Self-organized criticality

Self-organized criticality is one of a number of important discoveries made in statistical physics and related fields over the latter half of the 20th century, discoveries which relate particularly to the study of complexity in nature. For example, the study of cellular automata, from the early discoveries of Stanislaw Ulam and John von Neumann through to John Conway's Game of Life and the extensive work of Stephen Wolfram, made it clear that complexity could be generated as an emergent feature of extended systems with simple local interactions. Over a similar period of time, Benoît Mandelbrot's large body of work on fractals showed that much complexity in nature could be described by certain ubiquitous mathematical laws, while the extensive study of phase transitions carried out in the 1960s and 1970s showed how scale invariant phenomena such as fractals and power laws emerged at the critical point between phases.

The term self-organized criticality was firstly introduced by Bak, Tang and Wiesenfeld's 1987 paper, which clearly linked together those factors: a simple cellular automaton was shown to produce several characteristic features observed in natural complexity (fractal geometry, pink (1/f) noise and power laws) in a way that could be linked to critical-point phenomena. Crucially, however, the paper emphasized that the complexity observed emerged in a robust manner that did not depend on finely tuned details of the system: variable parameters in the model could be changed widely without affecting the emergence of critical behavior: hence, self-organized criticality. Thus, the key result of BTW's paper was its discovery of a mechanism by which the emergence of complexity from simple local interactions could be spontaneous—and therefore plausible as a source of natural complexity—rather than something that was only possible in artificial situations in which control parameters are tuned to precise critical values. The publication of this research sparked considerable interest from both theoreticians and experimentalists, producing some of the most cited papers in the scientific literature.

Due to BTW's metaphorical visualization of their model as a "sandpile" on which new sand grains were being slowly sprinkled to cause "avalanches", much of the initial experimental work tended to focus on examining real avalanches in granular matter, the most famous and extensive such study probably being the Oslo ricepile experiment. Other experiments include those carried out on magnetic-domain patterns, the Barkhausen effect and vortices in superconductors.

Early theoretical work included the development of a variety of alternative SOC-generating dynamics distinct from the BTW model, attempts to prove model properties analytically (including calculating the critical exponents[7][8]), and examination of the conditions necessary for SOC to emerge. One of the important issues for the latter investigation was whether conservation of energy was required in the local dynamical exchanges of models: the answer in general is no, but with (minor) reservations, as some exchange dynamics (such as those of BTW) do require local conservation at least on average. In the long term, key theoretical issues yet to be resolved include the calculation of the possible universality classes of SOC behavior and the question of whether it is possible to derive a general rule for determining if an arbitrary algorithm displays SOC.

Alongside these largely lab-based approaches, many other investigations have centered around large-scale natural or social systems that are known (or suspected) to display scale-invariant behavior. Although these approaches were not always welcomed (at least initially) by specialists in the subjects examined, SOC has nevertheless become established as a strong candidate for explaining a number of natural phenomena, including: earthquakes (which, long before SOC was discovered, were known as a source of scale-invariant behavior such as the Gutenberg–Richter law describing the statistical distribution of earthquake size, and the Omori law describing the frequency of aftershocks[9][3]); solar flares; fluctuations in economic systems such as financial markets (references to SOC are common in econophysics); landscape formation; forest fires; landslides; epidemics; neuronal avalanches in the cortex;[5][10] 1/f noise in the amplitude of electrophysiological signals;[4] and biological evolution (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "punctuated equilibria" put forward by Niles Eldredge and Stephen Jay Gould). These "applied" investigations of SOC have included both modelling (either developing new models or adapting existing ones to the specifics of a given natural system) and extensive data analysis to determine the existence and/or characteristics of natural scaling laws.

In addition, SOC has been applied to computational algorithms. Recently, it has been found that the avalanches from an SOC process, like the BTW model, make effective patterns in a random search for optimal solutions on graphs.[11] An example of such an optimization problem is graph coloring. The SOC process apparently helps the optimization from getting stuck in a local optimum without the use of any annealing scheme, as suggested by previous work on extremal optimization.

The recent excitement generated by scale-free networks has raised some interesting new questions for SOC-related research: a number of different SOC models have been shown to generate such networks as an emergent phenomenon, as opposed to the simpler models proposed by network researchers where the network tends to be assumed to exist independently of any physical space or dynamics. While many single phenomena have been shown to exhibit scale-free properties over narrow ranges, a phenomenon offering by far a larger amount of data is solvent-accessible surface areas in globular proteins.[12] These studies quantify the differential geometry of proteins, and resolve many evolutionary puzzles regarding the biological emergence of complexity.[13]

