Navigation in non-uniform density social networks

Recent empirical investigations suggest a universal scaling law for the spatial structure of social networks. It is found that the probability density distribution of an individual to have a friend at distance d scales as P(d) ∝ d−1. According to this spatial property, a scale invariant friendship (SIF) network model is introduced to deal with the navigation problem with non-uniform population density. We prove that the time cost of navigation in the 2-dimensional SIF network is at most O(log n). We also show that SIF network is navigable when taking account of the information dimension, through a numeric experiment and theoretic analysis.