Detection Thresholds for the $\beta$-Model on Sparse Graphs

TL;DR

This paper investigates the precise detection thresholds for sparse signals in $eta$-models on random graphs, revealing how graph and signal sparsity influence detectability and establishing sharp phase transitions.

Contribution

It provides new sharp thresholds for detection in $eta$-models, especially in sparse regimes, and introduces an improved Higher Criticism Test with proven optimality.

Findings

Detection thresholds depend on graph and signal sparsity levels.

Extreme graph sparsity renders all tests asymptotically powerless.

Denser graphs allow for sharp detection thresholds with near-optimal tests.

Abstract

In this paper we study sharp thresholds for detecting sparse signals in $\beta$-models for potentially sparse random graphs. The results demonstrate interesting interplay between graph sparsity, signal sparsity, and signal strength. In regimes of moderately dense signals, irrespective of graph sparsity, the detection thresholds mirror corresponding results in independent Gaussian sequence problems. For sparser signals, extreme graph sparsity implies that all tests are asymptotically powerless, irrespective of the signal strength. On the other hand, sharp detection thresholds are obtained, up to matching constants, on denser graphs. The phase transition mentioned above are sharp. As a crucial ingredient, we study a version of the Higher Criticism Test which is provably sharp up to optimal constants in the regime of sparse signals. The theoretical results are further verified by numerical simulations.