  Understanding the complex hierarchical topology of functional brain networks
is a key aspect of functional connectivity research. Such topics are obscured
by the widespread use of sparse binary network models which are fundamentally
different to the complete weighted networks derived from functional
connectivity. We introduce two techniques to probe the hierarchical complexity
of topologies. Firstly, a new metric to measure hierarchical complexity;
secondly, a Weighted Complex Hierarchy (WCH) model. To thoroughly evaluate our
techniques, we generalise sparse binary network archetypes to weighted forms
and explore the main topological features of brain networks- integration,
regularity and modularity- using curves over density. By controlling the
parameters of our model, the highest complexity is found to arise between a
random topology and a strict 'class-based' topology. Further, the model has
equivalent complexity to EEG phase-lag networks at peak performance.
Hierarchical complexity attains greater magnitude and range of differences
between different networks than the previous commonly used complexity metric
and our WCH model offers a much broader range of network topology than the
standard scale-free and small-world models at a full range of densities. Our
metric and model provide a rigorous characterisation of hierarchical
complexity. Importantly, our framework shows a scale of complexity arising
between 'all nodes are equal' topologies at one extreme and 'strict
class-based' topologies at the other.
