  Modularity is a key organizing principle in real-world large-scale complex
networks. Many real-world networks exhibit modular structures such as
transportation infrastructures, communication networks and social media. Having
the knowledge of the shortest paths length distribution (DSPL) between random
pairs of nodes in such networks is important for understanding many processes,
including diffusion or flow. Here, we provide analytical methods which are in
good agreement with simulations on large scale networks with an extreme modular
structure. By extreme modular, we mean that two modules or communities may be
connected by maximum one link. As a result of the modular structure of the
network, we obtain a distribution showing many peaks that represent the number
of modules a typical shortest path is passing through. We present theory and
results for the case where inter-links are weighted, as well as cases in which
the inter-links are spread randomly across nodes in the community or limited to
a specific set of nodes.
