Stefan Thurner, Rudolf Hanel, and Peter Klimek
OXFORD UNIVERSITY PRESS 2018, 448 PAGES
PRICE (HARDBACK) £49.99 ISBN 978-0-19-882193-9
As per the author’s comments and overall observation, the book is poised towards providing a framework for the comprehension of co-evolutionary behaviour of states and their corresponding interactions. The book is well written and structured nicely, with each chapter beginning with a very abstract introduction and with some history, sometimes followed by a more detailed set of sections and concludes with a list of exercises which serve as learning platforms for readers who wish to delve deeper into certain topics. Below is a summary of the chapters which make up the book.
The introductory chapter to the book starts off with definition of buzzwords used in the area of complexity studies followed by a list of 10 facts which characterise the nature of complex systems, and also key societal contributions made by complexity science such as – interactions in epidemiology, network theory, econophysics, genetic algorithms, risk in financial markets, and science of cities, to name a few.
Chapter 2 – randomness and statistics, for the description of stochastic processes. In this chapter the authors talk about the importance of probabilistic methods as a means of characterising the sample space of non-deterministic processes. Statistical distributions such as the Gaussian, log-normal, Cauchy, Levy and fat tailed distributions are introduced and contrasted as means of handling random numbers. This is followed with a description of the notion of stochastic processes which include the memoryless Bernoulli process and the one-step memory Markov process.
Chapter 3 – introduces scaling, power laws and their influence in statistical descriptions of complex systems. It begins with a brief background on Galileo’s work in this area and its link in physiological studies, fractals and complex systems. The authors emphasise that power law statistics is a useful mechanism for the understanding of the behaviour of certain complex systems.
Chapter 4 – is a rather comprehensive chapter which covers a key topic in complex systems; network theory and dynamical networks. Networks are described to be a key component of complex system studies due to their ability to make apparent systemic interactions in non-homogenous complex systems. The chapter commences with an abstraction of networks, followed by key works done in this area by Erdos, Renyi and Wigner, networks in data analytics and nodal interactions, extension to complex structures and its application in epidemiology and understanding financial risks.
Chapter 5 – covers evolutionary dynamics which ultimately gives rise to new entities or sees the fall of existing ones. This chapter explores the prospect of developing statistical and mathematical frameworks which can describe: combinatorial co-evolution of species in their environments; the magnitude and timescale of disruptive restructuring events; duration of equilibrium phases; systemic risks and innovation rates. A non-trivial evolutionary system within this framework is considered for the creation of a novel entity in a unique environment. The outcome of the inevitable interaction between the new entity and its environment determines if it survives or gets eradicated and surviving entities interact with already existing entities and may even modify them in the process. The computation of fitness in an evolutionary landscape can be framed as an optimisation problem.
Chapter 6 – describes methods that can be used in the analysis of non-markovian and non-ergodic complex systems which renders classical statistical and information theory methods as inapplicable due to their stochastic nature. Due to this, probabilistic methods based on 3 variants of entropy are employed. These are statistical mechanics for the investigation of system properties, information theory for the quantification of information produced by a system and maximum entropy principle for the inference of statistical distributions.
Chapter 7 – concludes with a summary of how the intention of the book is to provide theoretical and analytical frameworks for complex systems under the headings discussed in the various chapters. The authors provide clarification that computational and data driven methods, most especially machine learning which is useful for pattern recognition tasks, will not replace the need for complexity science. Instead it will be combined with it to produce a scientific understanding of the cause and effect of co-evolving stochastic systems and then recognise their patterns.
Ejay Nsugbe CMath CSci MIMA
Book review first published in Mathematics Today October 2020
