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Special Issue "Complex Systems"

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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: closed (20 September 2013)

Special Issue Editor

Guest Editor
Prof. Dr. Stefan Thurner (Website)

Medical University of Vienna, Spitalgasse 23, 1090 Vienna, Austria
Interests: statistical mechanics of complex systems; theory of evolutionary processes; entropy formulations; network theory; scaling theory; annomalous diffusion

Special Issue Information

Dear Colleagues,

Complex Systems, when seen as statistical systems of strongly interacting components, are far from being understood on a fundamental level. These systems cover an immense range of important phenomena in the natural-, life- and social sciences. In many of these fields an incremental understanding of underlying principles would mean significant progress. The role of statistical mechanics in understanding Complex Systems is fundamental, in particular one has to understand to what extend thermodynamical descriptions are sensible for specific systems, and where new paths have to be taken to achieve reasonable ways to manage the typically large numbers of variables and parameters. The role of entropy has to be understood for non-ergodic Complex Systems, and new ways to practically characterize high dimensional phase diagrams have to be explored. Big progress has been made in network theory, one of the fundamental building blocks of strongly interacting systems, however a new generation of problems lies ahead: the understanding of mutual influences of different networks linking the same set of nodes (multiplex networks), or the understanding of networks of networks. The purpose of this special issue is to try to localize the status quo of the statistical mechanical understanding of Complex Systems and its applications, and to sketch innovative paths into the future, both in fundamental understanding and applications.

Specific topics of interest include:

  • entropies of non-ergodic systems
  • statistical mechanics of networks
  • multiplex networks
  • collective dynamics
  • evolutionary dynamics
  • generalized statistics - superstatistics
  • multiscale-analysis
  • non-ergodic systems
  • applications and agent based models
Prof. Dr. Stefan Thurner
Guest Editor

Submission

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. Papers will be published continuously (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are refereed through a peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed Open Access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1400 CHF (Swiss Francs).

Published Papers (10 papers)

