Complex systems present problems both in mathematical modelling and philosophical foundations. The study of complex systems represents a new approach to science that investigates how relationships between parts give rise to the collective behaviors of a system and how the system interacts and forms relationships with its environment. The equations from which models of complex systems are developed generally derive from statistical physics, information theory and non-linear dynamics, and represent organized but unpredictable behaviors of natural systems that are considered fundamentally complex.
Complex systems may have billion components making consensus formation slow and difficult. Recently several overlapping stories emerged from various disciplines, including protein structures, neuroscience and social networks, showing that fast responses to known stimuli involve a network core of few, strongly connected nodes. In unexpected situations the core may fail to provide a coherent response, thus the stimulus propagates to the periphery of the network. Here the final response is determined by a large number of weakly connected nodes mobilizing the collective memory and opinion, i.e. the slow democracy exercising the 'wisdom of crowds'. This mechanism resembles to Kahneman's "Thinking, Fast and Slow" discriminating fast, pattern-based and slow, contemplative decision making. The generality of the response also shows that democracy is neither only a moral stance nor only a decision making technique, but a very efficient general learning strategy developed by complex systems during evolution. The duality of fast core and slow majority may increase our understanding of metabolic, signaling, ecosystem, swarming or market processes, as well as may help to construct novel methods to explore unusual network responses, deep-learning neural network structures and core-periphery targeting drug design strategies.
(Illustrative videos can be downloaded from here:this http URL)
Fast and slow thinking -- of networks: The complementary 'elite' and 'wisdom of crowds' of amino acid, neuronal and social networks Peter Csermely
Temporal order memories are critical for everyday animal and human functioning. Experiments and our own experience show that the binding or association of various features of an event together and the maintaining of multimodality events in sequential order are the key components of any sequential memories—episodic, semantic, working, etc. We study a robustness of binding sequential dynamics based on our previously introduced model in the form of generalized Lotka-Volterra equations. In the phase space of the model, there exists a multi-dimensional binding heteroclinic network consisting of saddle equilibrium points and heteroclinic trajectories joining them. We prove here the robustness of the binding sequential dynamics, i.e., the feasibility phenomenon for coupled heteroclinic networks: for each collection of successive heteroclinic trajectories inside the unified networks, there is an open set of initial points such that the trajectory going through each of them follows the prescribed collection staying in a small neighborhood of it. We show also that the symbolic complexity function of the system restricted to this neighborhood is a polynomial of degree L − 1, where L is the number of modalities.
Recently a concept of self-excited and hidden attractors was suggested: an attractor is called a self-excited attractor if its basin of attraction overlaps with neighborhood of an equilibrium, otherwise it is called a hidden attractor. For example, hidden attractors are attractors in systems with no equilibria or with only one stable equilibrium (a special case of multistability and coexistence of attractors). While coexisting self-excited attractors can be found using the standard computational procedure, there is no standard way of predicting the existence or coexistence of hidden attractors in a system. In this plenary survey lecture the concept of self-excited and hidden attractors is discussed, and various corresponding examples of self-excited and hidden attractors are considered.
Over the last decades, in disciplines as diverse as economics, geography, and complex systems, a perspective has arisen proposing that many properties of cities are quantitatively predictable due to agglomeration or scaling effects. Using new harmonized definitions for functional urban areas, we examine to what extent these ideas apply to European cities. We show that while most large urban systems in Western Europe (France, Germany, Italy, Spain, UK) approximately agree with theoretical expectations, the small number of cities in each nation and their natural variability preclude drawing strong conclusions. We demonstrate how this problem can be overcome so that cities from different urban systems can be pooled together to construct larger datasets. This leads to a simple statistical procedure to identify urban scaling relations, which then clearly emerge as a property of European cities. We compare the predictions of urban scaling to Zipf's law for the size distribution of cities and show that while the former holds well the latter is a poor descriptor of European cities. We conclude with scenarios for the size and properties of future pan-European megacities and their implications for the economic productivity, technological sophistication and regional inequalities of an integrated European urban system.
