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  • Journal article
    Aschwanden MJ, Crosby NB, Dimitropoulou M, Georgoulis MK, Hergarten S, McAteer J, Milovanov AV, Mineshige S, Morales L, Nishizuka N, Pruessner G, Sanchez R, Sharma AS, Strugarek A, Uritsky Vet al., 2016,

    25 Years of Self-Organized Criticality: Solar and Astrophysics

    , Space Science Reviews, Vol: 198, Pages: 47-166, ISSN: 1572-9672

    Shortly after the seminal paper “Self-Organized Criticality: An explanation of 1/f noise” by Bak et al. (1987), the idea has been applied to solar physics, in “Avalanches and the Distribution of Solar Flares” by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. The novel applications stimulated also vigorous debates about the discrimination between SOC models, SOC-like, and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC studies from the last 25 years and highlight new trends, open questions, and future challenges, as discussed during two recent ISSI workshops on this theme.

  • Journal article
    Clough JR, Evans TS, 2016,

    What is the dimension of citation space?

    , Physica A, 448 (2016) 235-247

    Citation networks represent the flow of information between agents. They areconstrained in time and so form directed acyclic graphs which have a causalstructure. Here we provide novel quantitative methods to characterise thatstructure by adapting methods used in the causal set approach to quantumgravity by considering the networks to be embedded in a Minkowski spacetime andmeasuring its dimension using Myrheim-Meyer and Midpoint-scaling estimates. Weillustrate these methods on citation networks from the arXiv, supreme courtjudgements from the USA, and patents and find that otherwise similar citationnetworks have measurably different dimensions. We suggest that thesedifferences can be interpreted in terms of the level of diversity or narrownessin citation behaviour.

  • Journal article
    Goldberg SR, Anthony H, Evans TS, 2015,

    Modelling citation networks

    , SCIENTOMETRICS, Vol: 105, Pages: 1577-1604, ISSN: 0138-9130
  • Journal article
    Zand J, Tirnakli U, Jensen HJ, 2015,

    On the relevance of q-distribution functions: The return time distribution of restricted random walker

    , Journal of Physics A - Mathematical and Theoretical, Vol: 48, ISSN: 1751-8113

    There exist a large literature on the application of q-statistics to the out-of-equilibrium non-ergodic systems in which some degree of strong correlations exists. Here we study the distribution of first return times to zero, PR(0; t), of a random walk on the set of integers {0, 1, 2, ..., L} with a position dependent transition probability given by |n/L|^a. We find that for all values of a ∈ [0, 2] P_R(0, t) can be fitted by q-exponentials, but only for a = 1 is P_R(0, t) given exactly by a q-exponential in the limit L → ∞. This is a remarkable result since the exact analytical solution of the corresponding continuum model represents P_R(0, t) as a sum of Bessel functions witha smooth dependence on a from which we are unable to identify a = 1 as of special significance. However, from the high precision numerical iteration of the discrete Master Equation, we do verify that only for a = 1 is P_R(0, t) exactly a q-exponential and that a tiny departure from this parameter value makes the distribution deviate from q-exponential. Further research is certainly required to identify the reason for this result and also the applicability of q-statistics and its domain.

  • Journal article
    Goldberg SR, Anthony H, Evans TS,

    Modelling citation networks

    , Scientometrics, ISSN: 1588-2861
  • Journal article
    Vázquez P, Del Río JA, Cedano KG, Martínez M, Jensen HJet al., 2015,

    An Entangled Model for Sustainability Indicators.

    , PLOS One, Vol: 10, Pages: e0135250-e0135250, ISSN: 1932-6203

    Nowadays the challenge for humanity is to find pathways towards sustainable development. Decision makers require a set of sustainability indicators to know if the sustainability strategies are following those pathways. There are more than one hundred sustainability indicators but they differ on their relative importance according to the size of the locality and change on time. The resources needed to follow these sustainability indicators are scarce and in some instances finite, especially in smaller regions. Therefore strategies to select set of these indicators are useful for decision makers responsible for monitoring sustainability. In this paper we propose a model for the identification and selection of a set of sustainability indicators that adequately represents human systems. In developing this model, we applied evolutionary dynamics in a space where sustainability indicators are fundamental entities interconnected by an interaction matrix. we used a fixed interaction that simulates the current context for the city of Cuernavaca, México as an example. We were able to identify and define relevant sets indicators for the system by using the Pareto principle. In this case we identified a set of sixteen sustainability indicators with more than 80% of the total strength. This set presents resilience to perturbations. For the Tangled Nature framework we provided a manner of treating different contexts (i.e., cities, counties, states, regions, countries, continents or the whole planet), dealing with small dimensions. This model provides decision makers with a valuable tool to select sustainability indicators set for towns, cities, regions, countries, continents or the entire planet according to a coevolutionary framework. The social legitimacy can arise from the fact that each individual indicator must be selected from those that are most important for the subject community.

