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\"Imagine trying to understand a stained glass window by breaking it into pieces and examining it one shard at a time. While you could probably learn a lot about each piece, you would have no idea about what the entire picture looks like. This is reductionism--the idea that to understand the world we only need to study its pieces--and it is how most social scientists approach their work. In [this book], social scientist and economist John H. Miller shows why we need to start looking at whole pictures. For one thing, whether we are talking about stock markets, computer networks, or biological organisms, individual parts only make sense when we remember that they are part of larger wholes. And perhaps more importantly, those wholes can take on behaviors that are strikingly different from that of their pieces\"--Amazon.com.

Many physical and social systems are best described by networks. And the structural properties of these networks often critically determine the properties and function of the resulting mathematical models. An important method to infer the correlations between topology and function is the detection of community structure, which plays a key role in the analysis, design, and optimization of many complex systems. The nonnegative matrix factorization has been used prolifically to that effect in recent years, although it cannot guarantee balanced partitions, and it also does not allow a proactive computation of the number of communities in a network. This indicates that the nonnegative matrix factorization does not satisfy all the nonnegative low-rank approximation conditions. Here we show how to resolve this important open problem by optimizing the identifiability of community structure. We propose a new form of nonnegative matrix decomposition and a probabilistic surrogate learning function that can be solved according to the majorization-minimization principle. Extensive in silico tests on artificial and real-world data demonstrate the efficient performance in community detection, regardless of the size and complexity of the network.

The so-called “pinched disk” model of the Mandelbrot set is due to A. Douady, J. H. Hubbard and W. P. Thurston. It can be described in the language of geodesic laminations. The combinatorial model is the quotient space of the unit disk under an equivalence relation that, loosely speaking, “pinches” the disk in the plane (whence the name of the model). The significance of the model lies in particular in the fact that this quotient is planar and therefore can be easily visualized. The conjecture that the Mandelbrot set is actually homeomorphic to this model is equivalent to the celebrated MLC conjecture stating that the Mandelbrot set is locally connected. For parameter spaces of higher degree polynomials no combinatorial model is known. One possible reason may be that the higher degree analog of the MLC conjecture is known to be false. We investigate to which extent a geodesic lamination is determined by the location of its critical sets and when different choices of critical sets lead to essentially the same lamination. This yields models of various parameter spaces of laminations similar to the “pinched disk” model of the Mandelbrot set.

Background A growing body of research highlights the limitations of traditional methods for studying the process of change in psychotherapy. The science of complex systems offers a useful paradigm for studying patterns of psychopathology and the development of more functional patterns in psychotherapy. Some basic principles of change are presented from subdisciplines of complexity science that are particularly relevant to psychotherapy: dynamical systems theory, synergetics, and network theory. Two early warning signs of system transition that have been identified across sciences (critical fluctuations and critical slowing) are also described. The network destabilization and transition (NDT) model of therapeutic change is presented as a conceptual framework to import these principles to psychotherapy research and to suggest future research directions. Discussion A complex systems approach has a number of implications for psychotherapy research. We describe important design considerations, targets for research, and analytic tools that can be used to conduct this type of research. Conclusions A complex systems approach to psychotherapy research is both viable and necessary to more fully capture the dynamics of human change processes. Research to date suggests that the process of change in psychotherapy can be nonlinear and that periods of increased variability and critical slowing might be early warning signals of transition in psychotherapy, as they are in other systems in nature. Psychotherapy research has been limited by small samples and infrequent assessment, but ambulatory and electronic methods now allow researchers to more fully realize the potential of concepts and methods from complexity science.

Despite a profusion of literature on complex adaptive system (CAS) definitions, it is still challenging to definitely answer whether a given system is or is not a CAS. The challenge generally lies in deciding where the boundaries lie between a complex system (CS) and a CAS. In this work, we propose a novel definition for CASs in the form of a concise, robust, and scientific algorithmic framework. The definition allows a two-stage evaluation of a system to first determine whether it meets complexity-related attributes before exploring a series of attributes related to adaptivity, including autonomy, memory, self-organisation, and emergence. We demonstrate the appropriateness of the definition by applying it to two case studies in the medical and supply chain domains. We envision that the proposed algorithmic approach can provide an efficient auditing tool to determine whether a system is a CAS, also providing insights for the relevant communities to optimise their processes and organisational structures.

