  We examine how well people learn when information is noisily relayed from
person to person; and we study how communication platforms can improve learning
without censoring or fact-checking messages. We analyze learning as a function
of social network depth (how many times information is relayed) and breadth
(the number of relay chains accessed). Noise builds up as depth increases, so
learning requires greater breadth. In the presence of mutations (deliberate or
random) and transmission failures of messages, we characterize sharp thresholds
for breadths above which receivers learn fully and below which they learn
nothing. When there is uncertainty about mutation rates, optimizing learning
requires either capping depth, or if that is not possible, limiting breadth by
capping the number of people to whom someone can forward a message. Limiting
breadth cuts the number of messages received but also decreases the fraction
originating further from the receiver, and so can increase the signal to noise
ratio. Finally, we extend our model to study learning from message survival:
e.g., people are more likely to pass messages with one conclusion than another.
We find that as depth grows, all learning comes from either the total number of
messages received or from the content of received messages, but the learner
does not need to pay attention to both.
