide a general andautomated approach to deriving a set of ordinary differential equations, ormean-field model, that describes, to a high degree of accuracy, the expectedvalues of system-level quantities, such as the prevalence of infection. Ourapproach offers a previously unattainable degree of control over thearrangement of subgraphs and network characteristics such as classical nodedegree, variance and clustering. The combination of these features makes itpossible to generate families of networks with different subgraph compositionswhile keeping classical network metrics constant. Using our approach, we showthat higher-order structure realised either through the introduction of loopsof different sizes or by generating clustered networks based on differentsubgraphs, leads to significant differences in epidemic dynamics despitecontrolling for basic network metrics.

Beyond clustering: Mean-field dynamics on networks with arbitrary subgraph composition