We propose and analyze two algorithms for maintaining approximate Personalized PageRank (PPR) vectors on a dynamic graph, where edges are added or deleted. Our algorithms are natural dynamic versions of two known local variations of power iteration. One, Forward Push, propagates probability mass forwards along edges from a source node, while the other, Reverse Push, propagates local changes backwards along edges from a target. In both variations, we maintain an invariant between two vectors, and when an edge is updated, our algorithm first modifies the vectors to restore the invariant, then performs any needed local push operations to restore accuracy.

For Reverse Push, we prove that for an arbitrary directed graph in a random edge model, or for an arbitrary undirected graph, given a uniformly random target node t, the cost to maintain a PPR vector to t of additive error ε as k edges are updated is O(k + d/ε), where d is the average degree of the graph. This is O(1) work per update, plus the cost of computing a reverse vector once on a static graph. For Forward Push, we show that on an arbitrary undirected graph, given a uniformly random start node s, the cost to maintain a PPR vector from s of degree-normalized error ε as k edges are updated is O(k + 1/ε), which is again O(1) per update plus the cost of computing a PPR vector once on a static graph.