Almost sure convergence to a constant for a mean-aggregated term language

With motivation coming from machine learning, we define a term language on graphs generalising many graph neural networks. Our main result is that the closed terms of this language converge almost surely to constants. This probabilistic result holds for Erdős–Rényi graphs for a variety of sparsity levels, as well as the Barabási–Albert preferential attachment graph distribution. The key technique is a kind of almost sure quantifier elimination. A natural extension of this language generalises first-order logic, and a similar convergence result can be obtained there.