Unbiased Estimation Based on Biased Sampled Data of Communication Networks

Random walks are the first choice for gathering random samples from communication networks. In its simplest form, which we will call classical random-walk (RW) sampling, random walkers select an outgoing link with equal probability to visit an adjacent node. Although classical RW sampling is preferable because of its simplicity, it does not perform uniform sampling when the degree of the network nodes is not uniformly distributed, in that case, it introduces a large amount of bias to the nodes of large degree. We propose a technique that uses an idea from probability theory, change of measure, to correct the bias in the data obtained by classical RW sampling. For this purpose, we introduce two probability measures defined on a common space: one is a probability measure that characterizes the observation by using a uniformly sampled node, and the other characterizes the observation by using a node sampled by classical RW sampling. Using the relationship between the two probability measures, the statistics for the data from the uniform sampling can be inferred from the statistics for the data from the classical RW sampling. We present a mathematical framework for the above-mentioned approach and derive several formulas for transforming between the two types of statistics. Simulation experiments based on the data of real networks verify the effectiveness of this proposal.