Information networks, such as social media and email networks, often contain sensitive information. Releasing such network data could seriously jeopardize individual privacy. Therefore, we need to sanitize network data before the release. In this paper, we present a novel data sanitization solution that infers a network's structure in a differentially private manner. We observe that, by estimating the connection probabilities between vertices instead of considering the observed edges directly, the noise scale enforced by differential privacy can be greatly reduced. Our proposed method infers the network structure by using a statistical hierarchical random graph (HRG) model. The guarantee of differential privacy is achieved by sampling possible HRG structures in the model space via Markov chain Monte Carlo (MCMC). We theoretically prove that the sensitivity of such inference is only O(log n), where n is the number of vertices in a network. This bound implies less noise to be injected than those of existing works. We experimentally evaluate our approach on four real-life network datasets and show that our solution effectively preserves essential network structural properties like degree distribution, shortest path length distribution and influential nodes.