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Cellular Automata and Self-Organized Criticality
Abstract
The notion of “phase transition” is a key concept in the theory of complex systems. Right at the point of a continuous transition between one phase and another, systems behave in a very special fashion; they are said to be “critical”. Criticality is reached normally when tuning an external parameter, such as the temperature for physical phase transitions. The central question discussed in this chapter is whether “self-organized criticality” is possible in complex adaptive systems, i.e. whether a system can autonomously adapt its own parameters in a way to move towards criticality on its own, as a consequence of a suitable adaptive dynamics. Possible self-organized states in nature involve life as it is, where one speaks of “life at the edge of chaos”, and the neural dynamics of the human brain. We will introduce in this chapter the Landau theory of phase transitions and then discuss cellular automata, an important and popular class of standardized dynamical systems. Cellular automata allow a very intuitive construction of models, such as the forest fire mode, the game of life and the sandpile model, which exhibits self-organized criticality. The chapter then concludes with a discussion of whether self-organized criticality occurs in the most adaptive dynamical system of all, namely in the context of long-term evolution.
... x -the rule specifying orthogonal movement, 2 x -the rule for positioning accuracy control, 3 x -the rule for assembling process scheduling. Using discrete-event system sequential design, proposed in [13], one can obtain supervisor S that does not contain any blockings. For control software development and for intuitive image of discrete-event system logic workflow finite state machines (FSM) are used. ...
... In [12][13][14] two-level model of structured discrete-event system proposed. The first level contains actuators ( ) 12 ...
In this paper, design method and implementation of Cyber-Physical System (CPS) of mobile robots group control for general mission accomplishment are regarded. Picture segments are attached to each robot, thus mission accomplishment provides full picture assembling like a puzzle solving. Software and hardware for moving and positioning control in constrained rectangular region were implemented. Mathematical technique of set theory, control theory, digital image processing, discrete-event systems description methods and their integration with different Matlab and Simulink Toolboxes were utilized. Hardware implemented on the robotic mobile platform by LEGO. The presented hardware and software successfully utilized by teachers of National Research Tomsk Polytechnic University for laboratory works and seminars within such disciplines as Control Theory, System Modeling, Computer Vision Methods, etc. Obtained results fully or partially can be utilized for development of advanced algorithms and control methods for technical systems and processes as well as planning approaches and decision making for complex systems.
... The key idea to achieve self-management without deceasing the dependability is using simplicity to control complexity [5] and simplifying the management (control) based on selfsimilarity [46]. To achieve this, we need to take full advantage of the characteristic of SoS and design a systematic framework and self-similar subsystems to enable recursive composition for CPS. ...
Cyber Physical Systems (CPSs) need to interact with the changeable environment under various interferences. To provide continuous and high quality services, a self-managed CPS should automatically reconstruct itself to adapt to these changes and recover from failures. Such dynamic adaptation behavior introduces systemic challenges for CPS design, advice evaluation and decision process arrangement. In this paper, a formal compositional framework is proposed to systematically improve the dependability of the decision process. To guarantee the consistent observation of event orders for causal reasoning, this work first proposes a relative time-based method to improve the composability and compositionality of the timing property of events. Based on the relative time solution, a formal reference framework is introduced for self-managed CPSs, which includes a compositional FSM-based actor model (subsystems of CPS), actor-based advice and runtime decomposable decisions. To simplify self-management, a self-similar recursive actor interface is proposed for decision (actor) composition. We provide constraints and seven patterns for the composition of reliability and process time requirements. Further, two decentralized decision process strategies are proposed based on our framework, and we compare the reliability with the static strategy and the centralized processing strategy. The simulation results show that the one-order feedback strategy has high reliability, scalability and stability against the complexity of decision and random failure. This paper also shows a way to simplify the evaluation for dynamic system by improving the composability and compositionality of the subsystem.
We show that dynamical systems with spatial degrees of freedom naturally evolve into a self-organized critical point. Flicker noise, or 1/f noise, can be identified with the dynamics of the critical state. This picture also yields insight into the origin of fractal objects.
A simple dynamical model for Darwinian evolution on its slowest time scale is analyzed. Its mean field theory is formulated and solved. A random neighbor version of the model is simulated, as is a one-dimensional version. In one dimension, the dynamics can be described in terms of a ``repetitious random walker'' and anomalous diffusion with exponent 0.4. In all cases the model self-organizes to a robust critical attractor.
A simple and robust model of biological evolution of an ecology of interacting species is introduced. The model self-organizes into a critical steady state with intermittent coevolutionary avalanches of all sizes; i.e., it exhibits ``punctuated equilibrium'' behavior. This collaborative evolution is much faster than non-cooperative scenarios since no large and coordinated, and hence prohibitively unlikely, mutations are involved.
this article. A nice review can be found in Ref.
We discuss mean-field theories for self-organized criticality and the connection with the general theory of branching processes. We point out that the nature of the self-organization is not addressed properly by the previously proposed mean-field theories. We introduce a new mean-field model that explicitly takes the boundary conditions into account; in this way, the local dynamical rules are coupled to a global equation that drives the control parameter to its critical value. We study the model numerically, and analytically we compute the avalanche distributions. Comment: 4 pages + 4 ps figures
This review addresses recent developments in nonequilibrium statistical physics. Focusing on phase transitions from fluctuating phases into absorbing states, the universality class of directed percolation is investigated in detail. The survey gives a general introduction to various lattice models of directed percolation and studies their scaling properties, field-theoretic aspects, numerical techniques, as well as possible experimental realizations. In addition, several examples of absorbing-state transitions which do not belong to the directed percolation universality class will be discussed. As a closely related technique, we investigate the concept of damage spreading. It is shown that this technique is ambiguous to some extent, making it impossible to define chaotic and regular phases in stochastic nonequilibrium systems. Finally, we discuss various classes of depinning transitions in models for interface growth which are related to phase transitions into absorbing states.
