Ingredients for robustness
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REVIEW
Ingredients for robustness Nihat Ay1,2,3 Received: 12 October 2020 / Accepted: 12 November 2020 / Published online: 2 December 2020 © The Author(s) 2020
Abstract A core property of robust systems is given by the invariance of their function against the removal of some of their structural components. This intuition has been formalised in the context of input–output maps, thereby introducing the notion of exclusion independence. We review work on how this formalisation allows us to derive characterisation theorems that provide a basis for the design of robust systems. Keywords Robustness · Knockouts · Neutrality · Interaction order
Introduction: What are robust systems? The robustness of a system, be it biological or artificial, always refers to a set of perturbations. A biological system has to cope with a number of perturbations, including a constantly changing environment (Wagner 2007 gives a general introduction to the robustness of biological systems). One way to do so is by shaping and controlling its environment in a way that reduces perturbations. This is often referred to as niche construction, which can take place in the ecological as well as the social domain (Flack et al. 2006; Krakauer et al. 2009). Another way to deal with perturbations is by an intrinsic adaptation, change of the system itself and not the environment, so that the system’s function remains unchanged. It is this second way of coping with perturbations that we are addressing in this article. The setting in which we want to study this kind of robustness is kept very simple, as shown in Fig. 1. It is given by a (stochastic) map that receives n inputs and generates a, potentially highdimensional, output. It is surprising how far-reaching this minimalistic setting is. It is general enough for studying the robustness of a number of biologically relevant mappings, such as the genotype-phenotype map, the genetic code, and neurons. However, addressing such application fields is * Nihat Ay [email protected] 1
Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
2
Leipzig University, Leipzig, Germany
3
Santa Fe Institute, Santa Fe NM, USA
beyond the scope of this article. Our aim is rather moderate. We review previous work on this subject and highlight the main insights in a more direct, instructive and conceptually complete way. The presented theory has been initially proposed by Ay and Krakauer (2007) and further developed in a number of works, including (Ay et al. 2007; Krakauer et al. 2010; Boldhaus et al. 2010; Rauh 2013; Rauh and Ay 2014). It touches upon important subjects related to robustness, such as niche construction, adaptation and neutrality, which have been addressed from various perspectives in a large number of publications. This article is by no means complete in providing a review of these publications. Instead, it restricts attention to the mentioned works in which connections to other works are outlined more thoroughly. In order to study edge or node deletion, which we also call knockout
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