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Brain structure reflects the influence of evolutionary processes that pit the costs of its anatomical wiring against the computational advantages conferred by its complexity. We show that cost-neutral ‘mutations’ of the human connectome almost inevitably degrade its complexity and disconnect high-strength connections to prefrontal network hubs. Conversely, restoring the peripheral location and strong connectivity of empirically observed hubs confers a wiring cost that the brain appears to minimize. Progressive cost-neutral randomization yields daughter networks that differ substantially from one another and results in a topologically unstable phenomenon consistent with a phase transition in complex systems. The fragility of hubs to disconnection shows a significant association with the acceleration of gray matter loss in schizophrenia. Together with effects on wiring cost, we suggest that fragile prefrontal hub …