  Given a time-evolving graph, how can we track similarity between nodes in a
fast and accurate way, with theoretical guarantees on the convergence and the
error? Random Walk with Restart (RWR) is a popular measure to estimate the
similarity between nodes and has been exploited in numerous applications. Many
real-world graphs are dynamic with frequent insertion/deletion of edges; thus,
tracking RWR scores on dynamic graphs in an efficient way has aroused much
interest among data mining researchers. Recently, dynamic RWR models based on
the propagation of scores across a given graph have been proposed, and have
succeeded in outperforming previous other approaches to compute RWR
dynamically. However, those models fail to guarantee exactness and convergence
time for updating RWR in a generalized form. In this paper, we propose OSP, a
fast and accurate algorithm for computing dynamic RWR with insertion/deletion
of nodes/edges in a directed/undirected graph. When the graph is updated, OSP
first calculates offset scores around the modified edges, propagates the offset
scores across the updated graph, and then merges them with the current RWR
scores to get updated RWR scores. We prove the exactness of OSP and introduce
OSP-T, a version of OSP which regulates a trade-off between accuracy and
computation time by using error tolerance {\epsilon}. Given restart probability
c, OSP-T guarantees to return RWR scores with O ({\epsilon} /c ) error in O
(log ({\epsilon}/2)/log(1-c)) iterations. Through extensive experiments, we
show that OSP tracks RWR exactly up to 4605x faster than existing static RWR
method on dynamic graphs, and OSP-T requires up to 15x less time with 730x
lower L1 norm error and 3.3x lower rank error than other state-of-the-art
dynamic RWR methods.
