## Abstract

Distances in a network capture relations between nodes and are the basis of centrality, similarity, and influence measures.Often, however, the relevance of a node $u$ to a node $v$ is more precisely measured not by the magnitude of the distance, but by the number of nodes that are closer to $v$ than $u$. That is, by the {\em rank} of $u$ in an ordering of nodes by increasing distance from $v$.

We identify and address fundamental challenges in rank-based graph mining. We first consider single-source computation of reverse-ranks and design a ``Dijkstra-like'' algorithm which computes nodes in order of increasing approximate reverse rank while only traversing edges adjacent to returned nodes. We then define {\em reverse-rank influence}, which naturally extends reverse nearest neighbors influence [Korn and Muthukrishnan 2000] and builds on a well studied distance-based influence. We present near-linear algorithms for greedy approximate reverse-rank influence maximization. The design relies on our single-source algorithm. Our algorithms utilize near-linear preprocessing of the network to compute all-distance sketches. As a contribution of independent interest, we present a novel algorithm for computing these sketches, which have many other applications, on multi-core architectures.

We complement our algorithms by establishing the hardness of computing {\em exact} reverse-ranks for a single source and {\em exact} reverse-rank influence. This implies that when using near-linear algorithms, the small relative errors we obtain are the best we can currently hope for.

Finally, we conduct an experimental evaluation on graphs with tens of millions of edges, demonstrating both scalability and accuracy.