Measuring emergence via nonlinear Granger causality

The concept of emergence is central to artificial life and complexity science, yet quantitative, intuitive, and easy-to-apply measures of emergence are surprisingly lacking. Here, I introduce a just such a measure, G-emergence, which operationalizes the notion that an emergent process is both dependent upon and autonomous from its underlying causal factors. G-emergence is based on a nonlinear time series analysis adapted from ‘Granger causality’ and it provides a measure not only of emergence but also of apparent ‘downward causation’. I illustrate the measure by application to a canonical example of emergence, an agent-based simulation of bird flocking, and I discuss its potential impact on perhaps the most challenging of all scientific problems involving emergence: consciousness. The maturation of artificial life and complexity science over recent years has given rise to renewed interest in emergence. Although the concept of emergence has a long philosophical history (Broad, 1925; Kim, 1999), its essence is simple enough: An emergent property is somehow ‘more than the sum’ of its component parts. Emergent properties appear rife within complex systems of all kinds: biological, cognitive, social, and technological. Broadly speaking, artificial life and complexity science focus on explaining phenomena that seem to involve emergence, and models constructed under these auspices are often described as emergent (Bedau, 2003). It is therefore surprising and significant that quantitative and easy-to-apply measures of emergence are mostly lacking. This is unfortunate because the ability to measure a phenomenon is an essential step towards its effective scientific description (Chang, 2004). In this paper I will first differentiate several notions of emergence and by doing so briefly illustrate some relevant conceptual challenges. I will then introduce ‘G-emergence’, a new measure which operationalizes the intuition that an emergent process is simultaneously autonomous from and dependent upon its underlying causal factors. G-emergence is easy to apply, and I illustrate it by application to a canonical example of emergence: bird flocking. I end by discussing related measures, how it can defuse the metaphysically awkward notion of ‘downward causation’, and how it Figure 1: A flock of starlings about to roost. may shed new light one of the most recalcitrant problems in science: the relation between neural mechanism and phenomenal experience. Varieties of emergence Intuitively, emergence refers either to a macro-level property that is ‘more than the sum of’ the micro-level parts (‘property’ or ‘synchronic’ emergence) or to the appearance of a qualitatively distinctive new phenomenon over time (‘temporal’ or ‘diachronic’ emergence). A striking example of property emergence is a flock of starlings wheeling in the sky before they roost: the flock seems to have a shape and trajectory of its own, which appears to exceed those of the individual birds (Figure 1). Temporal emergence is well illustrated by the appearance of new morphological features during embryogenesis and development. This paper focuses on measuring property emergence, though new opportunities for measuring temporal emergence are also identified. Following Bedau (1997, 2003), both property emergence and temporal emergence can be differentiated into three categories: strong, weak, and nominal [similar decompositions can be found in (van Gulick, 2001; Bar-Yam, 2004)]. The least controversial of these is nominal emergence, which is simply the notion of a kind of property that can be possessed by macro-level objects or processes but not by their microlevel constituents. For example, a circle is nominally emerArtificial Life XI 2008 545 gent from the set of points from which it is constructed. Because nominally emergent properties can be derived trivially I will not discuss them any further. Most challenging and controversial is the notion of strong emergence, which involves two closely related claims. First, a macro-level property is in principle not identifiable from micro-level observations. Second, macro-level properties have irreducible causal powers. The first claim rejects mechanistic explanations altogether, apparently calling a halt to scientific advance in the absence of new fundamental principles of nature (Chalmers, 2006). The second raises the difficult notion of ‘downward causation’. Downward causation is problematic firstly because it contravenes the plausible doctrine that ‘the macro is the way it is in virtue of the how things are at the micro’, an idea that has been expressed variously as ‘causal fundamentalism’ (Jackson and Pettit, 1992) or ‘supervenience’ (Kim, 1999). A second challenge raised by downward causation is that of resolving conflicts between micro-level and macro-level causes (Bedau, 2003). Even so, the main problem with strong emergence may lie in its scientific irrelevance (Bedau, 2003). The only recurrent example of strong emergence in the scientific literature is that of the emergence of conscious states (e.g., qualia) from neurobiological processes (Sperry, 1969; Chalmers, 2006), which may speak more to our lack of understanding of consciousness than to our grasp of deep principles of emergence. I will return to this possibility later on. In between strong emergence and nominal emergence lies the useful notion of weak emergence (Bedau, 1997, 2003), according to which a macro-level property is derived from the interaction of micro-level components but in complicated ways such that the macro-level property has no simple micro-level explanation. In contrast to strong emergence, weakly emergent properties are in principle identifiable from micro-level components, and in contrast to nominal emergence, the micro-to-macro inferential pathways must be non-trivial. According to Bedau, weakly emergent macro-level properties are ontologically dependent on and reducible to micro-level causal factors, but at the same time they are epistemologically irreducible due to the complexity of the micro-to-macro inferential pathways. What exactly does it mean for a macro-level property to be epistemologically irreducible? Bedau’s answer is that a weakly emergent (epistemologically irreducible) property is underivable from its micro-level parts except by simulation. This is an all-or-none classification. Either a macro-level property can be derived by some explanatory short-cut, in which case weak emergence does not apply, or it cannot, in which case the micro-level causal factors need to be simulated explicitly in order to derive the macro-level property. In this paper I consider a continuous version of weak emergence, in which a macro property is weakly emergent to the extent that it is not identifiable from micro-level observations. This variation is valuable firstly because for many systems it may not be possible to prove ‘underivability except by simulation’, and secondly because from the perspective of measurement, a continuous value is much more useful than a binary classification. Measuring weak emergence To derive a continuous measure of weak emergence, I take as a starting point the idea that a weakly emergent macrolevel property is simultaneously (i) autonomous from and (ii) dependent upon its underlying causal factors (Bedau, 1997). To operationalize this notion statistically, I propose that a macro-variable M can be measured as weakly emergent from a set of micro-variables m (m = m1 . . . mN ) to the extent that: (i) past observations of M help predict future observations of M with greater accuracy than predictions based on past observations of m alone, and (ii) past observations of m help predict future observations of M with greater accuracy than predictions based on past observations of M alone. The first condition provides an objective measure of the non-triviality of micro-to-macro inferential pathways, and the second checks for micro-to-macro causal dependence. This definition is relative to a choice of macro and micro description levels and is also relative to a choice of prediction method. As described below, an appropriate framework for prediction is provided by a statistical definition of causality first introduced by Granger (1969), in recognition of which the present measure is called G-emergence.