Fundamental physics is core area of research at SFI. It spans the principles of quantum and statistical mechanics, information theory, nonlinear dynamics and chaos, and discrete systems.  These fields have provided techniques and approaches to problem solving that are useful across the sciences, and served as points of departure for the recognition of new principles. For instance, the application of self-organization to dynamical critical states arose from the study of granular systems, and agent-based simulation introduced a process-based generalization of Monte Carlo methods.  Current and future SFI research in physics occupies four main areas: statistical physics with emphases on self-organized states and non-conventional statistics; foundations of quantum mechanics and quantum information and control; network structure and dynamics with a wide variety of applications; and scaling in social and biological systems. Significant progress has been made in understanding phenomena as varied as criticality in rainfall, modularity in complex networks, and metabolic scaling with body mass. Future directions in the physics of complex systems include universality in dissipative systems, quantum simulation and the feedback control of decoherence, and the structure of optimal distribution networks. The wide-ranging sciences brought together at SFI utilize more than merely existing methods and models from physics.  Many dynamical properties in chemical, biological and engineered systems present new paradigms for organization that will expand the conceptual scope of physics. For a comprehensive list of SFI researchers working on these and related topics, click here.

• Nonequilibrium Statistical Physics and Self-Organization

• Quantum Information and Decoherence

• Networks: Social, Biological, and Technological

• Scaling, Universality, and Quantitative Laws of Life

Nonequilibrium Statistical Physics and Self-Organization

Self-organized states, a long-standing topic of study at SFI, remain on the forefront of many-body physics, as problems of both theoretical foundation and empirical validation.  The paradox of self-organization is that it results in typical configurations, which, when viewed as a subset within an equilibrium ensemble, have an entropy smaller than the equilibrium entropy.  Thus in classical thermodynamics such states are expected to be improbable and ephemeral rather than self-generating and stable. The very general principles of inference leading to the equilibrium entropy have not produced a counterpart for self-organized systems, resulting in widespread recourse to phenomenological models. Nonetheless, explicit microdynamic models have suggested that self-organization to the critical point of a continuous, dynamical phase transition can account for both complex spatial structure and heavy-tailed distributions associated with many presumed self-organizing systems in nature.  It has been disputed, however, that self-organized criticality (SOC) is a valid description of complex natural phenomena and the limited theoretical linkage of SOC to concepts such as universality and classical phase transitions have made it unclear whether power-law correlations were sufficient evidence of criticality. Current work in these areas at SFI is directed toward developing new entropy principles for driven systems, deeper connections between self-organized and dissipative dynamical criticality, including the connection to universality classification, and studying a striking suite of empirical observations that suggest SOC as the explanation for moisture balance in the global weather system.

SFI Visiting Scientist Ole Peters and collaborators have argued that rainfall is an SOC phenomenon. Using satellite data from NASA’s Tropical Rainfall Measuring Mission, they present evidence that tropical convection and precipitation place the atmosphere near the critical point of a non-equilibrium phase transition between dry quiescence and strong deep convection with precipitation. Knowledge of the critical properties of this transition can help inform climate modeling by providing more realistic convective closure schemes, based on energy constraints and relations between atmospheric observables such as column-integrated water vapor and precipitation rate.

As part of a larger project to understand how self-organized chemical states can be driven into existence by non-equilibrium energetic boundary conditions, SFI Professor Eric Smith has developed ensemble descriptions and entropy methods for driven quantum ensembles.  In particular, he has shown that maximization of the true entropy of a driven ensemble can lead to the spontaneous formation of stable currents, and that the entropy production principles of Onsager arise as context-dependent approximations if the equilibrium-form entropy instead of the exact entropy is used. Smith has also shown that the onset of order in thermodynamically reversible nonlinear systems has the same statistics as classical second-order phase transition, and that the ordered state has reduced impedance to the energy stresses created by the environment, relative to that of the disordered near-equilibrium state.

