  The standard definition of genuine multipartite entanglement stems from the
need to assess the quantum control over an ever-growing number of quantum
systems. We argue that this notion is easy to hack: in fact, a source capable
of distributing bipartite entanglement can, by itself, generate genuine
$k$-partite entangled states for any $k$. We propose an alternative definition
for genuine multipartite entanglement, whereby a quantum state is genuinely
network $k$-entangled if it cannot be produced by applying local
trace-preserving maps over several $k$-partite states distributed among the
parties, even with the aid of global shared randomness. We provide analytic and
numerical witnesses of genuine network entanglement, and we reinterpret many
past quantum experiments as demonstrations of this feature.
