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We present a Bayesian graph neural network (BGNN) that can estimate the weaklensing convergence ($\kappa$) from photometric measurements of galaxies alonga given line of sight. The method is of particular interest in stronggravitational time delay cosmography (TDC), where characterizing the "externalconvergence" ($\kappa_{\rm ext}$) from the lens environment and line of sightis necessary for precise inference of the Hubble constant ($H_0$). Startingfrom a large-scale simulation with a $\kappa$ resolution of $\sim$1$'$, weintroduce fluctuations on galaxy-galaxy lensing scales of $\sim$1$''$ andextract random sightlines to train our BGNN. We then evaluate the model on testsets with varying degrees of overlap with the training distribution. For eachtest set of 1,000 sightlines, the BGNN infers the individual $\kappa$posteriors, which we combine in a hierarchical Bayesian model to yieldconstraints on the hyperparameters governing the population. For a test fieldwell sampled by the training set, the BGNN recovers the population mean of$\kappa$ precisely and without bias, resulting in a contribution to the $H_0$error budget well under 1\%. In the tails of the training set with sparsesamples, the BGNN, which can ingest all available information about eachsightline, extracts more $\kappa$ signal compared to a simplified version ofthe traditional method based on matching galaxy number counts, which is limitedby sample variance. Our hierarchical inference pipeline using BGNNs promises toimprove the $\kappa_{\rm ext}$ characterization for precision TDC. Theimplementation of our pipeline is available as a public Python package, Node toJoy.

Hierarchical Inference of the Lensing Convergence from Photometric Catalogs with Bayesian Graph Neural Networks