Among the tools used to study dissipation in quantum systems are the Schwinger-Keldysh and Doi-Peliti (DP) path integral methods. Smith, SFI External Professor Supriya Krishnamurthy , SFI Professor David Krakauer , and SFI External Professor Walter Fontana , are using DP methods to study the statistics of cellular signal transduction, have advanced the use of instanton methods to predict properties of second-order irreversible phase transitions.  (The application in this work was to the stochastic creation of bistable switches for cell states, see Information Processing & Computation in Complex Systems ). To further understand the statistical basis of self-organization, particularly its applications to chemistry, Smith will expand the existing quantum models of driven ensembles to dissipative field theories capable of representing the transition states of chemical reactions.  To do this, it will be necessary to generalize Doi-Peliti techniques, currently applied to classical stochastic processes, to quantum systems with thermodynamic regularities in non-commuting observables.  The immediate goal is a statistical description of steady-state driven chemical reactions.  The longer-term goal, which will require aggregation of individual reactions into networks, is to derive the phase transition structure of metabolism from graph properties such as network autocatalysis

In addition to the application of standard statistical mechanics to driven systems, SFI researchers have studied the application of nonstandard statistics to systems with long-range correlations among components.  An important nonstandard form of entropy is “q-entropy” for q not equal to one, as discussed by SFI Visiting Scientist Constantino Tsallis , SFI Professor Murray Gell-Mann and collaborators, and labeled “nonextensive.” At SFI, Tsallis, Gell-Mann, and former SFI Postdoctoral Fellow Yuzuru Sato showed that whereas q-entropy is nonextensive for separable systems (where it should probably not be used), it is extensive when phase space is unevenly occupied in suitable ways. Tsallis, Gell-Mann and two collaborators, Steinberg and Umarov, have established the q-equivalent of the central limit theorem, thus justifying the use of q-statistics in some interesting problems in physics and other disciplines.

Universality is an important tool in statistical physics, providing a principled way to discard irrelevant details of a system and focus on simple mechanisms that determine the macroscopic behavior for all systems in a class. However, the origin of universality in models with self-organized non-equilibrium phase transitions, such as in the Oslo Christensen, BTW, or Manna models, is still a matter of debate. SFI Postdoctoral Fellow Ole Peters and collaborators have shown that previously suggested mechanisms of coupling tuning and order parameters are incapable of producing universality. SFI Postdoctoral Fellow Michelle Girvan and Peters have developed a new mechanism that imposes a maximization condition on the tuning parameter, thus targeting critical singularities. Future research will take into account the fact that in applications to large-scale real-world systems such as the atmosphere, the assumption of homogeneous tuning fields is not justified. In this case, the spectrum of scales of the inhomogeneities interacts with the diverging correlation length near criticality. Peters, SFI Postdoctoral Fellow Michael Gastner , SFI international fellow Beata Oborny , and collaborators are addressing the question how noisy tuning fields with different types of Fourier spectra affect critical behavior.

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Quantum Information and Decoherence

The nature of quantum superposition and the proper representation of measurement have been conceptual concerns since the first successes of quantum theory.  They remain fundamental to the proper description of the classical world, and to the formulation of a quantum cosmology. Especially within the last decade, they have also become pressing practical problems, as quantum computing on a useful scale has been suggested to be feasible, although subject to qualitatively new engineering and control problems. The counterintuitive phenomenon of quantum superposition in macroscopic systems, not usually encountered in nature, is expressed as entanglement in quantum computers, where it allows two or more subsystems to possess more information about each other than is classically possible. Entanglement is responsible for the power of quantum computing, but also for the difficulty of operating and analyzing macroscopic quantum systems. SFI researchers have played important roles in the studies of decoherence as the foundation of classical physics, in the study of quantum computing, and in the use of quantum simulators to study quantum systems. Research described over the next several paragraphs focuses on physical aspects of quantum information; research on quantum computation, including quantum algorithms and cryptography, is summarized in Information Processing and Computation in Complex Systems .

When the dynamics of a quantum system generates large amounts of entanglement between its component parts, it becomes practically impossible to simulate its behavior using a classical computer. Indeed, the only known way to simulate the behavior of a quantum system consisting of three hundred two-level components, such as nuclear spins, is to use a classical computer with 2300 ^ 1090 bits—in other words, a classical computer the size of the universe itself!

