  We investigate the emergence of complex dynamics in networks with
heavy-tailed connectivity by developing a non-Hermitian random matrix theory.
We uncover the existence of an extended critical regime of spatially
multifractal fluctuations between the quiescent and active phases. This
multifractal critical phase combines features of localization and
delocalization and differs from the edge of chaos in classical networks by the
appearance of universal hallmarks of Anderson criticality over an extended
region in phase space. We show that the rich nonlinear response properties of
the extended critical regime can account for a variety of neural dynamics such
as the diversity of timescales, providing a computational advantage for
persistent classification in a reservoir setting.

Disordered Systems and Neural Networks