  We study an asynchronous online learning setting with a network of agents. At
each time step, some of the agents are activated, requested to make a
prediction, and pay the corresponding loss. The loss function is then revealed
to these agents and also to their neighbors in the network. Our results
characterize how much knowing the network structure affects the regret as a
function of the model of agent activations. When activations are stochastic,
the optimal regret (up to constant factors) is shown to be of order
$\sqrt{\alpha T}$, where $T$ is the horizon and $\alpha$ is the independence
number of the network. We prove that the upper bound is achieved even when
agents have no information about the network structure. When activations are
adversarial the situation changes dramatically: if agents ignore the network
structure, a $\Omega(T)$ lower bound on the regret can be proven, showing that
learning is impossible. However, when agents can choose to ignore some of their
neighbors based on the knowledge of the network structure, we prove a
$O(\sqrt{\overline{\chi} T})$ sublinear regret bound, where $\overline{\chi}
\ge \alpha$ is the clique-covering number of the network.
