Network-induced oscillatory behavior in material flow networks and irregular business cycles

Network theory is rapidly changing our understanding of complex systems, but the relevance of topological features for the dynamic behavior of metabolic networks, food webs, production systems, information networks, or cascade failures of power grids remains to be explored. Based on a simple model of supply networks, we offer an interpretation of instabilities and oscillations observed in biological, ecological, economic, and engineering systems. We find that most supply networks display damped oscillations, even when their units—and linear chains of these units—behave in a nonoscillatory way. Moreover, networks of damped oscillators tend to produce growing oscillations. This surprising behavior offers, for example, a different interpretation of business cycles and of oscillating or pulsating processes. The network structure of material flows itself turns out to be a source of instability, and cyclical variations are an inherent feature of decentralized adjustments.

Figure 1

(Color) Properties of our dynamic model of supply networks for a characteristic input matrix specified as average input matrix of macroeconomic commodity flows of several countries (top) and for a synthetic input matrix generated by random changes of input matrix entries until the number of complex eigenvalues was eventually reduced to zero (bottom). Subfigures (a) and (e) illustrate the color-coded input matrices $\mathbf{A}$, (b) and (f) the corresponding network structures, when only the strongest links (commodity flows) are shown, (c) and (g) the eigenvalues ${\omega }_i=\mathrm{Re}\left({\omega }_i\right)+i\mathrm{Im}\left({\omega }_i\right)$ of the respective input matrix $\mathbf{A}$, and (d) and (h) the phase diagrams indicating the stability behavior of the model equations (1, 2, 3, 4) on a double-logarithmic scale as a function of the model parameters ${\alpha }_i=\alpha$ and ${\nu }_i∕\mu _i{}^2=\nu ∕{\mu }^2=V∕M^2$. The other model parameters were set to ${\nu }_i=C_i=D_i=P_i^0=N_i^0=Y_i^0=1$. Surprisingly, for empirical input matrices $\mathbf{A}$, one never finds an overdamped, exponential relaxation to the stationary equilibrium state, but network-induced oscillations due to complex eigenvalues ${\omega }_i$.

Figure 2

Typical simulation result of the time-dependent gross domestic product ${\sum }_i{Q}_i\left(t\right)P_i\left(t\right)$ in percent, i.e., relative to the initial value. The input matrix was chosen as in Figs. 1a, 1b, 1c, 1d, but $Y_i^0$ was determined from averaged input-output data. $Q_i^0$ was obtained from the equilibrium condition, and the fluctuations ${\xi }_i\left(t\right)$ were specified as a Gaussian white noise with mean value 0 and variance $\sigma =10.000$ (about 10% of the average final consumption). The initial prices $P_i\left(0\right)$ were selected from the interval [0.9;1.1]. Moreover, in this example we have assumed $f_i\left(P_i\right)=\max [0,1+d(P_i-P_i^0)]$ with $d=f^{\prime }\left(P_i^0\right)=-10$ and the parameters ${\nu }_i=0.1$, ${\mu }_i=0.0001$, ${\alpha }_i=1=P_i^0$, and $N_i^0=Y_i^0$. Although this implies a growth of small oscillations [cf. Fig. 1d], the oscillation amplitudes are rather limited. This is due to the nonlinearity of model equations (1, 2, 3, 4) and due to the phase shifts between oscillations of different economic sectors $i$. Note that irregular oscillations with frequencies between 4 and $6\mathrm{yr}$ and amplitudes of about 2.5% are qualitatively compatible with empirical business cycles. Our material flow model can explain $w$-shaped, nonperiodic oscillations without having to assume technological shocks or externally induced perturbations. The long-term growth of national economies was intentionally not included in the model in order to separate this effect from network-induced instability effects.