Planted Bipartite Graph Detection

We consider the task of detecting a hidden bipartite subgraph in a given random graph. This is formulated as a hypothesis testing problem, under the null hypothesis, the graph is a realization of an Erdős-Rényi random graph over n vertices with edge density q. Under the alternative, there exists a p...

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Vydáno v:IEEE transactions on information theory Ročník 70; číslo 6; s. 4319 - 4334
Hlavní autoři: Rotenberg, Asaf, Huleihel, Wasim, Shayevitz, Ofer
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York IEEE 01.06.2024
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:
Algorithms
Apexes
Bipartite graph
Computational modeling
Density
Detection
Graph theory
hidden structures
Hypotheses
Hypothesis testing
Image edge detection
Information theory
Lower bounds
Null hypothesis
Phase transitions
Polynomials
Stars
statistical and computational limits
statistical inference
Task analysis
Testing
ISSN:0018-9448, 1557-9654
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Shrnutí:We consider the task of detecting a hidden bipartite subgraph in a given random graph. This is formulated as a hypothesis testing problem, under the null hypothesis, the graph is a realization of an Erdős-Rényi random graph over n vertices with edge density q. Under the alternative, there exists a planted <inline-formula> <tex-math notation="LaTeX">k_{ \mathsf {R}} \times k_{ \mathsf {L}} </tex-math></inline-formula> bipartite subgraph with edge density <inline-formula> <tex-math notation="LaTeX">p>q </tex-math></inline-formula>. We characterize the statistical and computational barriers for this problem. Specifically, we derive information-theoretic lower bounds, and design and analyze optimal algorithms matching those bounds, in both the dense regime, where <inline-formula> <tex-math notation="LaTeX">p,q = \Theta \left ({1}\right) </tex-math></inline-formula>, and the sparse regime where <inline-formula> <tex-math notation="LaTeX">p,q = \Theta \left ({n^{-\alpha }}\right), \alpha \in \left ({0,2}\right] </tex-math></inline-formula>. We also consider the problem of testing in polynomial-time. As is customary in similar structured high-dimensional problems, our model undergoes an "easy-hard-impossible" phase transition and computational constraints penalize the statistical performance. To provide an evidence for this statistical computational gap, we prove computational lower bounds based on the low-degree conjecture, and show that the class of low-degree polynomials algorithms fail in the conjecturally hard region.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2024.3382228