Hybrid graphs as a framework for the small-world effect

Abstract

In this paper we formalize the small-world effect which describes the surprising fact that a hybrid graph composed of a local graph component and a very sparse random graph has a diameter of O(ln n) whereby the diameter of both components alone is much higher. We show that a large family of these hybrid graphs shows this effect and that this generalized family also includes classic small-world models proposed by various authors although not all of them are captured by the small-world definition given by Watts and Strogatz. Furthermore, we give a detailed upper bound of the hybrid's graph diameter for different choices of the expected number of random edges by applying a new kind of proof pattern that is applicable to a large number of hybrid graphs. The focus in this paper is on presenting a flexible family of hybrid graphs showing the small-world effect that can be tuned closely to real-world systems.