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We consider a seller offering a large network of $N$ products over a timehorizon of $T$ periods. The seller does not know the parameters of theproducts' linear demand model, and can dynamically adjust product prices tolearn the demand model based on sales observations. The seller aims to minimizeits pseudo-regret, i.e., the expected revenue loss relative to a clairvoyantwho knows the underlying demand model. We consider a sparse set of demandrelationships between products to characterize various connectivity propertiesof the product network. In particular, we study three different sparsityframeworks: (1) $L_0$ sparsity, which constrains the number of connections inthe network, and (2) off-diagonal sparsity, which constrains the magnitude ofcross-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 theoptimism-in-the-face-of-uncertainty and PAC-Bayesian approaches, and show thatthis policy achieves asymptotically optimal performance in terms of $N$ and$T$. We also show that in the case of spectral and off-diagonal sparsity, theseller can have a pseudo-regret linear in $N$, even when the network is dense.

Dynamic Pricing and Demand Learning on a Large Network of Products: A PAC-Bayesian Approach