An alternative approach to determining average distance in a class of scale-free modular networks

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.