  We characterize the computational power of neural networks that follow the
graph neural network (GNN) architecture, not restricted to aggregate-combine
GNNs or other particular types. We establish an exact correspondence between
the expressivity of GNNs using diverse activation functions and arithmetic
circuits over real numbers. In our results the activation function of the
network becomes a gate type in the circuit. Our result holds for families of
constant depth circuits and networks, both uniformly and non-uniformly, for all
common activation functions.
