The small-world phenomenon: a model, explanations, characterizations and examples

TL;DR

This paper formalizes three types of small-world networks based on different distance measures, explores their relationships, and introduces alpha-sequences to characterize small-world properties.

Contribution

It defines three types of small-world networks, establishes their relationships, and introduces alpha-sequences for analyzing small-world properties.

Findings

Each SWD network is also an SWA network.

Each SWA network is also an SWMd network.

Most networks exhibit small-world properties in at least one of the three ways.

Abstract

We introduce and define three types of small worlds: small worlds based on the diameter of the network (SWD), those based on the average geodesic distance between nodes (SWA), and those based on the median geodesic distance (SWMd). These types of networks are defined as limiting properties of sequences of sets. We show the exact relation between these three types, namely that each SWD network is also an SWA network and that each SWA network is also an SWMd network. Yet, having the small-world property is rather evident, in the sense that most networks are small-world networks in one of the three ways. We introduce sequences of distance frequencies, so-called alpha-sequences, and prove a relation between the majorization property between alpha-sequences and small-world properties.