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Research

Jump to: Review

Open AccessArticle Relative Entropy, Interaction Energy and the Nature of Dissipation
Entropy 2014, 16(6), 3173-3206; doi:10.3390/e16063173
Received: 10 February 2014 / Revised: 20 April 2014 / Accepted: 23 May 2014 / Published: 6 June 2014
Cited by 4 | PDF Full-text (314 KB) | HTML Full-text | XML Full-text
Abstract
Many thermodynamic relations involve inequalities, with equality if a process does not involve dissipation. In this article we provide equalities in which the dissipative contribution is shown to involve the relative entropy (a.k.a. Kullback-Leibler divergence). The processes considered are general time evolutions [...] Read more.
Many thermodynamic relations involve inequalities, with equality if a process does not involve dissipation. In this article we provide equalities in which the dissipative contribution is shown to involve the relative entropy (a.k.a. Kullback-Leibler divergence). The processes considered are general time evolutions both in classical and quantum mechanics, and the initial state is sometimes thermal, sometimes partially so. By calculating a transport coefficient we show that indeed—at least in this case—the source of dissipation in that coefficient is the relative entropy. Full article
(This article belongs to the Special Issue Complex Systems)
Open AccessArticle How Do Life, Economy and Other Complex Systems Escape the Heat Death?
Entropy 2014, 16(3), 1687-1727; doi:10.3390/e16031687
Received: 2 December 2013 / Revised: 25 January 2014 / Accepted: 27 February 2014 / Published: 21 March 2014
PDF Full-text (2794 KB) | HTML Full-text | XML Full-text
Abstract
The primordial confrontation underlying the existence of our Universe can be conceived as the battle between entropy and complexity. The law of ever-increasing entropy (Boltzmann H-theorem) evokes an irreversible, one-directional evolution (or rather involution) going uniformly and monotonically from birth to death. [...] Read more.
The primordial confrontation underlying the existence of our Universe can be conceived as the battle between entropy and complexity. The law of ever-increasing entropy (Boltzmann H-theorem) evokes an irreversible, one-directional evolution (or rather involution) going uniformly and monotonically from birth to death. Since the 19th century, this concept is one of the cornerstones and in the same time puzzles of statistical mechanics. On the other hand, there is the empirical experience where one witnesses the emergence, growth and diversification of new self-organized objects with ever-increasing complexity. When modeling them in terms of simple discrete elements one finds that the emergence of collective complex adaptive objects is a rather generic phenomenon governed by a new type of laws. These “emergence” laws, not connected directly with the fundamental laws of the physical reality, nor acting “in addition” to them but acting through them were called “More is Different” by Phil Anderson, “das Maass” by Hegel etc. Even though the “emergence laws” act through the intermediary of the fundamental laws that govern the individual elementary agents, it turns out that different systems apparently governed by very different fundamental laws: gravity, chemistry, biology, economics, social psychology, end up often with similar emergence laws and outcomes. In particular the emergence of adaptive collective objects endows the system with a granular structure which in turn causes specific macroscopic cycles of intermittent fluctuations. Full article
(This article belongs to the Special Issue Complex Systems)
Open AccessArticle Entropy and the Predictability of Online Life
Entropy 2014, 16(1), 543-556; doi:10.3390/e16010543
Received: 1 December 2013 / Revised: 16 December 2013 / Accepted: 30 December 2013 / Published: 16 January 2014
Cited by 5 | PDF Full-text (271 KB) | HTML Full-text | XML Full-text
Abstract
Using mobile phone records and information theory measures, our daily lives have been recently shown to follow strict statistical regularities, and our movement patterns are, to a large extent, predictable. Here, we apply entropy and predictability measures to two datasets of the [...] Read more.
Using mobile phone records and information theory measures, our daily lives have been recently shown to follow strict statistical regularities, and our movement patterns are, to a large extent, predictable. Here, we apply entropy and predictability measures to two datasets of the behavioral actions and the mobility of a large number of players in the virtual universe of a massive multiplayer online game. We find that movements in virtual human lives follow the same high levels of predictability as offline mobility, where future movements can, to some extent, be predicted well if the temporal correlations of visited places are accounted for. Time series of behavioral actions show similar high levels of predictability, even when temporal correlations are neglected. Entropy conditional on specific behavioral actions reveals that in terms of predictability, negative behavior has a wider variety than positive actions. The actions that contain the information to best predict an individual’s subsequent action are negative, such as attacks or enemy markings, while the positive actions of friendship marking, trade and communication contain the least amount of predictive information. These observations show that predicting behavioral actions requires less information than predicting the mobility patterns of humans for which the additional knowledge of past visited locations is crucial and that the type and sign of a social relation has an essential impact on the ability to determine future behavior. Full article
(This article belongs to the Special Issue Complex Systems)
Open AccessArticle Multiscale Model Selection for High-Frequency Financial Data of a Large Tick Stock by Means of the Jensen–Shannon Metric
Entropy 2014, 16(1), 567-581; doi:10.3390/e16010567
Received: 20 October 2013 / Revised: 18 November 2013 / Accepted: 11 December 2013 / Published: 16 January 2014
PDF Full-text (368 KB) | HTML Full-text | XML Full-text
Abstract
Modeling financial time series at different time scales is still an open challenge. The choice of a suitable indicator quantifying the distance between the model and the data is therefore of fundamental importance for selecting models. In this paper, we propose a [...] Read more.
Modeling financial time series at different time scales is still an open challenge. The choice of a suitable indicator quantifying the distance between the model and the data is therefore of fundamental importance for selecting models. In this paper, we propose a multiscale model selection method based on the Jensen–Shannon distance in order to select the model that is able to better reproduce the distribution of price changes at different time scales. Specifically, we consider the problem of modeling the ultra high frequency dynamics of an asset with a large tick-to-price ratio. We study the price process at different time scales and compute the Jensen–Shannon distance between the original dataset and different models, showing that the coupling between spread and returns is important to model return distribution at different time scales of observation, ranging from the scale of single transactions to the daily time scale. Full article
(This article belongs to the Special Issue Complex Systems)
Open AccessArticle Entropy and Equilibria in Competitive Systems
Entropy 2014, 16(1), 1-22; doi:10.3390/e16010001
Received: 28 October 2013 / Revised: 16 December 2013 / Accepted: 16 December 2013 / Published: 24 December 2013
Cited by 5 | PDF Full-text (719 KB) | HTML Full-text | XML Full-text | Supplementary Files
Abstract
This paper investigates the applicability of thermodynamic concepts and principles to competitive systems. We show that Tsallis entropies are suitable for the characterisation of systems with transitive competition when mutations deviate from Gibbs mutations. Different types of equilibria in competitive systems are [...] Read more.
