Scaling in ordered and critical random Boolean networks

Despite their apparent simplicity, random Boolean networks display a rich variety of dynamical behaviors. Much work has been focused on the properties and abundance of attractors. We here derive an expression for the number of attractors in the special case of one input per node. Approximating some other non-chaotic networks to be of this class, we apply the analytic results to them. For this approximation, we observe a strikingly good agreement on the numbers of attractors o...

Standard Random Boolean Networks display an order-disorder phase transition. We add to the standard Random Boolean Networks a disconnection rule which couples the control and order parameters. By this way, the system is driven to the critical line transition. Under the influence of perturbations the system point out self-organized critical behavior. Several numerical simulations have been done and compared with a proposed analytical treatment.

Network structure strongly constrains the range of dynamic behaviors available to a complex system. These system dynamics can be classified based on their response to perturbations over time into two distinct regimes, ordered or chaotic, separated by a critical phase transition. Numerous studies have shown that the most complex dynamics arise near the critical regime. Here we use an information theoretic approach to study structure-dynamics relationships within a unified fram...

We discuss the complex dynamics of a non-linear random networks model, as a function of the connectivity k between the elements of the network. We show that this class of networks exhibit an order-chaos phase transition for a critical connectivity k = 2. Also, we show that both, pairwise correlation and complexity measures are maximized in dynamically critical networks. These results are in good agreement with the previously reported studies on random Boolean networks and ran...

Boolean networks, widely used to model gene regulation, exhibit a phase transition between regimes in which small perturbations either die out or grow exponentially. We show and numerically verify that this phase transition in the dynamics can be mapped onto a static percolation problem which predicts the long-time average Hamming distance between perturbed and unperturbed orbits.

We systematically study and compare damage spreading for random Boolean and threshold networks under small external perturbations (damage), a problem which is relevant to many biological networks. We identify a new characteristic connectivity $K_s$, at which the average number of damaged nodes after a large number of dynamical updates is independent of the total number of nodes $N$. We estimate the critical connectivity for finite $N$ and show that it systematically deviates ...

We study information processing in populations of Boolean networks with evolving connectivity and systematically explore the interplay between the learning capability, robustness, the network topology, and the task complexity. We solve a long-standing open question and find computationally that, for large system sizes $N$, adaptive information processing drives the networks to a critical connectivity $K_{c}=2$. For finite size networks, the connectivity approaches the critica...

Most models of complex systems have been homogeneous, i.e., all elements have the same properties (spatial, temporal, structural, functional). However, most natural systems are heterogeneous: few elements are more relevant, larger, stronger, or faster than others. In homogeneous systems, criticality -- a balance between change and stability, order and chaos -- is usually found for a very narrow region in the parameter space, close to a phase transition. Using random Boolean n...

For years, we have been building models of gene regulatory networks, where recent advances in molecular biology shed some light on new structural and dynamical properties of such highly complex systems. In this work, we propose a novel timing of updates in Random and Scale-Free Boolean Networks, inspired by recent findings in molecular biology. This update sequence is neither fully synchronous nor asynchronous, but rather takes into account the sequence in which genes affect ...

We propose a simple model that aims at describing, in a stylized manner, how local breakdowns due unbalances or congestion propagate in real dynamical networks. The model converges to a self-organized critical stationary state in which the network shapes itself as a consequence of avalanches of rewiring processes. Depending on the model's specification, we obtain either single scale or scale-free networks. We characterize in detail the relation between the statistical propert...