Symmetry in critical random Boolean network dynamics

Abstract

Using Boolean networks as prototypical examples, the role of symmetry in the dynamics of heterogeneous complex systems is explored. We show that symmetry of the dynamics, especially in critical states, is a controlling feature that can be used both to greatly simplify analysis and to characterize different types of dynamics. Symmetry in Boolean networks is found by determining the frequency at which the various Boolean output functions occur. There are classes of functions that consist of Boolean functions that behave similarly. These classes are orbits of the controlling symmetry group. We find that the symmetry that controls the critical random Boolean networks is expressed through the frequency by which output functions are utilized by nodes that remain active on dynamical attractors. This symmetry preserves canalization, a form of network robustness. We compare it to a different symmetry known to control the dynamics of an evolutionary process that allows Boolean networks to organize into a critical state. Our results demonstrate the usefulness and power of using the symmetry of the behavior of the nodes to characterize complex network dynamics, and introduce a novel approach to the analysis of heterogeneous complex systems.

Excitation and inhibition imbalance affects dynamical complexity through symmetries

“…For example, one might seek to understand symmetries in the output function on each node, and consider that the function f 1 ( x) = x 1 or x 2 is invariant under permutation of its argument. In fact, output function symmetries at the node level are a powerful way to characterize complex network dynamics [36,37]. Our work extends these observations by showing that symmetries on the global level can elucidate networks with complex dynamics.…”

Control of complex networks requires both structure and dynamics

“…the critical transition between ordered and chaotic dynamics in BNs depends both on structural (mean connectivity) and dynamical properties of nodes (bias and canalization). [11][12][13][14] Indeed, we already know that such dynamical properties strongly impact the stability, robustness, and controllability of existing models of gene regulation and biochemical signaling in a number of organisms. 7,[15][16][17][18] Therefore, a question of central importance remains: How well does network structure predict the dynamics of the underlying complex system, especially from the viewpoint of control?…”

“…On the other hand, there are important system attributes which depend on dynamical characteristics of variables and their interactions; e.g. the critical transition between ordered and chaotic dynamics in BNs depends both on structural (mean connectivity) and dynamical properties of nodes (bias and canalization) 11 12 13 14 . Indeed, we already know that such dynamical properties strongly impact the stability, robustness, and controllability of existing models of gene regulation and biochemical signaling in a number of organisms 7 15 16 17 18 .…”

Breaking reflection symmetry: evolving long dynamical cycles in Boolean systems

“…For example, one might seek to understand symmetries in the output function on each node and consider that the function is invariant under permutation of its argument. In fact, output function symmetries at the node level are a powerful way to characterize complex network dynamics [ 43 , 44 ]. Our work extends these observations by showing that symmetries at the global level can explain some properties of networks with complex dynamics.…”