  Although the interest in the the use of social and information networks has
grown, most inferences on networks assume the data collected represents the
complete. However, when ignoring missing data, even when missing completely at
random, this results in bias for estimators regarding inference network related
parameters. In this paper, we focus on constructing estimators for the
probability that a randomly selected node has node has at least one edge under
the assumption that nodes are missing completely at random along with their
corresponding edges. In addition, issues also arise in obtaining asymptotic
properties for such estimators, because linkage indicators across nodes are
correlated preventing the direct application of the Central Limit Theorem and
Law of Large Numbers. Using a subsampling approach, we present an improved
estimator for our parameter of interest that accommodates for missing data.
Utilizing the theory U-statistics, we derive consistency and asymptotic
normality of the proposed estimator. This approach decreases the bias in
estimating our parameter of interest. We illustrate our approach using the HIV
viral strains from a large cluster-randomized trial of a combination HIV
prevention intervention -- the Botswana Combination Prevention Project (BCPP).
