Cycles and dynamics on evolving signed digraphs

B.D. MacArthur, R.J. Sánchez-García, A. Ma'ayan

We consider a simple model which couples structural evolution of signed directed graphs to dynamic stability. The resulting systems self-organize to a complex dynamical state characterized by periods of stability punctuated by intermittent bursts of instability. By deriving analytical relationships between cyclic structure, cycle balance, and global stability in signed digraphs we show that the observed bursting dynamics result from the continual formation and breaking of transient directed cycles during the evolutionary process. At equilibrium bursts of instability develop heavy-tailed statistics suggesting a possible self-organized critical state.

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