Detectability under the edge-density-matched null distribution in regime III

Determine whether the noisy random hypergraph projection can be detected against the modified Erdős–Rényi null distribution whose edge density matches the expected edge density of the planted noisy projection, in the parameter regime where detection is possible against the simple null distribution but the detectability status under the modified null distribution is unknown.

Background

The paper studies detection of a noisy graph projection of a random d-uniform hypergraph. Its main detection threshold is established for a simple Erdős–Rényi null distribution with edge density q. The authors also consider a modified null distribution with edge density matched to the expected edge density of the planted model, motivated by whether this choice could close the detection–reconstruction gap.

The paper proves in the appendix that detection against the modified null distribution remains possible when both δ and β are below (d−1)/(d+1). Figure 1 nevertheless identifies a remaining parameter region, labeled regime III, in which detection against the simple null distribution is possible while detection against the better, edge-density-matched null distribution remains unresolved.