  We consider a seller offering a large network of $N$ products over a time
horizon of $T$ periods. The seller does not know the parameters of the
products' linear demand model, and can dynamically adjust product prices to
learn the demand model based on sales observations. The seller aims to minimize
its pseudo-regret, i.e., the expected revenue loss relative to a clairvoyant
who knows the underlying demand model. We consider a sparse set of demand
relationships between products to characterize various connectivity properties
of the product network. In particular, we study three different sparsity
frameworks: (1) $L_0$ sparsity, which constrains the number of connections in
the network, and (2) off-diagonal sparsity, which constrains the magnitude of
cross-product price sensitivities, and (3) a new notion of spectral sparsity,
which constrains the asymptotic decay of a similarity metric on network nodes.
We propose a dynamic pricing-and-learning policy that combines the
optimism-in-the-face-of-uncertainty and PAC-Bayesian approaches, and show that
this policy achieves asymptotically optimal performance in terms of $N$ and
$T$. We also show that in the case of spectral and off-diagonal sparsity, the
seller can have a pseudo-regret linear in $N$, even when the network is dense.