In contrast, if one uses quantum computers to simulate the behavior of quantum systems, only three hundred qubits are required. Thus quantum simulators can be used to simulate complex quantum systems which are far beyond the reach of classical simulation.  They can also be used to probe quantum chaos, decoherence, and the transition from quantum to classical behavior.  As a proof of concept, using solid-state nuclear magnetic resonance (NMR) quantum information processors, SFI External Professor Seth Lloyd and collaborators have succeeded in performing quantum simulations of quantum systems consisting of 1018 two-level components.
The kind of complexity that best represents what is usually meant by the term in conversation, and in many computing applications as well as scientific discourse, is the “effective complexity” of SFI Professor Murray Gell-Mann and Lloyd, the algorithmic information content of the regularities of the entity concerned. They are now studying further the connection they have discussed between effective complexity and mutual algorithmic information, which can be diagnostic of complexity.

Properly speaking, quantum mechanics must be applied to the universe as a whole.  Subsystems have wave functions only in an approximate sense.  Over the last twenty years, Gell-Mann and SFI External Professor James Hartle have formulated quantum mechanics so that it is compatible with quantum cosmology, making use of the concept of decoherence that is crucial in discussions of quantum computing. Elementary notions of consistency require that one use at least “medium decoherence,” in which histories are distinguished by orthogonal state vectors.  Probabilities are assigned to alternative decoherent coarse-grained histories of the universe. A set of such histories, forming a branching tree, is called a realm. A particular kind of coarse graining, related to hydrodynamics, yields a quasi-classical realm, in which certain variables approximately obey deterministic equations of motion, apart from frequent small fluctuations and occasional major branchings (as in measurement situations). It is striking that the coarse graining here is closely related to that used in the construction of physico-chemical (or thermodynamic) entropy. Gell-Mann and Hartle are pursuing this connection further.

A number of fundamental questions remain unresolved in the field of quantum communications; notably, the ultimate channel capacity of quantum channels such as the noisy, lossy bosonic channel that makes up virtually all of our current communication infrastructure in the form of wires, optical fibers, and wireless and cellular networks. Lloyd recently derived the capacity of the bosonic channel with pure loss. However, resolving the limiting communication capacity of such networks, and finding techniques for attaining that capacity, is a deep and important problem.

Lloyd and co-authors have shown how quantum information provides fundamental limits on the accuracy of atomic clocks and the global positioning system (GPS). However, in addition to its technological relevance, this work might also give us some insight into fundamental physical theories. Lloyd’s preliminary investigations indicate that a theory of quantum gravity based on quantum measurement can be made self-consistent and overcomes a number of the outstanding difficulties of quantizing gravity.  In this theory, gravity is not quantized directly: in other words, intervals of space and time are not associated with Hermitian observables.  Instead, space and time are derived from “GPS observables” such as the number of ticks of atomic clocks or the clicks of detectors. Accordingly, the spacetime metric exhibits fluctuations that it “inherits” from the quantum fluctuations of clocks and signals. Quantizing clocks and signals rather than the metric avoids the usual unrenormalizability of quantum gravity, and provides a direct method for investigating the behavior of black holes, spacetime singularities.

Finally, the construction of quantum machines is likely to need major advances in our uses of feedback techniques, which differ qualitatively from their classical counterparts because of information-disturbance theorems that preclude gaining information about a quantum system without introducing projection noise into its evolution. SFI Professor Tanmoy Bhattacharya and collaborators have demonstrated that condensing information from past measurements that are relevant to future evolution into a conditioned state vector considerably simplifies the control equations.  Even though the complete calculation of the conditioned state in real time is not possible with today’s computational resources, they find that approximate schemes are sufficient to control the system, and current experiments are within an order of magnitude of demonstrating this in atom-optics experiments. On a more theoretical side, the quantum back action can often be modeled as a noise process with an effective temperature. The systems being studied are, however, far out of equilibrium, and interesting questions as to the use of these effective thermodynamic description arise in this context.

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Networks: Social, Biological, and Technological

Many complex systems can be represented as a graph or network, i.e., a collection of vertices with edges between them indicating their interactions. These include networks of scientific citations and collaborations; technological networks such as the Internet, the World Wide Web, road networks, power grids, railways and airline routes; and biological networks, such as food webs, genetic regulatory networks, and metabolic networks.  Because of their ubiquity, networks are of deep interest to all these disciplines. Generic questions include: How is the structure of a network related to its function and behavior? What mathematical and statistical models accurately capture the important aspects of a network’s structure? And, can these models predict a network’s future behavior? SFI research on networks from a physics perspective is described in the following paragraphs. Research on network structure, function and dynamics in relation to biological and social processes is described under Innovation in Evolutionary Systems and Emergence, Organization, and Dynamics of Living Systems .

