Small World Graphs by the iterated "My Friends are Your Friends'' Principle

TL;DR

This paper investigates how local edge modifications in graphs can produce small world properties like logarithmic diameter and high clustering, revealing a phase transition based on initial conditions without using preferential attachment.

Contribution

It introduces a local edge formation process that generates small world graphs with realistic properties without relying on preferential attachment.

Findings

Graphs evolve to have logarithmic diameter and high clustering

A phase transition depends on initial graph conditions

No preferential attachment is necessary for small world features

Abstract

We study graphs obtained by successive creation and destruction of edges into small neighborhoods of the vertices. Starting with a circle graph of large diameter we obtain small world graphs with logarithmic diameter, high clustering coefficients and a fat tail distribution for the degree. Only local edge formation processes are involved and no preferential attachment was used. Furthermore we found an interesting phase transition with respect to the initial conditions.