Power laws and distributions with heavy tails are common features of many complex systems. Examples are the distribution of earthquake magnitudes, solar flare intensities and the sizes of neuronal avalanches. Previously, researchers surmised that a single general concept may act as an underlying generative mechanism, with the theory of self organized criticality being a weighty contender.
The power-law scaling observed in the primary statistical analysis is an important, but by far not the only feature characterizing experimental data. The scaling function, the distribution of energy fluctuations, the distribution of inter-event waiting times, and other higher order spatial and temporal correlations, have seen increased consideration over the last years. Leading to realization that basic models, like the original sandpile model, are often insufficient to adequately describe the complexity of real-world systems with power-law distribution.
Consequently, a substantial amount of effort has gone into developing new and extended models and, hitherto, three classes of models have emerged. The first line of models is based on a separation between the time scales of an external drive and a an internal dissipation, and includes the original sandpile model and its extensions, like the dissipative earthquake model. Within this approach the steady state is close to criticality in terms of an absorbing phase transition. The second line of models is based on external drives and internal dynamics competing on similar time scales and includes the coherent noise model, which has a non-critical steady state characterized by heavy-tailed distributions. The third line of models proposes a non-critical self-organizing state, being guided by an optimization principle, such as the concept of highly optimized tolerance.
We present a comparative overview regarding distinct modeling approaches together with a discussion of their potential relevance as underlying generative models for real-world phenomena. The complexity of physical and biological scaling phenomena has been found to transcend the explanatory power of individual paradigmal concepts. The interaction between theoretical development and experimental observations has been very fruitful, leading to a series of novel concepts and insights.
We review the properties of the self-organized critical (SOC) forest-fire model. The paradigm of self-organized criticality refers to the tendency of certain large dissipative systems to drive themselves into a critical state independent of the initial conditions and without fine tuning of the parameters. After an introduction, we define the rules of the model and discuss various large-scale structures which may appear in this system. The origin of the critical behaviour is explained, critical exponents are introduced and scaling relations between the exponents are derived. Results of computer simulations and analytical calculations are summarized. The existence of an upper critical dimension and the universality of the critical behaviour under changes of lattice symmetry or the introduction of immunity are discussed. A survey of interesting modifications of the forest-fire model is given. Finally, several other important SOC models are briefly described.
Timescale variation from astronomical sources has long been referred to
as a 'signal'. In other words, the variation exhibits some form of
regularity. In this paper, it is maintained that the stochastic property
of timescale variation is also common to a wide variety of physical
phenomena. Hence, integration and Fourier transformation of noise
functions, say, random white noise, can impart to the noise a semblance
of regularity, turning the noise into a 'signal'. Attention is given to
flicker noise (or low-frequency noises in general), and their appearance
in largely non-astronomical physical phenomena. Electrical and seismic
noise are presented as examples. Integrated and plotted onto a log
scale, random noise shows a remarkable fit with the power law. Finally,
the generation of flicker noise is dealt with mathematically by
superimposition and causal filtering.
A forest-fire model is introduced which contains a lightning probability f. This leads to a self-organized critical state in the limit f-->0 provided that the time scales of tree growth and burning down of forest clusters are separated. We derive scaling laws and calculate all critical exponents. The values of the critical exponents are confirmed by computer simulations. For a two-dimensional system, we show that the forest density in the critical state assumes its minimum possible value, i.e., that energy dissipation is maximum.
The structure of avalanches in the Abelian sandpile model on a square lattice is analyzed. It is shown that an avalanche can be considered as a sequence of waves of decreasing sizes. Being more simple objects, waves admit a representation in terms of spanning trees covering the lattice sites. The difference in sizes of subsequent waves follows a power law with the exponent α simply related to the basic exponent τ of the sandpile model. Using known exponents for the spanning trees, we derive from scaling arguments α=3/4 and τ=5/4.
By computer simulations, it was reported that the Bak-Tang-Wiesenfeld (BTW) model loses self-organized criticality (SOC) when some particles are annihilated in a toppling process in the bulk of system. We give a rigorous proof that the BTW model loses SOC as soon as the annihilation rate becomes positive. To prove this, a nonconservative Abelian sandpile model is defined on a square lattice, which has a parameter alpha (>/=1) representing the degree of breaking of the conservation law. This model is reduced to be the BTW model when alpha=1. By calculating the average number of topplings in an avalanche exactly, it is shown that for any alpha>1, <infinity even in the infinite-volume limit. The power-law divergence of with an exponent 1 as alpha-->1 gives a scaling relation 2nu(2-a)=1 for the critical exponents nu and a of the distribution function of T. The 1-1 height correlation C11(r) is also calculated analytically and we show that C11(r) is bounded by an exponential function when alpha>1, although C11(r) approximately r(-2d) was proved by Majumdar and Dhar for the d-dimensional BTW model. A critical exponent nu(11) characterizing the divergence of the correlation length xi as alpha-->1 is defined as xi approximately |alpha-1|(-nu(11)) and our result gives an upper bound nu(11)</=1/2.
The Bak-Tang-Wiesenfeld sandpile model provdes a simple and elegant system with which to demonstate self-organized criticality. This model has rather remarkable mathematical properties first elucidated by Dhar. I demonstrate some of these properties graphically with a simple computer simulation. Comment: Contribution to the Niels Bohr Summer Institute on Complexity and Criticality; to appear in a Per Bak Memorial Issue of PHYSICA A; 6 pages 3 figures
Forest fires and other examples of self-organized criticality
- S Clar
- B Drossel
- F Schwabl
Flicker noises in astronomy and elsewhere. Comments on Modern Physics
- W H Press