Traceroute sampling makes random graphs appear to have power law degree distributions

TL;DR

This paper demonstrates that traceroute sampling of the Internet's topology can create the illusion of power law degree distributions in random graphs, due to sampling bias from limited sources.

Contribution

It analytically explains how single-source traceroute sampling biases degree distribution measurements, making random graphs appear scale-free.

Findings

Sampling from a single source yields an apparent P(k) ~ 1/k degree distribution.

Sampling bias can cause Erdős-Rényi graphs to seem scale-free.

Analytical model using differential equations supports the bias explanation.

Abstract

The topology of the Internet has typically been measured by sampling traceroutes, which are roughly shortest paths from sources to destinations. The resulting measurements have been used to infer that the Internet's degree distribution is scale-free; however, many of these measurements have relied on sampling traceroutes from a small number of sources. It was recently argued that sampling in this way can introduce a fundamental bias in the degree distribution, for instance, causing random (Erdos-Renyi) graphs to appear to have power law degree distributions. We explain this phenomenon analytically using differential equations to model the growth of a breadth-first tree in a random graph G(n,p=c/n) of average degree c, and show that sampling from a single source gives an apparent power law degree distribution P(k) ~ 1/k for k < c.