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Life Finds A Way: Emergence of Cooperative Structures in Adaptive Threshold Networks

There has been a long debate on how new levels of organization have evolved. It might seem unlikely, as cooperation must prevail over competition. One well-studied example is the emergence of autocatalytic sets, which seem to be a prerequisite for the evolution of life. Using a simple model, we investigate how varying bias toward cooperation versus antagonism shapes network dynamics, revealing that higher-order organization emerges even amid pervasive antagonistic interactions. In general, we observe that a quantitative increase in the number of elements in a system leads to a qualitative transition. We present a random threshold-directed network model that integrates node-specific traits with dynamic edge formation and node removal, simulating arbitrary levels of cooperation and competition. In our framework, intrinsic node values determine directed links through various threshold rules. Our model generates a multi-digraph with signed edges (reflecting support/antagonism, labeled `help''/`harm''), which ultimately yields two parallel yet interdependent threshold graphs. Incorporating temporal growth and node turnover in our approach allows exploration of the evolution, adaptation, and potential collapse of communities and reveals phase transitions in both connectivity and resilience. Our findings extend classical random threshold and Erd\H{o}s-R\'enyi models, offering new insights into adaptive systems in biological and economic contexts, with emphasis on the application to Collective Affordance Sets. This framework should also be useful for making predictions that will be tested by ongoing experiments of microbial communities in soil.

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