Core-halo instability transition in complex systems Seth Lloyd Department of Mechanical Engineering Massachusetts Institute of Technology MIT 3-160, Cambridge, MA 02139 USA Santa Fe Institute 1399 Hyde Park Road, Santa Fe, NM 87501 USA Abstract: This paper proves a network instability theorem. As one adds interactions be- tween subystems in a complex system, structured or random, a threshold of connectivity is reached beyond which the overall dynamics inevitably goes unstable. The threshold occurs at the point at which flows and interactions between subsystems (`surface' effects) overwhelm internal stabilizing dynamics (`volume' effects). The theorem is used to identify instability thresholds in systems that possess a core-halo structure, including the grave- thermal catastrophe — i.e., star collapse and explosion — and the interbank payment net- work. The same dynamical instability gives rise both to gravitational collapse and to financial collapse. A wide variety of work addresses the stability of complex systems made up of networks of interacting subsystems [1-5]. A key ingredient of stability is network connectivity [5]. One of the best-known results in this field is May's theorem that differential equations described by random networks undergo a transition from stable to unstable behavior at a critical value of their connectivity [4]. Networks that occur in nature are rarely random, 1 EFTA00607095