Fast incremental and personalized PageRank

Abstract

In this paper, we analyze the efficiency of Monte Carlo methods for incremental computation of PageRank, personalized PageRank, and similar random walk based methods (with focus on SALSA), on large-scale dynamically evolving social networks. We assume that the graph of friendships is stored in distributed shared memory, as is the case for large social networks such as Twitter.For global PageRank, we assume that the social network has n nodes, and m adversarially chosen edges arrive in a random order. We show that with a reset probability of , the expected total work needed to maintain an accurate estimate (using the Monte Carlo method) of the PageRank of every node at all times is O( n ln m 2 ). This is significantly better than all known bounds for incremental PageRank. For instance, if we naively recompute the PageRanks as each edge arrives, the simple power iteration method needs Ω( ) total time and the Monte Carlo method needs O(mn/) total time; both are prohibitively expensive. We also show that we can handle deletions equally efficiently.We then study the computation of the top k personalized PageRanks starting from a seed node, assuming that personalized PageRanks follow a power-law with exponent α < 1. We show that if we store R > q ln n random walks starting from every node for large enough constant q (using the approach outlined for global PageRank), then the expected number of calls made to the distributed social network database is O(k/(R (1−α)/α )). We also present experimental results from the social networking site, Twitter, ver- * Work done while interning at Twitter † Work done while at Twitter.

Fast Distributed PageRank Computation

“…The correctness of the above approximation follows directly from the main result of [2] (see Algorithm 4 and Theorem 1) and also from [4] (Theorem 1). In particular, it is mentioned in [2,4] that the approximate PageRankvalue is quite good even for K = 1.…”

“…Let be a small constant which is fixed ( is called the reset probability, i.e., with probability , the random walk starts from a node chosen uniformly at random among all nodes in the network). The PageRank vector of a graph (e.g., see [2,4,5,9]) is the stationary distribution vector π of the following special type of random walk: at each step of the random walk, with probability the walk starts from a randomly chosen node and with remaining probability 1 − , the walk follows a randomly chosen outgoing (neighbor) edge from the current node and moves to that neighbor. 3 Therefore the PageRank transition matrix on the state space (or vertex set) V can be written as…”

“…Notice that nK is the (expected) total number of visits over all nodes of all the nK walks. The above idea of counting the number of visits is a standard technique to approximate PageRanks (see e.g., [2,4]). We want to note that our algorithm in this section does not require any direct communication between non-neighbors.…”

“…While there has been recent work on performing random walks efficiently in distributed networks [4,9], surprisingly, little provable results are known towards efficient distributed computation of PageRanks. This is perhaps because the traditional method of computing PageRanks is to apply iterative methods i.e., do matrix-vector multiplications till (near)-convergence.…”

“…Algorithm 1), that computes PageRanks accurately in O log n rounds with high probability 1 , where n is the number of nodes in the network and is the random reset probability in the PageRank random walk [2,4,9]. Our algorithm works for any arbitrary network (directed as well as undirected).…”