Symmetry and Universal Critical Properties of a Class of Discrete State Networks



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In this dissertation, we study the statistical mechanics of Boolean networks as a simple model in class of heterogeneous complex systems. Boolean networks are used as generic models of complex systems of many interacting units, such as gene and protein interaction systems, neural networks and economical systems. They are particularly good examples of complex systems to study because they are relatively simple, yet have a nontrivial dynamical phase transition. We investigate the statistical mechanics of how this model behaves and dynamical properties of random Boolean networks at criticality.

First we study the dynamics of critical random Boolean networks and find what the symmetry of the dynamics is. We propose a symmetry group, the \textit{canalization preserving group}, that describes the symmetry observed. The orbits of this symmetry group consist of Boolean functions that have the same canalization values. Canalization is a form of robustness in which a subset of the input values control the behavior of a node regardless of the remaining inputs. We show that the same symmetry governs critical random discrete multi-state networks dynamics with higher number of output values for each node. We investigate the criticality of random multi-state networks, and show that the same canalization preserving symmetry governs the critical multi-state networks dynamics as well.

We also study a particular dynamical property of critical random Boolean networks: their attractor length distribution. Using a known result that nodes relevant to the dynamics on attractors at criticality can be divided into separate components, we analyze the structure of these relevant components and how their dynamics combine to find the distribution of attractor lengths. This is accomplished by mapping the problem to the enumeration of binary Lyndon words. Using analytical arguments we show that the attractor length distribution becomes scale-free in the large network limit with a decay described by a critical exponent of 1. The universal nature of this behavior is demonstrated by comparison to that of the evolved critical state achieved through the playing of an adaptive game that selects for diversity of node's behavior, and that of the attractor length distribution of critical multi-state networks.



Symmetry, Discrete state networks


Portions of this document appear in: Hossein, Shabnam, Matthew D. Reichl, and Kevin E. Bassler. "Symmetry in critical random boolean network dynamics." Physical Review E 89, no. 4 (2014): 042808.