Alessandro Sarti, Giovanna Citti

The problem of the dynamics of living systems and their mathematical representation essentially unfolds on two levels. Firstly, the principles of autopoiesis and operational closure describe how a living system produces and maintains itself. Secondly, living phenomena are recognised as historical, involving the evolution of dynamics and spaces of possibility that change over time. Currently, no differential model can incorporate both of these aspects. We propose adopting the differential heterogenesis framework, which enables us to define differential operators that change in space and time, producing event-based and historical dynamics. In this construction, heterogeneous operator assemblages will be constrained by the principles of autopoiesis and operational closure, providing a theoretical framework for event-based and historical living systems. Finally, we provide a specific example of assembly with Gray-Scott kinetic equations, along with numerical simulations and results.

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David H. Wolpert, David C. Krakauer

“Emergence” refers to the fact that different mathematical languages and theories seem to be required to describe scientific theories that operate at different spatial and temporal scales. Highly circumscribed, special cases of emergence have been investigated using many different formal tools, including coarse-graining (over space and / or time), renormalization, dimension reduction, broken symmetry, and various formulations of causality. Here we extend the mathematics of model theory to unify and extend these earlier investigations. As a preliminary application of our framework, we investigate canonical examples of emergence, including the emergence of genetics from chemistry, of psychology from neurobiology, and of different forms of collective behavior from the laws governing their constituent parts. By exploiting the power of model theory we also uncover many aspects of emergence that were previously obscured. In particular, we uncover some of the rich hierarchical structure that relates different scientific fields. For example, we show that scientific fields can be related via a collection of multiple partial orders, all defined over the same set. On the other hand, under an alternative relation among scientific fields, one which allows cycles, we show that it is possible that the set of all of scientific fields must collapse into a single scientific field. Going forward, these investigations point to the formal tools that might need to be developed in order to arrive at a deeper understanding of this emergent hierarchy of scientific reality.

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Louis Boucherie , Yong-Yeol Ahn , Sune Lehmann
PNAS Nexus, Volume 5, Issue 11, November 2026, pgag344,

Hierarchical clustering and community detection are important problems in machine learning and complex network analysis. A common approach to identify clusters is to simply cut dendrograms at some threshold. However, single-level cuts are often suboptimal in terms of capturing underlying structure in the data, especially when the dendrogram is unbalanced. In this article, we present the adaptive cut, a method that uses the hierarchical structure of dendrograms through multilevel cuts to overcome the limitations of single-level approaches. The adaptive cut optimizes an objective function using a Markov chain Monte Carlo with simulated annealing, resulting in better partitions. We demonstrate the adaptive cut through applications to link clustering and modularity optimization, but note that the method is applicable to any clustering task that relies on a dendrogram and an objective function. Beyond the adaptive cut, we introduce the balancedness score, an information-theoretic metric that quantifies how balanced a dendrogram is. Balancedness predicts the potential benefits of using multilevel cuts. For the community detection examples, we evaluate our method on more than 200 real-world networks and multiple synthetic datasets, demonstrating improvements in partition density and modularity over traditional single-cut approaches. We also show the generality of the adaptive cut by applying it across hierarchical clustering techniques and objective functions. The adaptive cut improves clustering outcomes across a range of hierarchical clustering tasks.

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