  The organisation of a network in a maximal set of nodes having at least $k$
neighbours within the set, known as $k$-core decomposition, has been used for
studying various phenomena. It has been shown that nodes in the innermost
$k$-shells play a crucial role in contagion processes, emergence of consensus,
and resilience of the system. It is known that the $k$-core decomposition of
many empirical networks cannot be explained by the degree of each node alone,
or equivalently, random graph models that preserve the degree of each node
(i.e., configuration model). Here we study the $k$-core decomposition of some
empirical networks as well as that of some randomised counterparts, and examine
the extent to which the $k$-shell structure of the networks can be accounted
for by the community structure. We find that preserving the community structure
in the randomisation process is crucial for generating networks whose $k$-core
decomposition is close to the empirical one. We also highlight the existence,
in some networks, of a concentration of the nodes in the innermost $k$-shells
into a small number of communities.
