  Ranking on large-scale graphs plays a fundamental role in many high-impact
application domains, ranging from information retrieval, recommender systems,
sports team management, biology to neuroscience and many more. PageRank,
together with many of its random walk based variants, has become one of the
most well-known and widely used algorithms, due to its mathematical elegance
and the superior performance across a variety of application domains. Important
as it might be, state-of-the-art lacks an intuitive way to explain the ranking
results by PageRank (or its variants), e.g., why it thinks the returned top-k
webpages are most important ones in the entire graph; why it gives a higher
rank to actor John than actor Smith in terms of their relevance w.r.t. a
particular movie? In order to answer these questions, this paper proposes a
paradigm shift for PageRank, from identifying which nodes are most important to
understanding why the ranking algorithm gives a particular ranking result. We
formally define the PageRank auditing problem, whose central idea is to
identify a set of key graph elements (e.g., edges, nodes, subgraphs) with the
highest influence on the ranking results. We formulate it as an optimization
problem and propose a family of effective and scalable algorithms (AURORA) to
solve it. Our algorithms measure the influence of graph elements and
incrementally select influential elements w.r.t. their gradients over the
ranking results. We perform extensive empirical evaluations on real-world
datasets, which demonstrate that the proposed methods (AURORA) provide
intuitive explanations with a linear scalability.
