3,580 research outputs found
A categorical approach to open and interconnected dynamical systems
In his 1986 Automatica paper Willems introduced the influential behavioural approach to control theory with an investigation of linear time-invariant (LTI) discrete dynamical systems. The behavioural approach places open systems at its centre, modelling by tearing, zooming, and linking. We show that these ideas are naturally expressed in the language of symmetric monoidal categories.Our main result gives an intuitive sound and fully complete string diagram algebra for reasoning about LTI systems. These string diagrams are closely related to the classical notion of signal flow graphs, endowed with semantics as multi-input multi-output transducers that process discrete streams with an infinite past as well as an infinite future. At the categorical level, the algebraic characterisation is that of the prop of corelations.Using this framework, we derive a novel structural characterisation of controllability, and consequently provide a methodology for analysing controllability of networked and interconnected systems. We argue that this has the potential of providing elegant, simple, and efficient solutions to problems arising in the analysis of systems over networks, a vibrant research area at the crossing of control theory and computer science.<br/
Multilayer Networks
In most natural and engineered systems, a set of entities interact with each
other in complicated patterns that can encompass multiple types of
relationships, change in time, and include other types of complications. Such
systems include multiple subsystems and layers of connectivity, and it is
important to take such "multilayer" features into account to try to improve our
understanding of complex systems. Consequently, it is necessary to generalize
"traditional" network theory by developing (and validating) a framework and
associated tools to study multilayer systems in a comprehensive fashion. The
origins of such efforts date back several decades and arose in multiple
disciplines, and now the study of multilayer networks has become one of the
most important directions in network science. In this paper, we discuss the
history of multilayer networks (and related concepts) and review the exploding
body of work on such networks. To unify the disparate terminology in the large
body of recent work, we discuss a general framework for multilayer networks,
construct a dictionary of terminology to relate the numerous existing concepts
to each other, and provide a thorough discussion that compares, contrasts, and
translates between related notions such as multilayer networks, multiplex
networks, interdependent networks, networks of networks, and many others. We
also survey and discuss existing data sets that can be represented as
multilayer networks. We review attempts to generalize single-layer-network
diagnostics to multilayer networks. We also discuss the rapidly expanding
research on multilayer-network models and notions like community structure,
connected components, tensor decompositions, and various types of dynamical
processes on multilayer networks. We conclude with a summary and an outlook.Comment: Working paper; 59 pages, 8 figure
COMPUTER SIMULATION AND COMPUTABILITY OF BIOLOGICAL SYSTEMS
The ability to simulate a biological organism by employing a computer is related to the
ability of the computer to calculate the behavior of such a dynamical system, or the "computability" of the system.* However, the two questions of computability and simulation are not equivalent. Since the question of computability can be given a precise answer in terms of recursive functions, automata theory and dynamical systems, it will be appropriate to consider it first. The more elusive question of adequate simulation of biological systems by a computer will be then addressed and a possible connection between the two answers given will be considered. A conjecture is formulated that suggests the possibility of employing an algebraic-topological, "quantum" computer (Baianu, 1971b)
for analogous and symbolic simulations of biological systems that may include chaotic processes that are not, in genral, either recursively or digitally computable. Depending on the biological network being modelled, such as the Human Genome/Cell Interactome or a trillion-cell Cognitive Neural Network system, the appropriate logical structure for such simulations might be either the Quantum MV-Logic (QMV) discussed in recent publications (Chiara, 2004, and references cited therein)or Lukasiewicz Logic Algebras that were shown to be isomorphic to MV-logic algebras (Georgescu et al, 2001)
MuxViz: A Tool for Multilayer Analysis and Visualization of Networks
Multilayer relationships among entities and information about entities must
be accompanied by the means to analyze, visualize, and obtain insights from
such data. We present open-source software (muxViz) that contains a collection
of algorithms for the analysis of multilayer networks, which are an important
way to represent a large variety of complex systems throughout science and
engineering. We demonstrate the ability of muxViz to analyze and interactively
visualize multilayer data using empirical genetic, neuronal, and transportation
networks. Our software is available at https://github.com/manlius/muxViz.Comment: 18 pages, 10 figures (text of the accepted manuscript
Universal Constructions for (Co)Relations: categories, monoidal categories, and props
Calculi of string diagrams are increasingly used to present the syntax and
algebraic structure of various families of circuits, including signal flow
graphs, electrical circuits and quantum processes. In many such approaches, the
semantic interpretation for diagrams is given in terms of relations or
corelations (generalised equivalence relations) of some kind. In this paper we
show how semantic categories of both relations and corelations can be
characterised as colimits of simpler categories. This modular perspective is
important as it simplifies the task of giving a complete axiomatisation for
semantic equivalence of string diagrams. Moreover, our general result unifies
various theorems that are independently found in literature and are relevant
for program semantics, quantum computation and control theory.Comment: 22 pages + 3 page appendix, extended version of arXiv:1703.0824
A complex network perspective on clinical science
Contemporary classification systems for mental disorders assume that abnormal behaviors are expressions of latent disease entities. An alternative to the latent disease model is the complex network approach. Instead of assuming that symptoms arise from an underlying disease entity, the complex network approach holds that disorders exist as systems of interrelated elements of a network. This approach also provides a framework for the understanding of therapeutic change. Depending on the structure of the network, change can occur abruptly once the network reaches a critical threshold (the tipping point). Homogeneous and highly connected networks often recover more slowly from local perturbations when the network approaches the tipping point, potentially making it possible to predict treatment change, relapse, and recovery. In this article, we discuss the complex network approach as an alternative to the latent disease model and its implications for classification, therapy, relapse, and recovery.R34 MH086668 - NIMH NIH HHS; R01 AT007257 - NCCIH NIH HHS; R21 MH101567 - NIMH NIH HHS; R34 MH099311 - NIMH NIH HHS; R21 MH102646 - NIMH NIH HHS; K23 MH100259 - NIMH NIH HHS; R01 MH099021 - NIMH NIH HH
Newton vs. Leibniz: Intransparency vs. Inconsistency
We investigate the structure common to causal theories that attempt to
explain a (part of) the world. Causality implies conservation of identity,
itself a far from simple notion. It imposes strong demands on the
universalizing power of the theories concerned. These demands are often met by
the introduction of a metalevel which encompasses the notions of 'system' and
'lawful behaviour'. In classical mechanics, the division between universal and
particular leaves its traces in the separate treatment of cinematics and
dynamics. This analysis is applied to the mechanical theories of Newton and
Leibniz, with some surprising results
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