Random walk with restart over dynamic graphs

Random Walk with Restart (RWR) is an appealing measure of proximity between nodes based on graph structures. Since real graphs are often large and subject to minor changes, it is prohibitively expensive to recompute proximities from scratch. Previous methods use LU decomposition and degree reordering heuristics, entailing O(|V|³) time and O(|V|²) memory to compute all (|V|²) pairs of node proximities in a static graph. In this paper, a dynamic scheme to assess RWR proximities is proposed: (1) For unit update, we characterize the changes to all-pairs proximities as the outer product of two vectors. We notice that the multiplication of an RWR matrix and its transition matrix, unlike traditional matrix multiplications, is commutative. This can greatly reduce the computation of all-pairs proximities from O(|V|3) to O(|Δ|) time for each update without loss of accuracy, where |Δ| (<<|V|²) is the number of affected proximities. (2) To avoid O(|V|²) memory for all pairs of outputs, we also devise efficient partitioning techniques for our dynamic model, which can compute all pairs of proximities segment-wisely within O(l|V|) is a user-controlled trade-off between  memory an I/O costs. (3) For bulk update, we also devise aggregation and hashing methods, which can discard many unneccessary updates further and handle chuncks of unit updates simultaneously. Our experimental results on various datasets demonstrate that our methods can be 1-2 orders of magnitude faster than other competitors while securing scalability and exactness...