Urban Scaling in Europe Luis M. A. Bettencourt, Jose Lobo
This course of 25 lectures, filmed at Cornell University in Spring 2014, is intended for newcomers to nonlinear dynamics and chaos. It closely follows Prof. Strogatz's book, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering."
The mathematical treatment is friendly and informal, but still careful. Analytical methods, concrete examples, and geometric intuition are stressed. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.
A unique feature of the course is its emphasis on applications. These include airplane wing vibrations, biological rhythms, insect outbreaks, chemical oscillators, chaotic waterwheels, and even a technique for using chaos to send secret messages. In each case, the scientific background is explained at an elementary level and closely integrated with the mathematical theory. The theoretical work is enlivened by frequent use of computer graphics, simulations, and videotaped demonstrations of nonlinear phenomena.
The essential prerequisite is single-variable calculus, including curve sketching, Taylor series, and separable differential equations. In a few places, multivariable calculus (partial derivatives, Jacobian matrix, divergence theorem) and linear algebra (eigenvalues and eigenvectors) are used. Fourier analysis is not assumed, and is developed where needed. Introductory physics is used throughout. Other scientific prerequisites would depend on the applications considered, but in all cases, a first course should be adequate preparation.
Ecosystems exhibit surprising regularities in structure and function across terrestrial and aquatic biomes worldwide. We assembled a global data set for 2260 communities of large mammals, invertebrates, plants, and plankton. We find that predator and prey biomass follow a general scaling law with exponents consistently near ¾. This pervasive pattern implies that the structure of the biomass pyramid becomes increasingly bottom-heavy at higher biomass. Similar exponents are obtained for community production-biomass relations, suggesting conserved links between ecosystem structure and function. These exponents are similar to many body mass allometries, and yet ecosystem scaling emerges independently from individual-level scaling, which is not fully understood. These patterns suggest a greater degree of ecosystem-level organization than previously recognized and a more predictive approach to ecological theory.
The predator-prey power law: Biomass scaling across terrestrial and aquatic biomes Ian A. Hatton, Kevin S. McCann, John M. Fryxell, T. Jonathan Davies, Matteo Smerlak, Anthony R. E. Sinclair, Michel Loreau
Author Summary Given a body and an environment, what is the brain complexity needed in order to generate a desired set of behaviors? The general understanding is that the physical properties of the body and the environment correlate with the required brain complexity. More precisely, it has been pointed that naturally evolved intelligent systems tend to exploit their embodiment constraints and that this allows them to express complex behaviors with relatively concise brains. Although this principle of parsimonious control has been formulated quite some time ago, only recently one has begun to develop the formalism that is required for making quantitative statements on the sufficient brain complexity given embodiment constraints. In this work we propose a precise mathematical approach that links the physical and behavioral constraints of an agent to the required controller complexity. As controller architecture we choose a well-known artificial neural network, the conditional restricted Boltzmann machine, and define its complexity as the number of hidden units. We conduct experiments with a virtual six-legged walking creature, which provide evidence for the accuracy of the theoretical predictions.
The tremendous popular success of Chaos Theory shares some common points with the not less fortunate Relativity: they both rely on a misunderstanding. Indeed, ironically , the scientific meaning of these terms for mathematicians and physicists is quite opposite to the one most people have in mind and are attracted by. One may suspect that part of the psychological roots of this seductive appeal relies in the fact that with these ambiguous names, together with some superficial clichés or slogans immediately related to them ("the butterfly effect" or "everything is relative"), some have the more or less secret hope to find matter that would undermine two pillars of science, namely its ability to predict and to bring out a universal objectivity. Here I propose to focus on Chaos Theory and illustrate on several examples how, very much like Relativity, it strengthens the position it seems to contend with at first sight: the failure of predictability can be overcome and leads to precise, stable and even more universal predictions.
