## Abstract

We study the two inference problems of detecting and recovering an isolated community of <italic>general</italic> structure planted in a random graph. The detection problem is formalized as a hypothesis testing problem, where under the null hypothesis, the graph is a realization of an Erdős-Rényi random graph <inline-formula><tex-math notation="LaTeX">$\mathcal{G(n,q)}$</tex-math></inline-formula> with edge density <inline-formula><tex-math notation="LaTeX">$q\in (0,1)$</tex-math></inline-formula>; under the alternative, there is an unknown structure <inline-formula><tex-math notation="LaTeX">$\Gamma _{k}$</tex-math></inline-formula> on <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> nodes, planted in <inline-formula><tex-math notation="LaTeX">$\mathcal{G(n,q)}$</tex-math></inline-formula>, such that it appears as an <italic>induced subgraph</italic>. In case of a successful detection, we are concerned with the task of recovering the corresponding structure. For these problems, we investigate the fundamental limits from both the statistical and computational perspectives. Specifically, we derive lower bounds for detecting/recovering the structure <inline-formula><tex-math notation="LaTeX">$\Gamma _{k}$</tex-math></inline-formula> in terms of the parameters <inline-formula><tex-math notation="LaTeX">$(n,k,q)$</tex-math></inline-formula>, as well as certain properties of <inline-formula><tex-math notation="LaTeX">$\Gamma _{k}$</tex-math></inline-formula>, and exhibit computationally unbounded optimal algorithms that achieve these lower bounds. We also consider the problem of testing in polynomial-time. As is customary in many similar structured high-dimensional problems, our model undergoes an “easy-hard-impossible” phase transition and computational constraints can severely penalize the statistical performance. To provide an evidence for this phenomenon, we show that the class of low-degree polynomials algorithms match the statistical performance of the polynomial-time algorithms we develop.

Original language | English |
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Pages (from-to) | 1-13 |

Number of pages | 13 |

Journal | IEEE Transactions on Signal and Information Processing over Networks |

DOIs | |

State | Accepted/In press - 2022 |

## Keywords

- Computational modeling
- Hidden structures
- Image edge detection
- Inference algorithms
- Information processing
- random graphs and networks
- Stars
- statistical and computational limits
- statistical inference
- Task analysis
- Testing