## Abstract

We study the problem of detecting the presence of an underlying high-dimensional geometric structure in a random graph. Under the null hypothesis, the observed graph is a realization of an Erdős-Rényi random graph G(n, p). Under the alternative, the graph is generated from the G(n, p, d) model, where each vertex corresponds to a latent independent random vector uniformly distributed on the sphere S^{d-1}, and two vertices are connected if the corresponding latent vectors are close enough. In the dense regime (i.e., p is a constant), we propose a near-optimal and computationally efficient testing procedure based on a new quantity which we call signed triangles. The proof of the detection lower bound is based on a new bound on the total variation distance between a Wishart matrix and an appropriately normalized GOE matrix. In the sparse regime, we make a conjecture for the optimal detection boundary. We conclude the paper with some preliminary steps on the problem of estimating the dimension in G(n, p, d).

Original language | English (US) |
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Pages (from-to) | 503-532 |

Number of pages | 30 |

Journal | Random Structures and Algorithms |

Volume | 49 |

Issue number | 3 |

DOIs | |

State | Published - Oct 1 2016 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics

## Keywords

- high-dimensional geometric structure
- hypothesis testing
- random geometric graphs
- random graphs
- signed triangles