The Power of Local Information in PageRank

How large a fraction of a graph must one explore to rank a small set of nodes according to their PageRank scores? We show that the answer is quite nuanced, and depends crucially on the interplay between the correctness guarantees one requires and the way one can access the graph. On the one hand, assuming the graph can be accessed only via "natural" exploration queries that reveal small pieces of its topology, we prove that deterministic and Las Vegas algorithms must in the worst case perform $n - o(n)$ queries and explore essentially the entire graph, independently of the specific types of query employed. On the other hand we show that, depending on the types of query available, Monte Carlo algorithms can perform asymptotically better: if allowed to both explore the local topology around single nodes and access nodes at random in the graph they need $\Omega(n^{2/3})$ queries in the worst case, otherwise they still need $\Omega(n)$ queries similarly to Las Vegas algorithms. All our bounds generalize and tighten those already known, cover the different types of graph exploration queries appearing in the literature, and immediately apply also to the problem of approximating the PageRank score of single nodes.