OUP UNCORRECTED PROOF – FIRST PROOF, 7/6/2018, SPi Introduction to the Theory of Complex Systems Stefan Thurner, Rudolf Hanel, and Peter Klimek Medical University of Vienna, Austria 1 OUP UNCORRECTED PROOF – FIRST PROOF, 7/6/2018, SPi 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Stefan Thurner, Rudolf Hanel, and Peter Klimek 2018 The moral rights of the authors have been asserted First Edition published in 2018 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2018947065 Data available ISBN 978–0–19–882193–9 DOI: 10.1093/oso/9780198821939.001.0001 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work. OUP UNCORRECTED PROOF – FIRST PROOF, 7/6/2018, SPi Preface This book is for people who are interested in the science of complex adaptive systems and wish to have more than just a casual understanding of it. As with all the sciences, understanding of complex adaptive systems is reached solely in a quantitative, predictive, and ultimately experimentally testable manner. Complex adaptive systems are dynamical systems that are able to change their structure, their interactions, and, consequently, their dynamics as they evolve in time. This is not a book about complicated systems, even though most complex systems are complicated. Indeed, over the last 300 years, scientists have usually dealt with complicated systems that are neither complex nor adaptive. The theory of complex systems is the theory of generalized time-varying interactions between elements that are characterized by states. Interactions typically take place on networks that connect those elements. The interactions involved may cause the states of the elements themselves to alter over time. The essence of a complex system is that the interaction networks may change and rearrange as a consequence of changes in the states of the elements. Thus, complex systems are systems whose states change as a result of interactions and whose interactions change concurrently as a result of states. Due to this chicken–egg-type problem, complex systems show an extremely rich spectrum of behaviour: they are adaptive and co-evolutionary; they show path- dependence, emergence, power laws; they have rich phase diagrams; they produce and destroy diversity; they are inherently prone to collapse; they are resilient, and so on. The theory of complex systems tries to understand these properties based on its building blocks and on the interactions between those building blocks that take place on networks. It combines mathematical and physical principles with concepts borrowed from biology and the social sciences; it uses new computational techniques and, with the advent of comprehensive large-scale data sets, is becoming experimentally testable. The goal of the theory of complex systems is to understand the dynamical systemic outcomes of interconnected systems, and its ultimate goal is to eventually control and design systemic properties of systems such as the economy, the fnancial system, social processes, cities, the climate, and ecology. The theory of complex systems builds partly on previous attempts to understand systems that interact in non-trivial ways, such as game theory, cybernetics, or systems theory. However, in its current state, the science of complex systems goes well beyond these earlier developments, in so many ways, in fact, that it can be regarded as an independent scientifc branch, which—due to its quantitative, predictive, and testable nature—is a natural science. Even though it is fair to say that the theory of complex systems is not yet complete, in recent years, it has become quite clear just what the theory is going to look like. Its elements and structure are emerging. The current state of the theory of complex OUP UNCORRECTED PROOF – FIRST PROOF, 7/6/2018, SPi vi Preface systems is comparable perhaps to the state of quantum mechanics in the 1920s, before the famous Copenhagen meetings and Werner Heisenberg’s book. At that time, quantum mechanics was a collection of experimental and theoretical bits and pieces, which had not yet been seen within a fully comprehensive framework. Nevertheless, it was clear that, one day soon, such a framework would exist. The present situation can be compared to an archaeological project, where a mosaic foor has been discovered and is being excavated. While the mosaic is only partly visible and the full picture is still missing, several facts are becoming clear: the mosaic exists; it shows identifable elements (for instance, people and animals engaged in recognizable activities); there are large patches missing or still invisible, but experts can already tell that the mosaic represents a scene from,say,Homer’s Odyssey.Similarly,for dynamical complex adaptive systems, it is clear that a theory exists that, eventually, can be fully developed. There are those who say that complex systems will never be understood or that, by their very nature, they are incomprehensible. This book will demonstrate that such statements are incorrect. The elements of a theory of complex systems are becoming clear: dynamical multilayer networks, scaling, statistical mechanics of algorithmic dynamics, evolution and co-evolution, and information theory. The essence of this book is to focus on these components, clarify their meaning in the context of complex systems, and enable the reader with a mathematical skill set to apply them to concrete problems in the world of complex systems. The book is written in mathematical language because this is the only way to express facts in a quantitative and predictive manner and to make statements that are unambiguous. We aim for consistency. The book should be comprehensible so that no- one with an understanding of basic calculus, linear algebra, and statistics need refer to other works. The book is particularly designed for graduate students in physics or mathematics. We try to avoid ambiguous statements while, at the same time, being as general as possible. The hope is that this work will serve as a textbook and as a starting point for journeys into new and unexplored territory. Many complex systems are often sensitive to details in their internal setup, to initial and to boundary conditions. Concepts that proved to be extremely robust and effective in non-complex systems, such as the central limit theorem, classical statistical mechanics, or information theory, lose their predictive power when confronted with complex systems. Extreme care is thus needed in any attempt to apply these otherwise distinguished concepts to complex systems: doing so could end in confusion and nonsensical results. In several concrete examples, we will demonstrate the importance of understanding what these methods mean in the context of complex systems and whether they can or cannot be applied. We will discuss how some of these classical concepts can be generalized to become useful for understanding complex systems. The book is also a statement about our belief that the exact sciences may be entering a phase of transition from a traditional analytical description of nature, as used with tremendous success since Galileo and Newton, towards an algorithmic description. Whereas the analytical description of nature is, conceptually, based largely on differential equations and analytical equations of motion, the algorithmic view takes into account evolutionary and co-evolutionary aspects of dynamics. It provides a framework for OUP UNCORRECTED PROOF – FIRST PROOF, 7/6/2018, SPi Preface vii systems that can endogenously change their internal interaction networks, rules of functioning, dynamics, and even environment, as they evolve in time. Algorithmic dynamics, which is characteristic of complex dynamical systems, may be a key to the quantitative and predictive understanding of many natural and man-made systems. In contrast to physical systems, which typically evolve analytically, algorithmic dynamics describe certainly how living, social, environmental, and economic systems unfold. This algorithmic view is not new but has been advocated by authors like Joseph A. Schumpeter, Stuart Kauffman, and Brian Arthur. However, it has not, to date, been picked up by mainstream science, and it has never been presented in the context of the theory of complex systems. This book is