Statistical inference of a ranked community in a directed graph

TL;DR

This paper investigates the detection and recovery of a hidden ranked community within a directed graph, establishing thresholds for when such tasks are statistically and computationally feasible, and exploring various regimes including full tournaments and small ordered cliques.

Contribution

It introduces a novel model for detecting ranked communities in directed graphs and derives exact statistical and computational thresholds across different scaling regimes.

Findings

Established thresholds for detection and recovery in the planted ranked community model.

Identified regimes with statistical-computational gaps and detection-recovery gaps.

Provided detailed analysis for special cases like full tournaments and small ordered cliques.

Abstract

We study the problem of detecting or recovering a planted ranked subgraph from a directed graph, an analog for directed graphs of the well-studied planted dense subgraph model. We suppose that, among a set of $n$ items, there is a subset $S$ of $k$ items having a latent ranking in the form of a permutation $\pi$ of $S$, and that we observe a fraction $p$ of pairwise orderings between elements of $\{1, \dots, n\}$ which agree with $\pi$ with probability $\frac{1}{2} + q$ between elements of $S$ and otherwise are uniformly random. Unlike in the planted dense subgraph and planted clique problems where the community $S$ is distinguished by its unusual density of edges, here the community is only distinguished by the unusual consistency of its pairwise orderings. We establish computational and statistical thresholds for both detecting and recovering such a ranked community. In the log-density setting where $k$, $p$, and $q$ all scale as powers of $n$, we establish the exact thresholds in the associated exponents at which detection and recovery become statistically and computationally feasible. These regimes include a rich variety of behaviors, exhibiting both statistical-computational and detection-recovery gaps. We also give finer-grained results for two extreme cases: (1) $p = 1$, $k = n$, and $q$ small, where a full tournament is observed that is weakly correlated with a global ranking, and (2) $p = 1$, $q = \frac{1}{2}$, and $k$ small, where a small "ordered clique" (totally ordered directed subgraph) is planted in a random tournament.