  The emergence of order in nature manifests in different phenomena, with
synchronization being one of the most representative examples. Understanding
the role played by the interactions between the constituting parts of a complex
system in synchronization has become a pivotal research question bridging
network science and dynamical systems. Particular attention has been paid to
the emergence of chimera states, where subsets of synchronized oscillations
coexist with asynchronous ones. Such coexistence of coherence and incoherence
is a perfect example where order and disorder can persist in a long-lasting
regime. Although considerable progress has been made in recent years to
understand such coherent and (coexisting) incoherent states, how they manifest
in real-world networks remains to be addressed. Based on a symmetry-breaking
mechanism, in this paper, we shed light on the role that non-normality, a
ubiquitous structural property of real networks, has in the emergence of
several diverse dynamical phenomena, e.g., amplitude chimeras or oscillon
patterns. Specifically, we demonstrate that the prevalence of source or leader
nodes in networks leads to the manifestation of phase chimera states.
Throughout the paper, we emphasize that non-normality poses ongoing challenges
to global synchronization and is instrumental in the emergence of chimera
states.
