  Networks with fat-tailed degree distributions are omnipresent across many
scientific disciplines. Such systems are characterized by so-called hubs,
specific nodes with high numbers of connections to other nodes. By this
property, they are expected to be key to the collective network behavior, e.g.,
in Ising models on such complex topologies. This applies in particular to the
transition into a globally ordered network state, which thereby proceeds in a
hierarchical fashion, and with a non-trivial local structure. Standard
mean-field theory of Ising models on scale-free networks underrates the
presence of the hubs, while nevertheless providing remarkably reliable
estimates for the onset of global order. Here, we expose that a spurious
self-feedback effect, inherent to mean-field theory, underlies this apparent
paradox. More specifically, we demonstrate that higher order interaction
effects precisely cancel the self-feedback on the hubs, and we expose the
importance of hubs for the distinct onset of local versus global order in the
network. Due to the generic nature of our arguments, we expect the mechanism
that we uncover for the archetypal case of Ising networks of the
Barab\'asi-Albert type to be also relevant for other systems with a strongly
hierarchical underlying network structure.
