Non-normal networks, a new paradigmatic concept that I have recently introduced [1, 2, 3, 4, 5, 6], the ubiquity in the real world, ranging from biological/ecological networks (neuronal, genetic, metabolic, proteins, food webs) to human-made networks (social media, scientific citations, transport, communications).

• An immediate consequence of non-normality is a transient growth observed in the linear regime of stable non-normal networks. With T. Carletti and R. Lambiotte, I have shown that this particular property can strongly impact all kinds of nonlinear phenomena, which at first order can be described by the Master Stability Function (MSF) [2, 5], thus making many of the classical spectral methods to fail.

• I have also illustrated the role of non-normal networks in ecology concerning species diversity and their coexistence in a competitive environment, including the Allee effect, which strongly depends on the trophic’s non-normal nature relations [2, 3]. We have recently shown that the non-normality of real complex systems can be a suitable new mechanism for emergent self-organization beyond the Turing instability [3, 4].

• On the other hand, it can be an "ad hoc" instrument for amplifying demographic fluctuations, which can eventually induce a new dynamical behavior [5]. This way, non-normality further strengthens the role of the finite-size effect in spatial patterning.

• Furthermore, with J.D. O’Brien and K.A. Oliveira, we show that in the ensemble of real-world networks, when some given level of non-normality is reached, leader nodes (with only incoming or outgoing degrees) simultaneously emerge [6].