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

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... nded 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.