# The Cosmic Graph: Optimal Information Extraction from Large-Scale Structure using Catalogues

**Authors:** T. Lucas Makinen, Tom Charnock, Pablo Lemos, Natalia Porqueres, Alan Heavens, Benjamin D. Wandelt  
**arXiv:** [2207.05202](https://arxiv.org/abs/2207.05202)  
**Date:** 2022-07-11  
**Summary page:** https://www.summarizepaper.com/en/paper/2207.05202v1/

## Key points

- Implicit likelihood approach to quantifying cosmological information over discrete catalogue data
- Utilization of mock dark matter halo catalogues and Information Maximizing Neural Networks (IMNNs)
- High sensitivity of modular graph structure to underlying cosmology in noise-free limit
- Automatic combination of mass and clustering information through comparisons to traditional statistics
- Extraction of information from noisy survey cuts using graph neural networks
- Nonlinear IMNN summaries as asymptotically optimal compressed statistics for Bayesian implicit likelihood inference
- Reduction of parameter constraints with small halo catalogues by a factor of 42 over two-point correlation function
- Introduction of new IMNN implementation over graph data in Jax, taking advantage of numerical or auto-differentiability
- Successful compression of simulations far from the fiducial model using graph IMNNs
- Limitations of traditional 2-point statistics in catalogue-based analyses and proposal for nonlinear IMNN summaries as informative alternatives
- Exploration of cosmological information extraction from halo catalogues represented as graphs
- Superiority of nonlinear graph neural network summaries over traditional 2-point statistics
- Usability of network-produced summaries for simulation-based inference.

## Summary

In this study, the authors present an implicit likelihood approach to quantifying cosmological information over discrete catalogue data, specifically focusing on graphs. They utilize mock dark matter halo catalogues and employ Information Maximizing Neural Networks (IMNNs) to extract Fisher information as a function of graph representation. The authors demonstrate the high sensitivity of modular graph structure to the underlying cosmology in the noise-free limit. They show that networks automatically combine mass and clustering information through comparisons to traditional statistics. Additionally, they demonstrate that graph neural networks can still extract information even when catalogues are subject to noisy survey cuts. One key finding is that nonlinear IMNN summaries can be used as asymptotically optimal compressed statistics for Bayesian implicit likelihood inference. The authors reduce the area of joint $\Omega_m, \sigma_8$ parameter constraints with small ($\sim$100 object) halo catalogues by a factor of 42 over the two-point correlation function. This highlights the ability of the networks to automatically combine mass and clustering information. The study also introduces a new IMNN implementation over graph data in Jax, which can take advantage of numerical or auto-differentiability. The authors show that graph IMNNs successfully compress simulations far from the fiducial model at which the network is fitted, indicating a promising alternative to $n$-point statistics in catalogue-based analyses. In terms of context, modern cosmological analyses typically focus on obtaining theory and parameter constraints from compressed summary statistics obtained from field data such as the Cosmic Microwave Background or weak lensing mass-maps. However, data collected by telescopes is often instantly compressed into discrete catalogues of sources like galaxies and their underlying dark matter halos or cosmic voids. Analyses of these catalogues usually focus on 2-point information in real or Fourier space. The authors suggest that these statistics are only sufficient when the underlying field is Gaussian. To address this limitation, they propose using nonlinear IMNN summaries as informative summaries in Approximate Bayesian Computation (ABC) or density estimation schemes. In conclusion, this study explores cosmological information extraction from halo catalogues represented as graphs. The authors demonstrate the superiority of nonlinear graph neural network summaries over traditional 2-point statistics, showing that mass information increases the information yield and decreases sampling variance. They also highlight the usability of these network-produced summaries for simulation-based inference.