Randomness and Complexity in Networks

TL;DR

This paper explores how random walks and copying mechanisms in networks explain the emergence of long-tailed distributions, providing a mathematical framework for understanding complex network growth.

Contribution

It introduces a novel model linking random walks and copying to network evolution, with exact solutions and analysis of variations.

Findings

Random walks can explain long-tailed distributions in networks.

The proposed model allows exact solutions for network growth dynamics.

Variations of the basic model reveal diverse network behaviors.

Abstract

I start by reviewing some basic properties of random graphs. I then consider the role of random walks in complex networks and show how they may be used to explain why so many long tailed distributions are found in real data sets. The key idea is that in many cases the process involves copying of properties of near neighbours in the network and this is a type of short random walk which in turn produce a natural preferential attachment mechanism. Applying this to networks of fixed size I show that copying and innovation are processes with special mathematical properties which include the ability to solve a simple model exactly for any parameter values and at any time. I finish by looking at variations of this basic model.