  Networks that are organized as a hierarchy of modules have been the subject
of much research, mainly focusing on algorithms that can extract this community
structure from data. The question of why modular hierarchical organizations are
so ubiquitous in nature, however, has received less attention. One hypothesis
is that modular hierarchical topologies may provide an optimal structure for
certain dynamical processes. We revisit a modular hierarchical network model
that interpolates, using a single parameter, between two known network
topologies: from strong hierarchical modularity to an Erd\H{o}s-R\'enyi random
connectivity structure. We show that this model displays a similar small-world
effect as the Kleinberg model, where the connection probability between nodes
decays algebraically with distance. We find that there is an optimal structure,
in both models, for which the pair-averaged first passage time (FPT) and mean
cover time of a discrete-time random walk are minimal, and provide a heuristic
explanation for this effect. Finally, we show that analytic predictions for the
pair-averaged FPT based on an effective medium approximation fail to reproduce
these minima, which implies that their presence is due to a network structure
effect.
