Assessing the Resilience of Socio-Ecosystems: Coupling Viability Theory and Active Learning with kd-Trees. Application to Bilingual Societies

Abstract

This paper proposes a new algorithm to compute the resilience of a social system or an ecosystem when it is defined in the framework of the mathematical viability theory. It is applied to the problem of language coexistence: Although bilingual societies do exist, many languages have disappeared and some seem endangered presently. Mathematical models of language competition generally conclude that one language will disappear, except when the relative prestige of the languages can be modified. The viability theory provides concepts and tools that are suitable to study the resilience, but with severe computational limits since it uses extensive search on regular grids. The method we propose considers the computation of the viability output sets as an active learning problem with the objective of restraining the number of calls to the model and information storage. We adapt a kd-tree algorithm to approximate the level sets of the resilience value. We prove that this algorithm converges to the output sets defined by the viability theory (viability kernel and capture basin). The resilience value we compute can then be used to propose a policy of action in risky situations such as migration flows.