06/01/2012
Researchers in the social sciences currently employ a variety of mathematical/computational models for studying complex systems. Despite the diversity of these models, the majority can be grouped into one of three types: agent (rulebased) modeling, dynamical (equationbased) modeling and statistical (aggregatebased) modeling. The purpose of the current paper is to offer a fourth type: casebased modeling. To do so, we review the SACS Toolkit: a new method for quantitatively modeling complex social systems, based on a casebased, computational approach to data analysis. The SACS Toolkit is comprised of three main components: a theoretical blueprint of the major components of a complex system (social complexity theory); a set of casebased instructions for modeling complex systems from the ground up (assemblage); and a recommended list of casefriendly computational modeling techniques (casebased toolset). Developed as a variation on Byrne (in Sage Handbook of CaseBased Methods, pp. 260–268, 2009), the SACS Toolkit models a complex system as a set of kdimensional vectors (cases), which it compares and contrasts, and then condenses and clusters to create a lowdimensional model (map) of a complex system’s structure and dynamics over time/space. The assembled nature of the SACS Toolkit is its primary strength. While grounded in a defined mathematical framework, the SACS Toolkit is methodologically openended and therefore adaptable and amenable, allowing researchers to employ and bring together a wide variety of modeling techniques. Researchers can even develop and modify the SACS Toolkit for their own purposes. The other strength of the SACS Toolkit, which makes it a very effective technique for modeling large databases, is its ability to compress data matrices while preserving the most important aspects of a complex system’s structure and dynamics across time/space. To date, while the SACS Toolkit has been used to study several topics, a mathematical outline of its casebased approach to quantitative analysis (along with a case study) has yet to be written–hence the purpose of the current paper.
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01/01/2016
In the health informatics era, modeling longitudinal data remains problematic. The issue is method: health data are highly nonlinear and dynamic, multilevel and multidimensional, comprised of multiple major/minor trends, and causally complex—making curve fitting, modeling, and prediction difficult. The current study is fourth in a series exploring a case‐based density (CBD) approach for modeling complex trajectories, which has the following advantages: it can (1) convert databases into sets of cases (k dimensional row vectors; i.e., rows containing k elements); (2) compute the trajectory (velocity vector) for each case based on (3) a set of bio‐social variables called traces; (4) construct a theoretical map to explain these traces; (5) use vector quantization (i.e., k‐means, topographical neural nets) to longitudinally cluster case trajectories into major/minor trends; (6) employ genetic algorithms and ordinary differential equations to create a microscopic (vector field) model (the inverse problem) of these trajectories; (7) look for complex steady‐state behaviors (e.g., spiraling sources, etc) in the microscopic model; (8) draw from thermodynamics, synergetics and transport theory to translate the vector field (microscopic model) into the linear movement of macroscopic densities; (9) use the macroscopic model to simulate known and novel case‐based scenarios (the forward problem); and (10) construct multiple accounts of the data by linking the theoretical map and k dimensional profile with the macroscopic, microscopic and cluster models. Given the utility of this approach, our purpose here is to organize our method (as applied to recent research) so it can be employed by others.
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01/01/2016
Probability distributions have proven effective at modeling diversity in complex systems. The two most common are the Gaussian normal and skewed‐right. While the mechanics of the former are well‐known; the latter less so, given the significant limitations of the power‐law. Moving past the power‐law, we demonstrate that there exists, hidden‐in‐full‐view, a limiting law governing the diversity of complexity in skewed‐right systems; which can be measured using a case‐based version C c of Shannon entropy, resulting in a 60/40 rule. For our study, given the wide range of approaches to measuring complexity (i.e., descriptive, constructive, etc), we examined eight different systems, which varied significantly in scale and composition (from galaxies to genes). We found that skewed‐right complex systems obey the law of restricted diversity; that is, when plotted for a variety of natural and human‐made systems, as the diversity of complexity → ∞ (primarily in terms of the number of types; but also, secondarily, in terms of the frequency of cases) a limiting law of restricted diversity emerges, constraining the majority of cases to simpler types. Even more compelling, this limiting law obeys a scale‐free 60/40 rule: when measured using C c , 60%(or more) of the cases in these systems reside within the first 40% (or less) of the lower bound of equiprobable diversity types—with or without long‐tail and whether or not the distribution fits a power‐law. Furthermore, as an extension of the Pareto Principle, this lower bound accounts for only a small percentage of the total diversity; that is, while the top 20% of cases constitute a sizable percentage of the total diversity in a system, the bottom 60% are highly constrained. In short, as the central limit theorem governs the diversity of complexity in normal distributions, restricted diversity seems to govern the diversity of complexity in skewed‐right distributions.
