## Abstract

We review the class of continuous latent space (statistical) models for network data, paying particular attention to the role of the geometry of the latent space. In these models, the presence/absence of network dyadic ties are assumed to be conditionally independent given the dyads' unobserved positions in a latent space. In this way, these models provide a probabilistic framework for embedding network nodes in a continuous space equipped with a geometry that facilitates the description of dependence between random dyadic ties. Specifically, these models naturally capture homophilous tendencies and triadic clustering, among other common properties of observed networks. In addition to reviewing the literature on continuous latent space models from a geometric perspective, we highlight the important role the geometry of the latent space plays on properties of networks arising from these models via intuition and simulation. Finally, we discuss results from spectral graph theory that allow us to explore the role of the geometry of the latent space, independent of network size. We conclude with conjectures about how these results might be used to infer the appropriate latent space geometry from observed networks.

Original language | English |
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Pages (from-to) | 428-453 |

Number of pages | 26 |

Journal | Statistical Science |

Volume | 34 |

Issue number | 3 |

DOIs | |

State | Published - Aug 1 2019 |

### Bibliographical note

Publisher Copyright:© Institute of Mathematical Statistics, 2019.

### Funding

Smith and Calder were partially supported by grants from the National Science Foundation (NSF DMS-1209161), the National Institutes of Health (NIH R01 HD088545) and The Ohio State University Institute for Population Research (NIH P2CHD058484).

Funders | Funder number |
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National Science Foundation Arctic Social Science Program | 1209161, DMS-1209161 |

National Science Foundation Arctic Social Science Program | |

National Institutes of Health (NIH) | R01 HD088545 |

National Institutes of Health (NIH) | |

Institute for Population Research, Ohio State University | P2CHD058484 |

Institute for Population Research, Ohio State University |

## Keywords

- Geometric curvature
- Graph Laplacian
- Latent variable
- Network model

## ASJC Scopus subject areas

- Statistics and Probability
- General Mathematics
- Statistics, Probability and Uncertainty