  Analysis of social networks with limited data access is challenging for third
parties. To address this challenge, a number of studies have developed
algorithms that estimate properties of social networks via a simple random
walk. However, most existing algorithms do not assume private nodes that do not
publish their neighbors' data when they are queried in empirical social
networks. Here we propose a practical framework for estimating properties via
random walk-based sampling in social networks involving private nodes. First,
we develop a sampling algorithm by extending a simple random walk to the case
of social networks involving private nodes. Then, we propose estimators with
reduced biases induced by private nodes for the network size, average degree,
and density of the node label. Our results show that the proposed estimators
reduce biases induced by private nodes in the existing estimators by up to
92.6% on social network datasets involving private nodes.
