On the Bias of Traceroute Sampling; or, Power-law Degree Distributions in Regular Graphs

TL;DR

This paper rigorously analyzes how traceroute sampling introduces bias in observed network degree distributions, often creating the illusion of power-law behavior even in networks with different underlying distributions.

Contribution

It provides a systematic, mathematical framework to quantify traceroute sampling bias across various degree distributions, extending previous empirical findings.

Findings

Traceroute sampling can produce power-law degree distributions from non-power-law networks.

The expected degree distribution of traceroute trees can be explicitly calculated for general degree distributions.

The analysis confirms and generalizes prior empirical observations of sampling bias in network topology studies.

Abstract

Understanding the structure of the Internet graph is a crucial step for building accurate network models and designing efficient algorithms for Internet applications. Yet, obtaining its graph structure is a surprisingly difficult task, as edges cannot be explicitly queried. Instead, empirical studies rely on traceroutes to build what are essentially single-source, all-destinations, shortest-path trees. These trees only sample a fraction of the network's edges, and a recent paper by Lakhina et al. found empirically that the resuting sample is intrinsically biased. For instance, the observed degree distribution under traceroute sampling exhibits a power law even when the underlying degree distribution is Poisson. In this paper, we study the bias of traceroute sampling systematically, and, for a very general class of underlying degree distributions, calculate the likely observed distributions explicitly. To do this, we use a continuous-time realization of the process of exposing the BFS tree of a random graph with a given degree distribution, calculate the expected degree distribution of the tree, and show that it is sharply concentrated. As example applications of our machinery, we show how traceroute sampling finds power-law degree distributions in both delta-regular and Poisson-distributed random graphs. Thus, our work puts the observations of Lakhina et al. on a rigorous footing, and extends them to nearly arbitrary degree distributions.