Separating Internal and External Dynamics of Complex Systems

The observable behavior of a complex system reflects the mechanisms governing the internal interactions between the system's components and the effect of external perturbations. Here we show that by capturing the simultaneous activity of several of the system's components we can separate the internal dynamics from the external fluctuations. The method allows us to systematically determine the origin of fluctuations in various real systems, finding that while the Internet and the computer chip have robust internal dynamics, highway and Web traffic are driven by external demand. As multichannel measurements are becoming the norm in most fields, the method could help uncover the collective dynamics of a wide array of complex systems.

Figure 1

Splitting a measured signal into its external and internal contributions for the model system. We study the traffic on the nodes of a scale-free network with ${10}^3$ nodes, generated by $M(t)$ random walkers whose number follow a sinusoidal signal with amplitude $\Delta M$ and period 60. (a) The activity measured on a typical node when the amplitude of the external fluctuations is $\Delta M=1000$. (b) The external contribution provided by Eq. (3) recovers the periodic signal imposed on the system. (c) The internal contribution, predicted by Eq. (4), captures the random pattern of the diffusion process. (d)–(f) The same as in (a)–(c) but with a small external amplitude $\Delta M=10$, demonstrating that the method works even when the amplitude of external fluctuations are comparable to fluctuations of the internal dynamics.

Figure 2

Distribution of ${\eta }_i={\sigma }_i^{\mathrm{e}\mathrm{x}\mathrm{t}}/{\sigma }_i^{\mathrm{i}\mathrm{n}\mathrm{t}}$ ratios of external and internal fluctuations for model (a),(b) and selected real systems (c),(d). Distribution of the ${\eta }_i$ ratios for the random walk model: (a) For smaller external fluctuations, the distribution is centered around a small value of $\eta \ll 1$, indicating that internal fluctuations overcome external ones, dominating the system's dynamics. (b) When $\Delta M$ is increased, however, such fluctuations overshadow the system's internal dynamics, and the $P(\eta )$ distribution shifts towards larger values of $\eta$. (c) $P(\eta )$ distributions for the Internet and the microchip, centered around $\eta \sim 0.1$, indicate that external fluctuations do not affect the system's overall dynamics significantly. (d) The World Wide Web and the highway networks, with $P(\eta )$ peaked around $\eta \sim 1$, are strongly influenced by fluctuations in the total number of web surfers and the number of cars, respectively.

Figure 3

Scaling of the external and internal fluctuations with the average flux. Internal fluctuations ${\sigma }_i^{\mathrm{i}\mathrm{n}\mathrm{t}}$ on the microchip (a) and on the Internet (b), both belonging to the $\alpha =1/2$ class, are significantly larger than external fluctuations ${\sigma }_i^{\mathrm{e}\mathrm{x}\mathrm{t}}$ and scale with a different exponent. External and internal fluctuations are comparable in magnitude on the World Wide Web (c) and the highway network (d), and they also follow the same scaling, indicating that in these systems' external fluctuations should have strong impact on systems' overall dynamics.