Complexity Science: Space-time phases

Abstract Complexity science is the study of systems with many interdependent components. One of the main concepts is “emergence”: the whole may be greater than the sum of the parts. The objective of this chapter is to put emergence on a firm mathematical foundation in the context of dynamics of large networks. Both stochastic and deterministic dynamics are treated. To minimise technicalities, attention is restricted to dynamics in discrete time, in particular to probabilistic cellular automata and coupled map lattices. The key notion is space-time phases : probability distributions for state as a function of space and time that can arise in systems that have been running for a long time. What emerges from a complex dynamic system is one or more space-time phases. The amount of emergence in a space-time phase is its distance from the set of product distributions over space, using an appropriate metric. A system exhibits strong emergence if it has more than one space-time phase. Strong emergence is the really interesting case. This chapter is based on MSc or PhD courses given at Warwick in 2006/7, Paris in April 2007, Warwick in Spring 2009 and Autumn 2009, and Brussels in Autumn 2010. It was written up during study leave in 2010/11 at the Universite Libre de Bruxelles, to whom I am grateful for hospitality, and finalised in 2012. The chapter provides an introduction to the theory of space-time phases, via some key examples of complex dynamic system. I am most grateful to Dayal Strub for transcribing the notes into LaTeX and for preparing the figures.