  Various real-life networks of current interest are simultaneously scale-free
and modular. Here we study analytically the average distance in a class of
deterministically growing scale-free modular networks. By virtue of the
recursive relations derived from the self-similar structure of the networks, we
compute rigorously this important quantity, obtaining an explicit closed-form
solution, which recovers the previous result and is corroborated by extensive
numerical calculations. The obtained exact expression shows that the average
distance scales logarithmically with the number of nodes in the networks,
indicating an existence of small-world behavior. We present that this
small-world phenomenon comes from the peculiar architecture of the network
family.