Information is a precise concept that can be defined mathematically, but its relationship to what we call "knowledge" is not always made clear. Furthermore, the concepts "entropy" and "information", while deeply related, are distinct and must be used with care, something that is not always achieved in the literature. In this elementary introduction, the concepts of entropy and information are laid out one by one, explained intuitively, but defined rigorously. I argue that a proper understanding of information in terms of prediction is key to a number of disciplines beyond engineering, such as physics and biology.
The SIR-network model, introduced in [S. Boatto et al., SIR-Network Model for Epidemics Dynamics in a City, in preparation] and [L. Stolerman, Spreading of an Epidemic over a City: A Model on Networks, Master's thesis, 2012 (in Portuguese)], deals with the propagation of disease epidemics in highly populated cities. The nodes, or vertices, are the city's neighborhoods, in which the local populations are assumed to be well-mixed. The directed edges represent the fractions of people moving from their neighborhoods of residence to those of daily activities. First, we present some fundamental properties of the basic reproduction number ($R_o$) for this model. In particular, we focus on how $R_o$ depends upon the geometry and the heterogeneity (different infection rates in each vertex) of the network. This allows us to conclude whether an epidemic outbreak can be expected or not. Second, we submit the SIR-network model to data fitting, using data collected during the 2008 Rio de Janeiro dengue fever epidemic. Important conclusions are drawn from the fitted parameters, and we show that improved results are found when a time-dependent infection parameter is introduced.
“The study of cas is a difficult, exciting task. The returns are likely to be proportionate to the difficulty.” Holland ()
On August 9, 2015, cancer took Prof. John Henry Holland away from us. Prof. Holland was a pioneer of Complex Adaptive Systems (CAS) research and a true inspiration. He is known not only for his work on CAS, Holland (, )—which he would fondly write as “cas”—but also for his seminal work on adaptation in natural and artificial systems leading to the creation of genetic algorithms and eventually the fields of evolutionary computation, Holland () and Learning Classifier Systems, Holland and Holyoak ().
Holland was a truly interdisciplinary academic. He had an undergraduate degree in Physics from MIT (1950), an M.A. in Mathematics (1954) and possibly the first ever PhD in Computer Science (1959), both from the University of Michigan—a place where he also subsequently served as a Professor of Psychology, Electrical Engineering and Computer Science.
Holland leaves behind his legacy in the form of a large number of thought-provoking articles, video lectures, books, and inspired people—ranging from colleagues, fellows and students to budding complexity enthusiasts. Two of his recent books summarize his views on CAS in both a longer, Holland () as well as a shorter form, Holland (). It is easy to foresee that these works will serve not only as a guide to CAS but also guidance for future generations. Holland will indeed be greatly missed.
This course provides an introduction to the study of environmental phenomena that exhibit both organized structure and wide variability—i.e., complexity. Through focused study of a variety of physical, biological, and chemical problems in conjunction with theoretical models, we learn a series of lessons with wide applicability to understanding the structure and organization of the natural world. Students also learn how to construct minimal mathematical, physical, and computational models that provide informative answers to precise questions. This course is appropriate for advanced undergraduates. Beginning graduate students are encouraged to register for 12.586 (graduate version of 12.086). Students taking the graduate version complete different assignments.
An introductory conclusion The Systems Approach: principles A few years ago I wrote a blog post about anti-planning as an alternative to the systems approach. Part of the post was devoted to a number of principles of deception-perception. Churchman discusses their importance in the concluding chapter of The Systems Approach (TSA), which…
The nexus concept aims at extending ‘integrated management thinking’, which has been applied with varying success in diverse disciplines and has become especially popular in water resources management. UNU-FLORES developed an interactive platform, the Nexus Tools Platform, for inter-model comparison of existing modeling tools related to Water-Soil-Waste Nexus providing detailed model information and advanced filtering based on real-time visualizations.
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