This paper investigates the applicability of thermodynamic concepts and principles to competitive systems. We show that Tsallis entropies are suitable for the characterisation of systems with transitive competition when mutations deviate from Gibbs mutations. Different types of equilibria in competitive systems are considered and analysed. As competition rules become more and more intransitive, thermodynamic analogies are eroded, and the behaviour of the system can become complex. This work analyses the phenomenon of punctuated evolution in the context of the competitive risk/benefit dilemma. Full article
(This article belongs to the Special Issue Complex Systems)
Open AccessArticle Global Inequality in Energy Consumption from 1980 to 2010
Entropy 2013, 15(12), 5565-5579; doi:10.3390/e15125565
Received: 21 October 2013 / Revised: 12 December 2013 / Accepted: 12 December 2013 / Published: 16 December 2013
Cited by 10 | PDF Full-text (2469 KB) | HTML Full-text | XML Full-text | Supplementary Files
Abstract
We study the global probability distribution of energy consumption per capita around the world using data from the U.S. Energy Information Administration (EIA) for 1980–2010. We find that the Lorenz curves have moved up during this time period, and the Gini coefficient, [...] Read more.
We study the global probability distribution of energy consumption per capita around the world using data from the U.S. Energy Information Administration (EIA) for 1980–2010. We find that the Lorenz curves have moved up during this time period, and the Gini coefficient, G, has decreased from 0.66 in 1980 to 0.55 in 2010, indicating a decrease in inequality. The global probability distribution of energy consumption per capita in 2010 is close to the exponential distribution withG = 0:5. We attribute this result to the globalization of the world economy, which mixes the world and brings it closer to the state of maximal entropy. We argue that global energy production is a limited resource that is partitioned among the world population. The most probable partition is the one that maximizes entropy, thus resulting in the exponential distribution function. A consequence of the latter is the law of 1/3: the top 1/3 of the world population consumes 2/3 of produced energy. We also find similar results for the global probability distribution of CO2 emissions per capita. Full article
(This article belongs to the Special Issue Complex Systems)
Open AccessArticle Structural Patterns in Complex Systems Using Multidendrograms
Entropy 2013, 15(12), 5464-5474; doi:10.3390/e15125464
Received: 9 October 2013 / Revised: 22 November 2013 / Accepted: 26 November 2013 / Published: 9 December 2013
Cited by 1 | PDF Full-text (279 KB) | HTML Full-text | XML Full-text
Abstract
Complex systems are usually represented as an intricate set of relations between their components forming a complex graph or network. The understanding of their functioning and emergent properties are strongly related to their structural properties. The finding of structural patterns is of [...] Read more.
Complex systems are usually represented as an intricate set of relations between their components forming a complex graph or network. The understanding of their functioning and emergent properties are strongly related to their structural properties. The finding of structural patterns is of utmost importance to reduce the problem of understanding the structure–function relationships. Here we propose the analysis of similarity measures between nodes using hierarchical clustering methods. The discrete nature of the networks usually leads to a small set of different similarity values, making standard hierarchical clustering algorithms ambiguous. We propose the use of multidendrograms, an algorithm that computes agglomerative hierarchical clusterings implementing a variable-group technique that solves the non-uniqueness problem found in the standard pair-group algorithm. This problem arises when there are more than two clusters separated by the same maximum similarity (or minimum distance) during the agglomerative process. Forcing binary trees in this case means breaking ties in some way, thus giving rise to different output clusterings depending on the criterion used. Multidendrograms solves this problem by grouping more than two clusters at the same time when ties occur. Full article
(This article belongs to the Special Issue Complex Systems)
Open AccessArticle Generalized (c,d)-Entropy and Aging Random Walks
Entropy 2013, 15(12), 5324-5337; doi:10.3390/e15125324
Received: 26 September 2013 / Revised: 12 November 2013 / Accepted: 25 November 2013 / Published: 3 December 2013
Cited by 12 | PDF Full-text (474 KB) | HTML Full-text | XML Full-text
Abstract
Complex systems are often inherently non-ergodic and non-Markovian and Shannon entropy loses its applicability. Accelerating, path-dependent and aging random walks offer an intuitive picture for non-ergodic and non-Markovian systems. It was shown that the entropy of non-ergodic systems can still be derived [...] Read more.
Complex systems are often inherently non-ergodic and non-Markovian and Shannon entropy loses its applicability. Accelerating, path-dependent and aging random walks offer an intuitive picture for non-ergodic and non-Markovian systems. It was shown that the entropy of non-ergodic systems can still be derived from three of the Shannon–Khinchin axioms and by violating the fourth, the so-called composition axiom. The corresponding entropy is of the form Sc,d ~ ∑iΓ(1 + d, 1 − cln pi) and depends on two system-specific scaling exponents, c and d. This entropy contains many recently proposed entropy functionals as special cases, including Shannon and Tsallis entropy. It was shown that this entropy is relevant for a special class of non-Markovian random walks. In this work, we generalize these walks to a much wider class of stochastic systems that can be characterized as “aging” walks. These are systems whose transition rates between states are path- and time-dependent. We show that for particular aging walks, Sc,d is again the correct extensive entropy. Before the central part of the paper, we review the concept of (c,d)-entropy in a self-contained way. Full article
(This article belongs to the Special Issue Complex Systems)
Open AccessArticle Co-Evolutionary Mechanisms of Emotional Bursts in Online Social Dynamics and Networks
Entropy 2013, 15(12), 5084-5120; doi:10.3390/e15125084
Received: 4 October 2013 / Revised: 6 November 2013 / Accepted: 12 November 2013 / Published: 26 November 2013
Cited by 11 | PDF Full-text (4021 KB) | HTML Full-text | XML Full-text
Abstract
Collective emotional behavior of users is frequently observed on various Web portals; however, its complexity and the role of emotions in the acting mechanisms are still not thoroughly understood. In this work, using the empirical data and agent-based modeling, a parallel analysis [...] Read more.
Collective emotional behavior of users is frequently observed on various Web portals; however, its complexity and the role of emotions in the acting mechanisms are still not thoroughly understood. In this work, using the empirical data and agent-based modeling, a parallel analysis is performed of two archetypal systems—Blogs and Internet-Relayed-Chats—both of which maintain self-organized dynamics but not the same communication rules and time scales. The emphasis is on quantifying the collective emotions by means of fractal analysis of the underlying processes as well as topology of social networks, which arise and co-evolve in these stochastic processes. The results reveal that two distinct mechanisms, which are based on different use of emotions (an emotion is characterized by two components, arousal and valence), are intrinsically associated with two classes of emergent social graphs. Their hallmarks are the evolution of communities in accordance with the excess of the negative emotions on popular Blogs, on one side, and smooth spreading of the Bot’s emotional impact over the entire hierarchical network of chats, on the other. Another emphasis of this work is on the understanding of nonextensivity of the emotion dynamics; it was found that, in its own way, each mechanism leads to a reduced phase space of the emotion components when the collective dynamics takes place. That a non-additive entropy describes emotion dynamics, is further confirmed by computing the q-generalized Kolmogorov-Sinai entropy rate in the empirical data of chats as well as in the simulations of interacting emotional agents and Bots. Full article
(This article belongs to the Special Issue Complex Systems)
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Review