  • Journal article
    Rochester CC, Pruessner G, Kornyshev AA, 2015,

    Statistical mechanics of 'Unwanted Electroactuation' in nanoporous supercapacitors

    , Electrochimica Acta, Vol: 174, Pages: 978-984, ISSN: 0013-4686

    Nanoporous electrodes have the potential to increase the surface electrode interfacial area and the stored energy density of a supercapacitor. However, structural deformation of the electrode can become apparent when the size of the pore is comparable to the size of a charging ion. After many cycles this could cause wear and degradation. We present a theoretical study of this ‘Unwanted Electroactuation’ in a carbon electrode wetted with an ionic liquid. We incorporate changes of the carbon-carbon bond length due to electrochemical doping of the pore walls and steric effects related to counterion insertion into the pore via a modified Ising model of charge storage. When considering the total electrode deformation these effects either complement or compete with each other, depending on the polarisation of the electrode. Our model shows qualitative agreement with the features of the experimentally observed expansion caused by variation of electrode potential.

  • Journal article
    Willis G, Pruessner G, Keelan J, 2015,

    Minimalistic real-space Renormalization of 4x4 Ising and Potts Models in two dimensions

    , Frontiers in Physics, Vol: 3, ISSN: 2296-424X

    We introduce and discuss a real-space renormalization group (RSRG) procedure on verysmall lattices, which in principle does not require any of the usual approximations, e.g., acut-off in the expansion of the Hamiltonian in powers of the field. The procedure is carriedout numerically on very small lattices (4×4 to 2×2) and implemented for the Ising Modeland the q = 3, 4, 5-state Potts Models. Nevertheless, the resulting estimates of thecorrelation length exponent and the magnetization exponent are typically within 3–7% ofthe exact values. The 4-state Potts Model generates a third magnetic exponent, whichseems to be unknown in the literature. A number of questions about the meaning ofcertain exponents and the procedure itself arise from its use of symmetry principles andits application to the q = 5 Potts Model.

  • Journal article
    Clough JR, Gollings J, Loach TV, Evans TSet al., 2015,

    Transitive reduction of citation networks

    , Journal of Complex Networks, Vol: 3, Pages: 189-203, ISSN: 2051-1310

    In many complex networks, the vertices are ordered in time, and edges represent causal connections. We propose methods of analysing such directed acyclic graphs taking into account the constraints of causality and highlighting the causal structure. We illustrate our approach using citation networks formed from academic papers, patents and US Supreme Court verdicts. We show how transitive reduction (TR) reveals fundamental differences in the citation practices of different areas, how it highlights particularly interesting work, and how it can correct for the effect that the age of a document has on its citation count. Finally, we transitively reduce null models of citation networks with similar degree distributions and show the difference in degree distributions after TR to illustrate the lack of causal structure in such models.

  • Journal article
    McAteer RTJ, Aschwanden MJ, Dimitropoulou M, Georgoulis MK, Pruessner G, Morales L, Ireland J, Abramenko Vet al., 2015,

    25 Years of self-organized criticality: numerical detection methods

    , Space Science Reviews, Vol: 198, Pages: 217-266, ISSN: 1572-9672

    The detection and characterization of self-organized criticality (SOC), in bothreal and simulated data, has undergone many significant revisions over the past 25 years. The explosive advances in the many numerical methods available for detecting, discriminating, and ultimately testing, SOC have played a critical role in developing our understanding of how systems experience and exhibit SOC. In this article, methods of detecting SOC are reviewed; from correlations to complexity to critical quantities. A description of the basicautocorrelation method leads into a detailed analysis of application-oriented methods developed in the last 25 years. In the second half of this manuscript space-based, time-based and spatial-temporal methods are reviewed and the prevalence of power laws in nature is described, with an emphasis on event detection and characterization. The search for numericalmethods to clearly and unambiguously detect SOC in data often leads us outside the comfort zone of our own disciplines—the answers to these questions are often obtained by studying the advances made in other fields of study. In addition, numerical detection methods often provide the optimum link between simulations and experiments in scientific research. We seek to explore this boundary where the rubber meets the road, to review this expandingfield of research of numerical detection of SOC systems over the past 25 years, and to iterate forwards so as to provide some foresight and guidance into developing breakthroughs in this subject over the next quarter of a century.

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