  We analyze the effect of interference on the convergence rate of average
consensus algorithms, which iteratively compute the measurement average by
message passing among nodes. It is usually assumed that these algorithms
converge faster with a greater exchange of information (i.e., by increased
network connectivity) in every iteration. However, when interference is taken
into account, it is no longer clear if the rate of convergence increases with
network connectivity. We study this problem for randomly-placed
consensus-seeking nodes connected through an interference-limited network. We
investigate the following questions: (a) How does the rate of convergence vary
with increasing communication range of each node? and (b) How does this result
change when each node is allowed to communicate with a few selected far-off
nodes? When nodes schedule their transmissions to avoid interference, we show
that the convergence speed scales with $r^{2-d}$, where $r$ is the
communication range and $d$ is the number of dimensions. This scaling is the
result of two competing effects when increasing $r$: Increased schedule length
for interference-free transmission vs. the speed gain due to improved
connectivity. Hence, although one-dimensional networks can converge faster from
a greater communication range despite increased interference, the two effects
exactly offset one another in two-dimensions. In higher dimensions, increasing
the communication range can actually degrade the rate of convergence. Our
results thus underline the importance of factoring in the effect of
interference in the design of distributed estimation algorithms.
