  Consensus dynamics support decentralized machine learning for data that is
distributed across a cloud compute cluster or across the internet of things. In
these and other settings, one seeks to minimize the time $\tau_\epsilon$
required to obtain consensus within some $\epsilon>0$ margin of error.
$\tau_\epsilon$ typically depends on the topology of the underlying
communication network, and for many algorithms $\tau_\epsilon$ depends on the
second-smallest eigenvalue $\lambda_2\in[0,1]$ of the network's normalized
Laplacian matrix: $\tau_\epsilon\sim\mathcal{O}(\lambda_2^{-1})$. Here, we
analyze the effect on $\tau_\epsilon$ of network community structure, which can
arise when compute nodes/sensors are spatially clustered, for example. We study
consensus machine learning over networks drawn from stochastic block models,
which yield random networks that can contain heterogeneous communities with
different sizes and densities. Using random matrix theory, we analyze the
effects of communities on $\lambda_2$ and consensus, finding that $\lambda_2$
generally increases (i.e., $\tau_\epsilon$ decreases) as one decreases the
extent of community structure. We further observe that there exists a critical
level of community structure at which $\tau_\epsilon$ reaches a lower bound and
is no longer limited by the presence of communities. We support our findings
with empirical experiments for decentralized support vector machines.
