Time-Optimal Directed q-Analysis

Abstract: Directed q-analysis is a recent extension of q-analysis, an established method for extracting structure from networks, to directed graphs. Until recently, a lack of efficient algorithms heavily restricted the application of this technique: Previous approaches scale with the square of the input size, which is also the maximal size of the output, rendering such approaches worst-case optimal. In practice, output sizes of relevant networks are usually far from the worst case, a fact that could be exploited by an (efficient) output-sensitive algorithm. We develop such an algorithm and formally describe it in detail. The key insight, obtained by carefully studying various approaches to directed q-analysis and how they relate to each other, is that inverting the order of computation leads to significant complexity gains. Targeted precomputation and caching tactics further reduce the introduced overhead, enough to achieve (under mild assumptions) a time complexity that is linear in output size. The resulting algorithm for performing directed q-analysis is shown to be time-optimal.