Much of SFI’s contribution to the physics of networks has focused on community structure, in which the network is organized into densely-connected clumps with sparse connections between clumps. These clumps can represent, for instance, genres of music in a network of recommendations, or functional modules in metabolic networks.  Although many algorithms now exist for detecting community structure in networks, they are often based on heuristic notions of “community,” and are often computationally slow, making them infeasible on the most interesting networks. SFI researchers have sought algorithms that are both mathematically principled and highly efficient. With SFI External Professor Mark Newman, SFI Postdoctoral Fellow Michelle Girvan developed an algorithm based on a quantity called “modularity” whose running time is O(n2) on networks of n nodes. SFI Postdoctoral Fellow Aaron Clauset , SFI Professor Cristopher Moore , and Newman improved this algorithm so that its running time is just O(n log2 n)on realistic networks. This made it possible for the first time to analyze networks with millions of nodes and edges, such as the recommendation network of Amazon, in which each node is a book or CD.

A more general way to represent the global organization of a network is with a hierarchical decomposition, in which the network is described as a set of groups of vertices, groups of groups, and so on, up through all levels of organization (Fig. 1). SFI Postdoctoral Fellow Clauset , Professor Cris Moore , and External Professor Newman recently developed a method for extracting the hierarchical organization of a network, using a Monte Carlo algorithm to find the maximum-likelihood structure within a large class of models.  They showed that this method can be used to automatically annotate real-world networks, for instance by labeling each node with the strength of its affiliation to its group, or labeling edges according to how surprising it is that those two nodes interact.  Many real-world networks are quite labor-intensive to measure; in protein networks, for instance, each pair of proteins must be individually tested to see if they interact. Often only a small fraction of the network’s edges have actually been observed. The statistical techniques of Clauset, Moore and Newman use the observed edges to predict, with high accuracy, unobserved edges that were not observed due to sampling constraints. 

Another focus of research at SFI on the physics of networks is Internet topology and routing. Even though the Internet is a critical part of the infrastructure of modern society, it is still relatively poorly understood as a complex system. Most previous studies of the Internet’s vulnerability idealized it as a random graph with a power-law degree distribution, and asked what happens when this graph becomes disconnected when nodes or edges are deleted.  However, this model is rather unrealistic; a “node” can represent a large collection of computers and human personnel, often spread over a large geographic area.  Thus deleting a node would correspond to, say, crashing all of AT&T’s computers at once.  In contrast, a real vulnerability lies in the Border Gateway Protocol (BGP), the method that all nodes use to connect users across the network. If a malicious or misconfigured node falsely advertises short paths to popular destinations, it can “hijack” the corresponding communications, and thereby disconnect that destination from part of the network. Unlike the simple topological model, little is known about the Internet’s vulnerability to this kind of attack, which has produced several large Internet outages. Clauset and Newman are developing a more realistic mathematical model of the Internet by combining a structured random graph, whose topology is statistically similar to the real Internet, with a realistic model of BGP’s routing behavior.

A third direction of SFI networks research is the relation between network topology and geographical structure. Geographical structure is important when a network’s topology is correlated with the positions of nodes on an underlying surface like road networks. Finding the optimal location of facilities, such as airports, stores, or post offices, in order to efficiently distribute a commodity (e.g., passengers, groceries, or mail) is a task of obvious practical importance for governments and the private sector. The classic facility location literature, such as Central Place Theory or the Hotelling model, assumes that demand is equal at every point in space. However, in reality the spatial density of customers varies by several orders of magnitude between different regions, and it is usually not obvious how to generalize existing theories to deal with nonuniform demand. SFI Postdoctoral Fellow Gastner and External Professor Mark Newman have studied the problem of locating facilities to minimize the average distance from a person’s home to the nearest facilities. Some previous approximate treatments of this problem by others had indicated that the optimal distribution of facilities should have a density that increases as the two-thirds power of the population density. Gastner and Newman have confirmed this result numerically and have proposed a novel density-dependent map projection simplifying the task of determining a nearly optimal facility allocation.

