Navigation in non-uniform density social networks

TL;DR

This paper introduces a scale-invariant friendship network model based on empirical social network data, analyzes its navigation complexity, and explores the relationship between high and low dimensional models.

Contribution

It presents a new scale-invariant social network model and proves its navigation complexity and dimensional projection relationships.

Findings

Navigation in 2D SIFN has at most O(log^4 n) complexity.

2D SIFN is a projection of 3D SIFN.

The model captures the universal scaling law of social networks.

Abstract

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 $P(d)\propto d^{-1}$. Since population density is non-uniform in real social networks, a scale invariant friendship network(SIFN) based on the above empirical law is introduced to capture this phenomenon. We prove the time complexity of navigation in 2-dimensional SIFN is at most $O(\log^4 n)$. In the real searching experiment, individuals often resort to extra information besides geography location. Thus, real-world searching process may be seen as a projection of navigation in a $k$-dimensional SIFN($k>2$). Therefore, we also discuss the relationship between high and low dimensional SIFN. Particularly, we prove a 2-dimensional SIFN is the projection of a 3-dimensional SIFN. As a matter of fact, this result can also be generated to any $k$-dimensional SIFN.