  We analyze the asynchronous version of the DeGroot dynamics: In a connected
graph $G$ with $n$ nodes, each node has an initial opinion in $[0,1]$ and an
independent Poisson clock. When a clock at a node $v$ rings, the opinion at $v$
is replaced by the average opinion of its neighbors. It is well known that the
opinions converge to a consensus. We show that the expected time $\mathbb
E(\tau_\varepsilon)$ to reach $\varepsilon$-consensus is poly$(n)$ in
undirected graphs and in Eulerian digraphs, but for some digraphs of bounded
degree it is exponential.
  Our main result is that in undirected graphs and Eulerian digraphs, if the
degrees are uniformly bounded and the initial opinions are i.i.d., then
$\mathbb E(\tau_\varepsilon)=\text{polylog}(n)$ for every fixed
$\varepsilon>0$. We give sharp estimates for the variance of the limiting
consensus opinion, which measures the ability to aggregate information
(``wisdom of the crowd''). We also prove generalizations to non-reversible
Markov chains and infinite graphs. New results of independent interest on
fragmentation processes and coupled random walks are crucial to our analysis.
