  In this paper, we study network linear equations subject to digital
communications with a finite data rate, where each node is associated with one
equation from a system of linear equations. Each node holds a dynamic state and
interacts with its neighbors through an undirected connected graph, where along
each link the pair of nodes share information. Due to the data-rate constraint,
each node builds an encoder-decoder pair, with which it produces transmitted
message with a zooming-in finite-level uniform quantizer and also generates
estimates of its neighbors' states from the received signals. We then propose a
distributed quantized algorithm and show that when the network linear equations
admit a unique solution, each node's state is driven to that solution
exponentially. We further establish the asymptotic rate of convergence, which
shows that a larger number of quantization levels leads to a faster convergence
rate but is still fundamentally bounded by the inherent network structure and
the linear equations. When a unique least-squares solution exists, we show that
the algorithm can compute such a solution with a suitably selected time-varying
step size inherited from the encoder and zooming-in quantizer dynamics. In both
cases, a minimal data rate is shown to be enough for guaranteeing the desired
convergence when the step sizes are properly chosen. These results assure the
applicability of various network linear equation solvers in the literature when
peer-to-peer communication is digital.
