  In this paper, we study distributed solvers for network linear equations over
a network with node-to-node communication messages compressed as scalar values.
Our key idea lies in a dimension compression scheme including a dimension
compressing vector that applies to individual node states to generate a
real-valued message for node communication as an inner product, and a data
unfolding step in the local computations where the scalar message is plotted
along the subspace generated by the compression vector. We first present a
compressed average consensus flow that relies only on such scalar
communication, and show that exponential convergence can be achieved with well
excited signals for the compression vector. We then employ such a compressed
consensus flow as a fundamental consensus subroutine to develop distributed
continuous-time and discrete-time solvers for network linear equations, and
prove their exponential convergence properties under scalar node
communications. With scalar communications, a direct benefit would be the
reduced node-to-node communication channel capacity requirement for distributed
computing. Numerical examples are presented to illustrate the effectiveness of
the established theoretical results.
