Average consensus algorithms are gossiping protocols for averaging original sensor measurements via near neighbor communications. In this paper, we consider the average consensus algorithm under communication rate constraints. Without any communication rate restrictions, the algorithm ideally allows every node state to converge to the initial average in the limit. Noting that brute force quantization does not guarantee convergence due to error propagation effects, in our recent work we proposed two source coding methods which use side information (predictive coding and Wyner-Ziv coding) to achieve convergence with vanishing quantization rates in the case of block coding. In this work, we focus on a simplified predictive coding scheme with variable quantization rates over the iterations and on a communication network with regular topology. We characterize the asymptotic rate which allows to achieve a bounded convergence in terms of the initial conditions (i. e, the rate at the first iteration, and the initial state correlation), and the connectivity of the network. Moreover, we study the optimal rate allocation among the average consensus iterations subject to the constraints that the total number of quantization bits is fixed.