  Estimating similarity between vertices is a fundamental issue in network
analysis across various domains, such as social networks and biological
networks. Methods based on common neighbors and structural contexts have
received much attention. However, both categories of methods are difficult to
scale up to handle large networks (with billions of nodes). In this paper, we
propose a sampling method that provably and accurately estimates the similarity
between vertices. The algorithm is based on a novel idea of random path, and an
extended method is also presented, to enhance the structural similarity when
two vertices are completely disconnected. We provide theoretical proofs for the
error-bound and confidence of the proposed algorithm. We perform extensive
empirical study and show that our algorithm can obtain top-k similar vertices
for any vertex in a network approximately 300x faster than state-of-the-art
methods. We also use identity resolution and structural hole spanner finding,
two important applications in social networks, to evaluate the accuracy of the
estimated similarities. Our experimental results demonstrate that the proposed
algorithm achieves clearly better performance than several alternative methods.
