  Modelling multiple network data is crucial for addressing a wide range of
applied research questions. However, there are many challenges, both
theoretical and computational, to address. Network cycles are often of
particular interest in many applications, such as ecological studies, and an
unexplored area has been how to incorporate networks' cycles within the
inferential framework in an explicit way. The recently developed Spherical
Network Family of models (SNF) offers a flexible formulation for modelling
multiple network data that permits any type of metric. This has opened up the
possibility to formulate network models that focus on network properties
hitherto not possible or practical to consider. In this article we propose a
novel network distance metric that measures similarities between networks with
respect to their cycles, and incorporate this within the SNF model to allow
inferences that explicitly capture information on cycles. These network motifs
are of particular interest in ecological studies. We further propose a novel
computational framework to allow posterior inferences from the intractable SNF
model for moderate sized networks. Lastly, we apply the resulting methodology
to a set of ecological network data studying aggressive interactions between
species of fish. We show our model is able to make cogent inferences concerning
the cycle behaviour amongst the species, and beyond those possible from a model
that does not consider this network motif.
