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Planted Bipartite Graph Detection

Asaf Rotenberg, Wasim Huleihel, Ofer Shayevitz

TL;DR

Computational lower bounds based on the low-degree conjecture are proved, and it is shown that the class of low-degree polynomials algorithms fail in the conjecturally hard region.

Abstract

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 $k_{\mathsf{R}} \times k_{\mathsf{L}}$ bipartite subgraph with edge density $p>q$. 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 $p,q = Θ\left(1\right)$, and the sparse regime where $p,q = Θ\left(n^{-α}\right), α\in \left(0,2\right]$. 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.

Planted Bipartite Graph Detection

TL;DR

Computational lower bounds based on the low-degree conjecture are proved, and it is shown that the class of low-degree polynomials algorithms fail in the conjecturally hard region.

Abstract

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 vertices with edge density . Under the alternative, there exists a planted bipartite subgraph with edge density . 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 , and the sparse regime where . 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.
Paper Structure (21 sections, 5 theorems, 80 equations, 2 figures, 1 table)

This paper contains 21 sections, 5 theorems, 80 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main contributions.
  3. Related Work.
  4. Notation.
  5. Problem Formulation
  6. Main Results
  7. Statistical lower bounds.
  8. Upper bounds.
  9. Computational lower bounds.
  10. Phase diagrams.
  11. Proofs
  12. Proof of Theorem \ref{['thm:sparse']}
  13. Proof of Theorem \ref{['thm:pdbs']}
  14. Scan test
  15. Count test
  16. ...and 6 more sections

Key Result

Theorem 1

Consider the $\mathsf{PDBS}{\left(n,k_{\mathsf{R}},k_{\mathsf{L}},p,q\right)}$ detection problem in Problem prob:pdbs. Then, strong detection is impossible with $\mathsf{R}_n^\star>1/2$, if:

Figures (2)

  • Figure 1: Phase diagram for detecting a planted clique of size $k$, as a function of $k$.
  • Figure 2: Phase diagram for detecting a planted $k_{\mathsf{R}} \times k_{\mathsf{L}}$ complete bipartite.

Theorems & Definitions (9)

  • Definition 1: Maximal subgraph density
  • Definition 2: Strong detection
  • Theorem 1: Statistical lower bounds
  • Theorem 2: Algorithmic upper bounds
  • Lemma 1: Optimally of $\mathsf{L}_{n,\leq \mathsf{D}}$ hopkins2017bayesianhopkins2017powerDmitriy19
  • Conjecture 1: Low-degree conj., informal
  • Theorem 3
  • Corollary 1: Computational lower bound
  • Remark 1