New tools are available for reconstructing the attractor basins of discrete dynamical networks where state-space is linked according the network's dynamics. In this thesis the computer software "Discrete Dynamics Lab" is applied to examine simple networks ranging from cellular automata (CA) to random Boolean networks (RBN), that have been widely applied as idealised models of physical and biological systems, to search for general principles underlying their dynamics. The algorithms and methods for generating pre-images for both CA and RBN, and reconstructing and representing attractor basins are described, and also considered in the mathematical context of random directed graphs. RBN and CA provide contrasting notions of self-organisation. RBN provide models of hierarchical categorisation in biology, for example memory in neural and genomic networks. CA provide models at the lower level of emergent complex pattern. New measures and results are presented on CA attractor basins and how they relate to measures on local dynamics and the Z parameter, characterising ordered to "complex" to chaotic behaviour. A method is described for classifying CA rules by an entropy-variance measure which allows glider rules and related complex rules to be found automatically giving a virtually unlimited sample for further study. The dynamics of RBN and intermediate network architectures are examined in the context of memory, where categorisation occurs at the roots of subtrees as well as at attractors. Learning algorithms are proposed for "sculpting" the basin of attraction field. RBN are proposed as a possible neural network model, and also discussed as a model of genomic regulatory networks, where cell types have been explained as attractors

<p>In this thesis several random Boolean networks are simulated. Both completely computer generated network and models for biological networks are simulated. Several different tools are used to gain knowledge about the robustness. These tools are Derrida plots, noise analysis and mean probability for canalizing rules. Some simulations on how entropy works as an indicator on if a network is robust are also included. The noise analysis works by measuring the hamming distance between the state of the network when noise is applied and when no noise is applied. For many of the simulated networks two types of rules are applied: nested canalizing and flat distributed rules. The computer generated networks consists of two types of networks: scale-free and ER-networks. One of the conclusions in this report is that nested canalizing rules are often more robust than flat distributed rules. Another conclusion is that the mean probability for canalizing rules has, for flat distributed rules, a very dominating effect on if the network is robust or not. Yet another conclusion is that the probability distribution for indegrees, for flat distributed rules, has a strong effect on if a network is robust due to the connection between the probability distribution for indegrees and the mean probability for canalizing rules.</p>

In this thesis several random Boolean networks are simulated. Both completely computer generated network and models for biological networks are simulated. Several different tools are used to gain knowledge about the robustness. These tools are Derrida plots, noise analysis and mean probability for canalizing rules. Some simulations on how entropy works as an indicator on if a network is robust are also included. The noise analysis works by measuring the hamming distance between the state of the network when noise is applied and when no noise is applied. For many of the simulated networks two types of rules are applied: nested canalizing and flat distributed rules. The computer generated networks consists of two types of networks: scale-free and ER-networks. One of the conclusions in this report is that nested canalizing rules are often more robust than flat distributed rules. Another conclusion is that the mean probability for canalizing rules has, for flat distributed rules, a very dominating effect on if the network is robust or not. Yet another conclusion is that the probability distribution for indegrees, for flat distributed rules, has a strong effect on if a network is robust due to the connection between the probability distribution for indegrees and the mean probability for canalizing rules.

With the transition from single-core to multi-core computing and CMOS technology reaching its physical limits, new computing architectures which are scalable, robust, and low-power are required. A promising alternative to conventional computing architectures are Cellular Automata (CA) networks and Random Boolean Networks (RBN), where simple computational nodes combine to form a network that is capable of performing a larger computational task. It has previously been shown that RBNs can offer superior characteristics over mesh networks in terms of robustness, information processing capabilities, and manufacturing costs while the locally connected computing elements of a CA network provide better scalability and low average interconnect length. This study presents a low level hardware analysis of these architectures using a framework which generates the HDL code and netlist of these networks for various network parameters. The HDL code and netlists are then used to simulate these new computing architectures to estimate the latency, area and power consumed when implemented on silicon and performing a pre-determined computation. We show that for RBNs, information processing is faster compared to a CA network, but CA networks are found to a have lower and better distribution of power dissipation than RBNs because of their regular structure. A well-established task to determine the latency of operation for these architectures is presented for a good understanding of the effect of non-local connections in a network. Programming the nodes for this purpose is done externally using a novel self-configuration algorithm requiring minimal hardware. Configuration for RBNs is done by sending in configuration packets through a randomly chosen node. Logic for identifying the topology for the network is implemented for the nodes in the RBN network to enable compilers to analyze and generate the configuration bit stream for that network. On the other hand, the configuration of the CA network is done by passing in configuration data through the inputs on one of the sides of the cell array and shifting it into the network. A study of the overhead of the network configuration and topology identification mechanisms are presented. An analysis of small-world networks in terms of interconnect power and information propagation capability has been presented. It has been shown that small-world networks, whose randomness lies between that of completely regular and completely irregular networks, are realistic while providing good information propagation capability. This study provides valuable information to help designers make decisions for various performance parameters for both RBN and CA networks, and thus to find the best design for the application under consideration.

The high cost of processor fabrication plants and approaching physical limits have started a new wave research in alternative computing paradigms. As an alternative to the top-down manufactured silicon-based computers, research in computing using natural and physical system directly has recently gained a great deal of interest. A branch of this research promotes the idea that any physical system with sufficiently complex dynamics is able to perform computation. The power of networks in representing complex interactions between many parts make them a suitable choice for modeling physical systems. Many studies used networks with a homogeneous structure to describe the computational circuits. However physical systems are inherently heterogeneous. We aim to study the effect of heterogeneity in the dynamics of physical systems that pertains to information processing. Two particularly well-studied network models that represent information processing in a wide range of physical systems are Random Boolean Networks (RBN), that are used to model gene interactions, and Liquid State Machines (LSM), that are used to model brain-like networks. In this thesis, we study the effects of function heterogeneity, in-degree heterogeneity, and interconnect irregularity on the dynamics and the performance of RBN and LSM. First, we introduce the model parameters to characterize the heterogeneity of components in RBN and LSM networks. We then quantify the effects of heterogeneity on the network dynamics. For the three heterogeneity aspects that we studied, we found that the effect of heterogeneity on RBN and LSM are very different. We find that in LSM the in-degree heterogeneity decreases the chaoticity in the network, whereas it increases chaoticity in RBN. For interconnect irregularity, heterogeneity decreases the chaoticity in LSM while its effects on RBN the dynamics depends on the connectivity. For {K} < 2, heterogeneity in the interconnect will increase the chaoticity in the dynamics and for {K} > 2 it decreases the chaoticity. We find that function heterogeneity has virtually no effect on the LSM dynamics. In RBN however, function heterogeneity actually makes the dynamics predictable as a function of connectivity and heterogeneity in the network structure. We hypothesize that node heterogeneity in RBN may help signal processing because of the variety of signal decomposition by different nodes.