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Open AccessReview Generalized Statistical Mechanics at the Onset of Chaos
Entropy 2013, 15(12), 5178-5222; doi:10.3390/e15125178
Received: 4 October 2013 / Revised: 9 November 2013 / Accepted: 11 November 2013 / Published: 27 November 2013
Cited by 4 | PDF Full-text (2468 KB) | HTML Full-text | XML Full-text
Abstract
Transitions to chaos in archetypal low-dimensional nonlinear maps offer real and precise model systems in which to assess proposed generalizations of statistical mechanics. The known association of chaotic dynamics with the structure of Boltzmann–Gibbs (BG) statistical mechanics has suggested the potential verification [...] Read more.
Transitions to chaos in archetypal low-dimensional nonlinear maps offer real and precise model systems in which to assess proposed generalizations of statistical mechanics. The known association of chaotic dynamics with the structure of Boltzmann–Gibbs (BG) statistical mechanics has suggested the potential verification of these generalizations at the onset of chaos, when the only Lyapunov exponent vanishes and ergodic and mixing properties cease to hold. There are three well-known routes to chaos in these deterministic dissipative systems, period-doubling, quasi-periodicity and intermittency, which provide the setting in which to explore the limit of validity of the standard BG structure. It has been shown that there is a rich and intricate behavior for both the dynamics within and towards the attractors at the onset of chaos and that these two kinds of properties are linked via generalized statistical-mechanical expressions. Amongst the topics presented are: (i) permanently growing sensitivity fluctuations and their infinite family of generalized Pesin identities; (ii) the emergence of statistical-mechanical structures in the dynamics along the routes to chaos; (iii) dynamical hierarchies with modular organization; and (iv) limit distributions of sums of deterministic variables. The occurrence of generalized entropy properties in condensed-matter physical systems is illustrated by considering critical fluctuations, localization transition and glass formation. We complete our presentation with the description of the manifestations of the dynamics at the transitions to chaos in various kinds of complex systems, such as, frequency and size rank distributions and complex network images of time series. We discuss the results. Full article
(This article belongs to the Special Issue Complex Systems)
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