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07/01/2016
This paper is part of a series addressing the empirical/statistical distribution of the diversity of complexity within and amongst complex systems. Here, we consider the problem of measuring the diversity of complexity in a system, given its ordered range of complexity types i and their probability of occurrence p_{i }, with the understanding that larger values of i mean a higher degree of complexity. To address this problem, we introduce a new complexity measure called casebased entropy C_{c} — a modification of the Shannon–Wiener entropy measure H. The utility of this measure is that, unlike current complexity measures–which focus on the macroscopic complexity of a single system–C_{c} can be used to empirically identify and measure the distribution of the diversity of complexity within and across multiple natural and humanmade systems, as well as the diversity contribution of complexity of any part of a system, relative to the total range of ordered complexity types.
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03/01/2015
This article introduces a new case‐based density approach to modeling big data longitudinally, which uses ordinary differential equations and the linear advection partial differential equations (PDE) to treat macroscopic, dynamical change as a transport issue of aggregate cases across continuous time. The novelty of this approach comes from its unique data‐driven treatment of cases: which are K dimensional vectors; where the velocity vector for each case is computed according to its particular measurements on some set of empirically defined social, psychological, or biological variables. The three main strengths of this approach are its ability to: (1) translate the data driven, nonlinear trajectories of microscopic constituents (cases) into the linear movement of macroscopic trajectories, which take the form of densities; (2) detect the presence of multiple, complex steady state behaviors, including sinks, spiraling sources, saddles, periodic orbits, and attractor points; and (3) predict the motion of novel cases and time instances. To demonstrate the utility of this approach, we used it to model a recognized cohort dynamic: the longitudinal relationship between a country's per capita gross domestic product (GDP) and its longevity rates. Data for the model came from the widely used Gapminder dataset. Empirical results, including the strength of the model's fit and the novelty of its results (particularly on a topic of such extensive study) support the utility of our new approach. © 2014 Wiley Periodicals, Inc. Complexity 20: 45–57, 2015
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11/01/2012
Recently, the continuity equation (also known as the advection equation) has been used to study stability properties of dynamical systems, where a linear transfer operator approach was used to examine the stability of a nonlinear equation both in continuous and discrete time (Vaidya and Mehta, IEEE Trans Autom Control 2008, 53, 307–323; Rajaram et al., J Math Anal Appl 2010, 368, 144–156). Our study, which conducts a series of simulations on residential patterns, demonstrates that this usage of the continuity equation can advance Haken's synergetic approach to modeling certain types of complex, self‐organizing social systems macroscopically. The key to this advancement comes from employing a case‐based approach that (1) treats complex systems as a set of cases and (2) treats cases as dynamical vsystems which, at the microscopic level, can be conceptualized as k dimensional row vectors; and, at the macroscopic level, as vectors with magnitude and direction, which can be modeled as population densities. Our case‐based employment of the continuity equation has four benefits for agent‐based and case‐based modeling and, more broadly, the social scientific study of complex systems where transport or spatial mobility issues are of interest: it (1) links microscopic (agent‐based) and macroscopic (structural) modeling; (2) transforms the dynamics of highly nonlinear vector fields into the linear motion of densities; (3) allows predictions to be made about future states of a complex system; and (4) mathematically formalizes the structural dynamics of these types of complex social systems.
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01/01/2015
The history of public health has focused on direct relationships between problems and solutions: vaccinations against diseases, ad campaigns targeting risky behaviors. But the accelerating pace and mounting intricacies of our lives are challenging the field to find new scientific methods for studying community health. The complexities of place (COP) approach is emerging as one such promising method. Place and Health as Complex Systems demonstrates how COP works, making an empirical case for its use in for designing and implementing interventions. This brief resource reviews the defining characteristics of places as dynamic and evolving social systems, rigorously testing them as well as the COP approach itself. The study, of twenty communities within one county in the Midwest, combines casebased methods and complexity science to determine whether COP improves upon traditional statistical methods of public health research. Its conclusions reveal strengths and limitations of the approach, immediate possibilities for its use, and challenges regarding future research. Included in the coverage:  Characteristics of places and the complexities of place approach.
 The Definitional Test of Complex Systems.
 Casebased modeling using the SACS toolkit.
 Methods, maps, and measures used in the study.
 Places as nodes within larger networks.
 Places as powerbased conflicted negotiations.
Place and Health as Complex Systems brings COP into greater prominence in public health research, and is also valuable to researchers in related fields such as demography, health geography, community health, urban planning, and epidemiology.
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