Previous contributions to research on the physics of networks include work by former SFI Postdoctoral Fellow Duncan Watts and External Professor Steve Strogatz on the small-world phenomenon.

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Scaling, Universality, and Quantitative Laws of Life

One of the grand challenges of 21st century science is the search for fundamental principles beyond the genetic code and Darwinian evolutionary process that govern how the complexity of life emerges from its underlying simplicity. This will almost certainly involve close trans-disciplinary collaboration between biologists, chemists, physicists and mathematicians. A basic question is: are there “universal laws of life” that can be mathematised so that biology can be formulated as a predictive, quantitative Science? Although it is unlikely that there are yet-to-be-discovered “Newton’s Laws of Biology” that lead to precise calculations of any biological phenomenon, it is not unreasonable that the generic coarse-grained behavior of living systems obeys quantifiable universal laws that capture their essential features. These provide a “zeroth order” point of departure for quantitatively understanding real biosystems which can be viewed as variations or perturbations around idealized norms due to local environmental conditions or historical evolutionary divergence.

Scaling is a potentially powerful tool in this process because scaling laws typically reflect generic features and principles that are independent of detailed dynamics or specific characteristics of particular models. Phase transitions, chaos, unification of forces, and the discovery of quarks are but a few of the more significant examples where scaling has illuminated important universal principles or structure. SFI Professor Geoffrey West , External Professor James Brown and collaborator Brian Enquist have been spearheading a major effort to understand the origin of scaling laws in biology and use the paradigm to formulate general principles of biological structure and organization. The work, which has received a great deal of attention in both the scientific and popular press, has recently been extended to investigate similar questions regarding social organizations, and in particular, cities.   

Although life is surely the most complex and diverse system in the universe, covering over 30 orders of magnitude in mass from the molecules of the genetic code and metabolic process up through microbes and mammals to macroscopic ecosystems, it manifests an extraordinary simplicity and universality in its scaling behavior over a broad spectrum of phenomena and an immense range of energy and mass. Understanding the origin of biological scaling might therefore reveal underlying dynamics and structure and lead to the possibility of constructing a quantitative, predictive, mathematical theory of biological structure, function and organization.

Scaling laws in biology are typically of a simple power law form: Y = Y0Mb, where Y is some observable, Y0 a normalization constant, and M the mass of the organism. Of even greater significance, the exponents, b, are invariably approximate simple multiples of ¼ (53, 270). Among the many variables obeying these allometric scaling laws are fundamental rates, times, and dimensions such as metabolic rate, lifespan, growth rate, heart rate, DNA nucleotide substitution rates, lengths of aortas and genomes, heights of trees, cerebral grey matter, mitochondrial densities, and RNA concentrations.

The predominance of scaling across all scales and life forms is surely telling us something fundamental about biological systems and opens a possible window onto fundamental underlying laws of biology. West, Brown and Enquist have proposed a set of “universal principles” for quantitatively understanding the origin of these scaling laws at all scales. The basic idea is that, in order to support the huge number of localized microscopic units of an organism, life at all scales is sustained by optimized hierarchical fractal-like branching networks whose terminal units are invariant. Functionally, biological systems are ultimately constrained by the rates at which energy, metabolites, and information can be supplied through these networks. The theory accounts, in a well-defined, testable fashion, for 1/4 power scaling in diverse biological phenomena and provides a quantitative framework for addressing many important problems, both basic and applied. The kinds of questions that either have been or will be addressed include: How many oxidase molecules and mitochondria are there in a cell? How many RNA molecules? Why do we stop growing and what weight will we reach? Why do we live approximately 100 years, and not a million or a few weeks, and how is this related to molecular scales? What are the flow rate, pulse rate, pressure, and dimensions in any vessel of any circulatory system? Why do we sleep eight hours a day, a mouse eighteen and an elephant three? How many trees of a given size are there in a forest, how far apart are they, how many leaves does each have and how much energy flows in each branch? What are the limits on the size of organisms?

Starting from these principles biological systems can effectively be described by field theoretic equations of motion with well-defined natural physical cut-offs. Detailed integrated analytic models of mammalian circulatory and respiratory systems, and of plant vascular systems and communities , have been successfully constructed.  Many scaling laws have been derived between organisms (e.g., 1/4-power allometry between mice and elephants), within an individual organism (e.g., from the aorta to capillaries), and during ontogeny (e.g., from seedling to giant sequoias). The theory puts severe physical constraints on the structural design and functional characteristics of organisms leading to a quantitative integrated description of the entire system. Where data exist, excellent agreement is generally found; where data are not available, the model provides testable predictions.

