  We present an implicit likelihood approach to quantifying cosmological
information over discrete catalogue data, assembled as graphs. To do so, we
explore cosmological parameter constraints using mock dark matter halo
catalogues. We employ Information Maximising Neural Networks (IMNNs) to
quantify Fisher information extraction as a function of graph representation.
We a) demonstrate the high sensitivity of modular graph structure to the
underlying cosmology in the noise-free limit, b) show that graph neural network
summaries automatically combine mass and clustering information through
comparisons to traditional statistics, c) demonstrate that networks can still
extract information when catalogues are subject to noisy survey cuts, and d)
illustrate how nonlinear IMNN summaries can be used as asymptotically optimal
compressed statistics for Bayesian simulation-based inference. We 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, and demonstrate that the networks automatically combine mass and
clustering information. This work utilises a new IMNN implementation over graph
data in Jax, which can take advantage of either numerical or
auto-differentiability. We also show that graph IMNNs successfully compress
simulations away from the fiducial model at which the network is fitted,
indicating a promising alternative to n-point statistics in catalogue
simulation-based analyses.
