  This work studies a well-known shared-cache coded caching scenario where each
cache can serve an arbitrary number of users, analyzing the case where there is
some knowledge about such number of users (i.e., the topology) during the
content placement phase. Under the assumption of regular placement and a
cumulative cache size that can be optimized across the different caches, we
derive the fundamental limits of performance by introducing a novel cache-size
optimization and placement scheme and a novel information-theoretic converse.
The converse employs new index coding techniques to bypass traditional
uniformity requirements, thus finely capturing the heterogeneity of the
problem, and it provides a new approach to handle asymmetric settings. The new
fundamental limits reveal that heterogeneous topologies can in fact outperform
their homogeneous counterparts where each cache is associated to an equal
number of users. These results are extended to capture the scenario of
topological uncertainty where the perceived/estimated topology does not match
the true network topology. This scenario is further elevated to the stochastic
setting where the user-to-cache association is random and unknown, and it is
shown that the proposed scheme is robust to such noisy or inexact knowledge on
the topology.