Despite the considerable interest and research output generated from the SOC hypothesis, there remains no general agreement with regards to its mechanisms in abstract mathematical form. Bak Tang and Wiesenfeld based their hypothesis on the behavior of their sandpile model.[1] However, it has been argued that this model would actually generate 1/f² noise rather than 1/f noise.[14] This claim was based on untested scaling assumptions, and a more rigorous analysis showed that sandpile models generally produce 1/f^a spectra, with a<2. [15] Other simulation models were proposed later that could produce true 1/f noise,[16] and experimental sandpile models were observed to yield 1/f noise.[17] In addition to the nonconservative theoretical model mentioned above, other theoretical models for SOC have been based upon information theory[18], mean field theory[19], the convergence of random variables[20], and cluster formation.[21] A continuous model of self-organised criticality is proposed by using tropical geometry.[22]

In chronological order of development:

• Stick-slip model of fault failure[9][3]
• Bak–Tang–Wiesenfeld sandpile
• Forest-fire model
• Olami–Feder–Christensen model
• Bak–Sneppen model

• 1/f noise
• Complex systems
• Critical brain hypothesis
• Critical exponents
• Detrended fluctuation analysis, a method to detect power-law scaling in time series.
• Dual-phase evolution, another process that contributes to self-organization in complex systems.
• Fractals
• Ilya Prigogine, a systems scientist who helped formalize dissipative system behavior in general terms.
• Power laws
• Red Queen hypothesis
• Scale invariance
• Self-organization
• Self-organized criticality control

• Adami, C. (1995). "Self-organized criticality in living systems". Physics Letters A. 203 (1): 29–32. arXiv:adap-org/9401001. Bibcode:1995PhLA..203...29A. CiteSeerX 10.1.1.456.9543. doi:10.1016/0375-9601(95)00372-A.

• Bak, P. (1996). How Nature Works: The Science of Self-Organized Criticality. New York: Copernicus. ISBN 978-0-387-94791-4.

• Bak, P. and Paczuski, M. (1995). "Complexity, contingency, and criticality". Proceedings of the National Academy of Sciences of the USA. 92 (15): 6689–6696. Bibcode:1995PNAS...92.6689B. doi:10.1073/pnas.92.15.6689. PMC 41396. PMID 11607561.CS1 maint: multiple names: authors list (link)

• Bak, P. and Sneppen, K. (1993). "Punctuated equilibrium and criticality in a simple model of evolution". Physical Review Letters. 71 (24): 4083–4086. Bibcode:1993PhRvL..71.4083B. doi:10.1103/PhysRevLett.71.4083. PMID 10055149.CS1 maint: multiple names: authors list (link)

• Bak, P., Tang, C. and Wiesenfeld, K. (1987). "Self-organized criticality: an explanation of ${\displaystyle 1/f}$ noise". Physical Review Letters. 59 (4): 381–384. Bibcode:1987PhRvL..59..381B. doi:10.1103/PhysRevLett.59.381. PMID 10035754.CS1 maint: multiple names: authors list (link)

• Bak, P., Tang, C. and Wiesenfeld, K. (1988). "Self-organized criticality". Physical Review A. 38 (1): 364–374. Bibcode:1988PhRvA..38..364B. doi:10.1103/PhysRevA.38.364. PMID 9900174.CS1 maint: multiple names: authors list (link) Papercore summary.

• Buchanan, M. (2000). Ubiquity. London: Weidenfeld & Nicolson. ISBN 978-0-7538-1297-6.

• Jensen, H. J. (1998). Self-Organized Criticality. Cambridge: Cambridge University Press. ISBN 978-0-521-48371-1.

• Katz, J. I. (1986). "A model of propagating brittle failure in heterogeneous media". Journal of Geophysical Research. 91 (B10): 10412. Bibcode:1986JGR....9110412K. doi:10.1029/JB091iB10p10412.

• Kron, T./Grund, T. (2009). "Society as a Selforganized Critical System". Cybernetics and Human Knowing. 16: 65–82.CS1 maint: multiple names: authors list (link)

• Paczuski, M. (2005). Networks as renormalized models for emergent behavior in physical systems. Complexity. The Science and Culture Series – Physics. pp. 363–374. arXiv:physics/0502028. Bibcode:2005cmn..conf..363P. CiteSeerX 10.1.1.261.9886. doi:10.1142/9789812701558_0042. ISBN 978-981-256-525-9.

• Turcotte, D. L. (1997). Fractals and Chaos in Geology and Geophysics. Cambridge: Cambridge University Press. ISBN 978-0-521-56733-6.

• Turcotte, D. L. (1999). "Self-organized criticality". Reports on Progress in Physics. 62 (10): 1377–1429. Bibcode:1999RPPh...62.1377T. doi:10.1088/0034-4885/62/10/201.
• Md. Nurujjaman/A. N. Sekar Iyengar (2007). "Realization of {SOC} behavior in a dc glow discharge plasma". Physics Letters A. 360 (6): 717–721. arXiv:physics/0611069. Bibcode:2007PhLA..360..717N. doi:10.1016/j.physleta.2006.09.005.
• Self-organized criticality on arxiv.org