Diffusion limited friendship network: A model for six degrees of separation

TL;DR

This paper introduces a dynamic society model where random walkers form a friendship network exhibiting a phase transition, demonstrating persistent six degrees of separation and conditions for disease containment.

Contribution

It presents a novel dynamic network model with percolation-like transition and analyzes its implications for social connectivity and disease spread.

Findings

Friendship network shows a percolation-like phase transition.

Six degrees of separation persists indefinitely under certain conditions.

Society remains healthy if population density is below a threshold.

Abstract

A dynamic model of a society is studied where each person is an uncorrelated and non-interacting random walker. A dynamical random graph represents the acquaintance network of the society whose nodes are the individuals and links are the pairs of mutual friendships. This network exhibits a novel percolation like phase transition in all dimensions. On introducing simultaneous death and birth rates in the population we show that the friendship network shows the six degrees of separation for ever after where the precise value of the network diameter depends on the death/birth rate. A SIS type model of disease spreading shows that this society remains always healthy if the population density is less than certain threshold value.