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

摘要

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