Complex systems research is becoming ever more important in both the natural and social sciences. It is commonly implied that there is such a thing as a complex system, different examples of which are studied across many disciplines. However, there is no concise definition of a complex system, let alone a definition on which all scientists agree. We review various attempts to characterize a complex system, and consider a core set of features that are widely associated with complex systems in the literature and by those in the field. We argue that some of these features are neither necessary nor sufficient for complexity, and that some of them are too vague or confused to be of any analytical use. In order to bring mathematical rigour to the issue we then review some standard measures of complexity from the scientific literature, and offer a taxonomy for them, before arguing that the one that best captures the qualitative notion of the order produced by complex systems is that of the Statistical Complexity. Finally, we offer our own list of necessary conditions as a characterization of complexity. These conditions are qualitative and may not be jointly sufficient for complexity. We close with some suggestions for future work.

The city represents a complex adaptive system consisting of a set of interconnected subsystems within self-organization and flexible structures that give it the ability to adapt to continue operating over time. This research discusses the adaptive flexibility of the complex urban system by measuring the degree of complexity and adaptation of secondary systems within the general system depending on its kinetic and visual system, making it self-organized and within a coherent and harmonious unit. Thus, the research problem arises from “the lack of knowledge on how to determine the appropriate degree of complexity and adaptation for urban systems, providing them with adaptive flexibility that allows them to maintain their energy and ensure their survival and continuity over time”. This research aims to determine the adaptive flexibility of the complex urban system, which keeps it self-organized and able to survive depending on its kinetic and visual system that links the secondary systems of the general system based on the degree of complexity and adaptation ideal. The research adopted quantitative analysis, which includes calculating the adaptive dimensions of the complex system based on the kinetic and visual system, as well as the nodes connecting the parts of the secondary system, in addition to determining both the area and the perimeter of the study area. The historic Karkh area in Baghdad was selected for implementation because it includes a model of the complex system that originated and developed within different time stages. The system’s adaptive flexibility was calculated to determine its ability to survive and continue. This research found that it is possible to determine the strength of the urban format and its ability to survive or not, as well as the possibility of developing it, by understanding the ideal degree of adaptive flexibility for the format. This ideal degree is 1, achieved through visual and kinetic coherence among the nodes that make up the parts of the format. The farther the format is from this degree, the less adaptive flexibility it has, which affects its ability to survive and continue. Therefore, it is necessary to maintain the strength of the interconnections between the parts of the kinetic and visual system and between the parts of the complex format, which directly affects its ability to self-regulate and survive.

A large number of complex systems find a natural abstraction in the form of weighted networks whose nodes represent the elements of the system and the weighted edges identify the presence of an interaction and its relative strength. In recent years, the study of an increasing number of large-scale networks has highlighted the statistical heterogeneity of their interaction pattern, with degree and weight distributions that vary over many orders of magnitude. These features, along with the large number of elements and links, make the extraction of the truly relevant connections forming the network's backbone a very challenging problem. More specifically, coarse-graining approaches and filtering techniques come into conflict with the multiscale nature of large-scale systems. Here, we define a filtering method that offers a practical procedure to extract the relevant connection backbone in complex multiscale networks, preserving the edges that represent statistically significant deviations with respect to a null model for the local assignment of weights to edges. An important aspect of the method is that it does not belittle small-scale interactions and operates at all scales defined by the weight distribution. We apply our method to realworld network instances and compare the obtained results with alternative backbone extraction techniques.

Many core marketing concepts (e.g., markets, relationships, customer experience, brand meaning, value) concern phenomena that are difficult to understand using linear and dyadic approaches, because they are emergent. That is, they arise, often unpredictably, from interactions within complex and dynamic contexts. This paper contributes to the marketing discipline through an explication of the concept of emergence as it applies to marketing theory. We accomplish this by first drawing on the existing literature on emergence in philosophy, sociology, and the theory of complex adaptive systems, and then link and extend this understanding to marketing using the theoretical framework of service-dominant (S-D) logic, particularly as enhanced by its service-ecosystems and institutionalization perspectives. Our work recognizes both emergence and institutionalization as integral or interrelated processes in the creation, maintenance, and disruption of markets and marketing phenomena. We conclude by discussing implications for marketing research and practice.

In this survey we stress the importance of the higher transcendental Mittag-Leffler function in the framework of the Fractional Calculus. We first start with the analytical properties of the classical Mittag-Leffler function as derived from being the solution of the simplest fractional differential equation governing relaxation processes. Through the sections of the text we plan to address the reader in this pathway towards the main applications of the Mittag-Leffler function that has induced us in the past to define it as the Queen Function of the Fractional Calculus. These applications concern some noteworthy stochastic processes and the time fractional diffusion-wave equation We expect that in the future this function will gain more credit in the science of complex systems. Finally, in an appendix we sketch some historical aspects related to the author’s acquaintance with this function.