Although originally applied to networks in macroscopic organisms (e.g. to mammalian and plant vasculature), the theoretical framework is much more general, as it is based on universal properties of complex hierarchical networks. West, Brown and Enquist showed that the ubiquitous 1/4 arises because organisms effectively function in four spatial dimensions even though they physically exist in three and applied it to intra-cellular organelles. This provided a “geometric” argument, independent of dynamical details, for why 1/4 dominates in network transport systems despite major differences in explicit structural designs

The theoretical framework has been extended to include temporal and temperature dependences of biological phenomena. One goal of ongoing and future work is to make corrections and extensions to the 3/4 power law for metabolic rate. This is important not only to check that they are indeed small but also because predictions of deviations provide possible new tests of the theory. West, Brown and Enquist originally pointed out that deviations from ¾ are expected for small mammals and that the ¾ result is an asymptotic prediction valid for large mammals. Ironically, the work was criticized by researchers who perceived such deviations in the data, but who did not appreciate that these had, in fact, been predicted by the theory. West and Former SFI Postdoctoral Fellow Van Savage propose to derive quantitative corrections to leading order results and compare them with data on scaling exponent and detailed dynamics and geometry of the circulatory system. A detailed analysis of the data was carried out by assembling the largest ever data set on mammalian metabolic rates, which showed overall consistency with ¾. West, Brown and Enquist will extend this to consider metabolic rate at maximal effort where the system is no longer optimized.

West, former Postdoctoral Fellow Van Savage and former Physics REU Alex Herman are exploring the role of scaling in the study tumor growth and vascularization. For tumors the theory must successfully describe the interface and integration of two coupled, but essentially autonomous, dynamical networks: the host and the tumor. Thus, in addition to addressing an important biomedical question, this potentially provides a sensitive test of the theory.

West and SFI External Professor Walter Fontana , and collaborators at Temple and Louisville medical schools have formed a large-scale collaboration on aging, morality and lifespan to address why humans live for ~100 years rather than a few months or thousands of years, and why mice live for only 2-3 years, even though they are made of essentially the same tissue? Where in the molecular structure of genes and respiratory enzymes, which operate at microscopic time scales, are time scales of years? Hints can be gleaned from scaling. Lifespan, like almost all life-history events, scales as ~M1/4 (53, 270). Additionally, both total lifetime energy needed to sustain unit mass and the number of turnovers of CytO molecules in mitochondria per lifetime are invariants (318). This suggests that a generic fundamental dynamical theory of longevity and senescence can be constructed. West, Fontana and collaborators are addressing this possibility by assuming that the origin of aging is dissipative forces (e.g. free radical damage via metabolism) and subsequent entropy production in supply networks, which ultimately produce irreversible damage at the molecular level.

The central and peripheral nervous systems are also hierarchical branching networks which exhibit scaling: the vertebrate brain scales non-linearly with body size, ~M3/4 (53, 270), and the grey to white matter ratio ( “cables to processors”) ~ M5/4 (341). Brain architecture has the remarkable property that computational power is increased simply by increasing the number of neurons. SFI Professor Geoffrey West and SFI External Professor Chuck Stevens are investigating how these can reveal new laws that neuronal components follow and decide what sorts of mathematical operations neuronal circuits carry out. This will involve both theoretical and experimental work (primarily on fish at the Salk Institute).

Finally, West, Savage, and SFI Postdoctoral Fellow Michelle Girvan are attempting to determine the universality class of networks that satisfy the general principles of the scaling theory and which lead to maximal fractal-like behavior. A major conceptual challenge presented by this work is why does it work so well? Is there some fixed point, or deep basin of attraction, operating within the general dynamical structure that ensures that, despite their enormous complexity, general features of biological systems are robust against significant perturbations? Can loops with local feedback mechanisms (as in the capillary bed) be included? How critical is space-filling? Preliminary investigations of specific examples in which local perturbations, such as loops and turbulence, are added suggest that there are fixed point theorems in “network space” which strongly favor the class of networks that have evolved.

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