  This work studies the operation of multi-agent networks engaged in binary
decision tasks, and derives performance expressions and performance operating
curves under challenging conditions with some revealing insights. One of the
main challenges in the analysis is that agents are only allowed to exchange
one-bit messages, and the information at each agent therefore consists of both
continuous and discrete components. Due to this mixed nature, the steady-state
distribution of the state of each agent cannot be inferred from direct
application of central limit arguments. Instead, the behavior of the continuous
component is characterized in integral form by using a log-characteristic
function, while the behavior of the discrete component is characterized by
means of an asymmetric Bernoulli convolution. By exploiting these results, the
article derives reliable approximate performance expressions for the network
nodes that match well with the simulated results for a wide range of system
parameters. The results also reveal an important interplay between continuous
adaptation under constant step-size learning and the binary nature of the
messages exchanged with neighbors.
