Physics, Stability and Dynamics of Supply Networks
  We show how to treat supply networks as physical transport problems governed by balance equations and equations for the adaptation of production speeds. Although the non-linear behaviour is different, the linearized set of coupled differential equations is formally related to those of mechanical or electrical oscillator networks. Supply networks possess interesting new features due to their complex topology and directed links. We derive analytical conditions for absolute and convective instabilities. The empirically observed "bull-whip effect" in supply chains is explained as a form of convective instability based on resonance effects. Moreover, it is generalized to arbitrary supply networks. Their related eigenvalues are usually complex, depending on the network structure (even without loops). Therefore, their generic behavior is characterized by oscillations. We also show that regular distribution networks possess two negative eigenvalues only, but perturbations generate a spectrum of complex eigenvalues. 
Dirk Helbing, Stefan Lammer, Thomas Seidel, Petr Seba, and Tadeusz   Platkowski