Breadth First Search (BFS) is a widely used approach for sampling large
unknown Internet topologies. Its main advantage over random walks and other
exploration techniques is that a BFS sample is a plausible graph on its own,
and therefore we can study its topological characteristics. However, it has
been empirically observed that incomplete BFS is biased toward high-degree
nodes, which may strongly affect the measurements. In this paper, we first
analytically quantify the degree bias of BFS sampling. In particular, we
calculate the node degree distribution expected to be observed by BFS as a
function of the fraction f of covered nodes, in a random graph RG(pk) with an
arbitrary degree distribution pk. We also show that, for RG(pk), all commonly
used graph traversal techniques (BFS, DFS, Forest Fire, Snowball Sampling, RDS)
suffer from exactly the same bias. Next, based on our theoretical analysis, we
propose a practical BFS-bias correction procedure. It takes as input a
collected BFS sample together with its fraction f. Even though RG(pk) does not
capture many graph properties common in real-life graphs (such as
assortativity), our RG(pk)-based correction technique performs well on a broad
range of Internet topologies and on two large BFS samples of Facebook and Orkut
networks. Finally, we consider and evaluate a family of alternative correction
procedures, and demonstrate that, although they are unbiased for an arbitrary
topology, their large variance makes them far less effective than the
RG(pk)-based technique.Comment: BFS, RDS, graph traversal, sampling bias correctio
