  Heterogeneity and geometry are key explanatory components underlying the
structure of real-world networks. The relationship between these components and
the statistical complexity of networks is not well understood. We introduce a
parsimonious normalised measure of statistical complexity for networks --
normalised hierarchical complexity. The measure is trivially 0 in regular
graphs and we prove that this measure tends to 0 in Erd\"os-R\'enyi random
graphs in the thermodynamic limit. We go on to demonstrate that greater
complexity arises from the combination of hierarchical and geometric components
to the network structure than either on their own. Further, the levels of
complexity achieved are similar to those found in many real-world networks. We
also find that real world networks establish connections in a way which
increases hierarchical complexity and which our null models and a range of
attachment mechanisms fail to explain. This underlines the non-trivial nature
of statistical complexity in real-world networks and provides foundations for
the comparative analysis of network complexity within and across disciplines.
