Modularity and the spread of perturbations in complex dynamical systems

We propose a method to decompose dynamical systems based on the idea that modules constrain the spread of perturbations. We find partitions of system variables that maximize “perturbation modularity,” defined as the autocovariance of coarse-grained perturbed trajectories. The measure effectively separates the fast intramodular from the slow intermodular dynamics of perturbation spreading (in this respect, it is a generalization of the “Markov stability” method of network community detection). Our approach captures variation of modular organization across different system states, time scales, and in response to different kinds of perturbations: aspects of modularity which are all relevant to real-world dynamical systems. It offers a principled alternative to detecting communities in networks of statistical dependencies between system variables (e.g., “relevance networks” or “functional networks”). Using coupled logistic maps, we demonstrate that the method uncovers hierarchical modular organization planted in a system's coupling matrix. Additionally, in homogeneously coupled map lattices, it identifies the presence of self-organized modularity that depends on the initial state, dynamical parameters, and type of perturbations. Our approach offers a powerful tool for exploring the modular organization of complex dynamical systems.

Figure 1

System of 80 coupled logistic maps ($\alpha =2,\gamma =0.04$) (a) The coupling matrix exhibits hierarchical modularity at three levels. (b) The system is chaotic and no strong correlations between variables are observed over 10 000 time steps. (c) Perturbation modularity of optimal decompositions (PM, solid black) at different time scales $t$ and NMI between optimal decompositions at successive times (dashed blue). Three time scale regions are robust ($\text{NMI}=1$, gray), corresponding to the three hierarchical levels of the coupling matrix (insets).

Figure 2

Two 100-variable CMLs are compared: one “modular” (top row; $\alpha =1.7,\gamma =0.1$) and one “diffusive” (bottom row; $\alpha =1.9,\gamma =0.6$). (a, d) Spacetime plots of the effect of a single-variable perturbation. A pixel is colored black if the absolute difference between perturbed and unperturbed trajectory at a variable (vertical axis) exceeds 1% of the size of the system-wide perturbation at a given time (horizontal axis). (b, e) Spacetime plots of the optimal decompositions at different time scales; color indicates each variable's subsystem. (c, f) Perturbation modularity of optimal decompositions (PM, solid black) at different time scales $t$ and NMI between optimal decompositions at successive times (dashed blue). Stable decompositions are observed in the modular CML (top row).

Figure 3

State dependence in the modular CML. For a $12000$-step trajectory starting from a random state, optimal decompositions (time scale $t=300$) are identified using states along this trajectory as initial conditions. (a) Optimal perturbation modularity (PM, solid black) grows with increasing trajectory steps (horizontal axis), indicating the emergence of robust modular structures. Trajectory step $10000$ (dotted green line) is the initial condition in Examples 2 and 5. (b) Optimal decompositions identified at different trajectory steps; color indicates the subsystem of each variable (vertical axis).

Figure 4

Parameter phase map of CML. Perturbation modularity (color) for optimal decompositions of 100-variable CMLs with different values of chaoticity ($\alpha$; horizontal axes) and coupling strength ($\gamma$; vertical axes). Perturbation modularity is computed at three time scales for two different classes of initial conditions: random states [(a) $t=100$, (b) $t=200$, and (c) $t=300$] as well as random states iterated for 10 000 time steps [(d) $t=100$, (e) $t=200$, and (f) $t=300$].

Figure 5

Perturbation dependence in the modular CML. (a) Spacetime plot of the effect of a perturbation to a 20-variable window (red arrows), as in Figs. 2 and 2. (b) Optimal decompositions (time scale $t=300$) for different perturbation sizes (horizontal axis).