Learning, Generalization, and Functional Entropy in Random Automata Networks

It has been shown citep{broeck90:physicalreview,patarnello87:europhys} that feedforward Boolean networks can learn to perform specific simple tasks and generalize well if only a subset of the learning examples is provided for learning. Here, we extend this body of work and show experimentally that random Boolean networks (RBNs), where both the interconnections and the Boolean transfer functions are chosen at random initially, can be evolved by using a state-topology evolution to solve simple tasks. We measure the learning and generalization performance, investigate the influence of the average node connectivity $K$, the system size $N$, and introduce a new measure that allows to better describe the networks learning and generalization behavior. We show that the connectivity of the maximum entropy networks scales as a power-law of the system size $N$. Our results show that networks with higher average connectivity $K$ (supercritical) achieve higher memorization and partial generalization. However, near critical connectivity, the networks show a higher perfect generalization on the even-odd task.

Emergent Criticality Through Adaptive Information Processing in Boolean Networks

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 critical value with a power-law of the system size $N$. We show that network learning and generalization are optimized near criticality, given task complexity and the amount of information provided threshold values. Both random and evolved networks exhibit maximal topological diversity near $K_{c}$. We hypothesize that this supports efficient exploration and robustness of solutions. Also reflected in our observation is that the variance of the values is maximal in critical network populations. Finally, we discuss implications of our results for determining the optimal topology of adaptive dynamical networks that solve computational tasks.

Assessing Random Dynamical Network Architectures for Nanoelectronics

Independent of the technology, it is generally expected that future nanoscale devices will be built from vast numbers of densely arranged devices that exhibit high failure rates. Other than that, there is little consensus on what type of technology and computing architecture holds most promises to go far beyond todays top-down engineered silicon devices. Cellular automata (CA) have been proposed in the past as a possible class of architectures to the von Neumann computing architecture, which is not generally well suited for future parallel and fine-grained nanoscale electronics. While the top-down engineered semi-conducting technology favors regular and locally interconnected structures, future bottom-up self-assembled devices tend to have irregular structures because of the current lack precise control over these processes. In this paper, we will assess random dynamical networks, namely Random Boolean Networks (RBNs) and Random Threshold Networks (RTNs), as alternative computing architectures and models for future information processing devices. We will illustrate that--from a theoretical perspective--they offer superior properties over classical CA-based architectures, such as inherent robustness as the system scales up, more efficient information processing capabilities, and manufacturing benefits for bottom-up designed devices, which motivates this investigation. We will present recent results on the dynamic behavior and robustness of such random dynamical networks while also including manufacturing issues in the assessment.

When correlations matter - response of dynamical networks to small perturbations

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 from the annealed approximation. Extending the approach followed in a previous study, we present new results indicating that internal dynamical correlations tend to increase not only the probability for small, but also for very large damage events, leading to a broad, fat-tailed distribution of damage sizes. These findings indicate that the descriptive and predictive value of averaged order parameters for finite size networks - even for biologically highly relevant sizes up to several thousand nodes - is limited.