Vicky Chuqiao Yang, James Holehouse, Hyejin Youn, José Ignacio Arroyo, Sidney Redner, Geoffrey B. West, and Christopher P. Kempes

PNAS 123 (7) e2509729123

Diversification and specialization are central to complex adaptive systems, yet overarching principles across domains remain elusive. We introduce a general theory that unifies diversity and specialization across disparate systems, including microbes, federal agencies, companies, universities, and cities, characterized by two key parameters. We show from extensive data that function diversity scales with system size as a sublinear power law-resembling Heaps’ law-in all but cities, where it is logarithmic. Our theory explains both behaviors and suggests that function creation depends on system goals and structure: federal agencies tend to ensure functional coverage; cities slow new function growth as old ones expand, and cells occupy an intermediate position. Once functions are introduced, their growth follows a remarkably universal pattern across all systems.

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Dashun Wang

As artificial-intelligence systems take on more of the scientific workflow, the central goal should not be complete automation, but designing platforms that preserve creativity, responsibility and surprise.

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Abbas K Rizi

PNAS Nexus, Volume 5, Issue 2, February 2026, pgag010,

The term emergence is increasingly used across scientific disciplines to describe phenomena that arise from interactions among a system’s components but cannot be readily inferred by examining those components in isolation. While often invoked to explain higher-level behaviors—such as flocking, synchronization, or collective intelligence—the term is frequently used without precision, sometimes giving rise to ambiguity or even mystique. In this perspective paper, I clarify the scientific meaning of emergence as a measurable and physically grounded phenomenon. Through concrete examples—such as temperature, magnetism, and herd immunity in social networks—I review how collective behavior can arise from local interactions that are constrained by global boundaries. By refining the concept of emergence, it is possible to gain a clearer and more grounded understanding of complex systems. My goal is to show that emergence, when properly framed, offers not mysticism, but rather insight.

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Lucas Lacasa
In network science, collective dynamics of complex systems are typically modelled as (nonlinear, often including many-body) vertex-level update rules evolving over a graph interaction structure. In recent years, frameworks that explicitly model such higher-order interactions in the interaction backbone (i.e. hypergraphs) have been advanced, somehow shifting the imputation of the effective nonlinearity from the dynamics to the interaction structure. In this work we discuss such structural–dynamical representation duality, and investigate how and when a nonlinear dynamics defined on the vertex set of a graph allows an equivalent representation in terms of a linear dynamics defined on the state space of a sufficiently richer, higher-order interaction structure. Using Carleman linearisation arguments, we show that finite polynomial dynamics defined in the |V| vertices of a graph admit an exact representation as linear dynamics on the state space of an hb-graph of order |V|, a combinatorial structure that extends hypergraphs by allowing vertex multiplicity, where the specific shape of the nonlinearity indicates whether the hb-graph is either finite or infinite (in terms of the number of hb-edges). For more general analytic nonlinearities, exact linear representation always require an hb-graph of infinite size, and its finite-size truncation provides an approximate representation of the original nonlinear graph-based dynamics.

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