Fernando Rodriguez-Vergara and Phil Husbands

Mathematics 2026, 14(3), 535

According to the Church–Turing thesis, the limit of what is computable is bounded by Turing machines. Following from this, given that general computable functions formally describe the notion of recursive mechanisms, it is sometimes argued that every organismic process that specifies consistent cognitive responses should be both limited to Turing machine capabilities and amenable to formalization. There is, however, a deep intuitive conviction permeating contemporary cognitive science, according to which mental phenomena, such as consciousness and agency, cannot be explained by resorting to this kind of framework. In spite of some exceptions, the overall tacit assumption is that whatever the mind is, it exceeds the reach of what is described by notions of computability. This issue, namely the nature of the relation between cognition and computation, becomes particularly pertinent and increasingly more relevant as a possible source of better understanding the inner workings of the mind, as well as the limits of artificial implementations thereof. Moreover, although it is often overlooked or omitted so as to simplify our models, it will probably define, or so we argue, the direction of future research on artificial life, cognitive science, artificial intelligence, and related fields.

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Alberto Robledo

Complexities 2026, 2(1), 3

We detail a procedure to transform the current empirical stage in the study of complex systems into a predictive phenomenological one. Our approach starts with the statistical-mechanical Landau-Ginzburg equation for dissipative processes, such as kinetics of phase change. Then, it imposes discrete time evolution to explicit back feeding, and adopts a power-law driving force to incorporate the onset of chaos, or, alternatively, criticality, the guiding principles of complexity. One obtains, in closed analytical form, a nonlinear renormalization-group (RG) fixed-point map descriptive of any of the three known (one-dimensional) transitions to or out of chaos. Furthermore, its Lyapunov function is shown to be the thermodynamic potential in q-statistics, because the regular or multifractal attractors at the transitions to chaos impose a severe impediment to access the system’s built-in configurations, leaving only a subset of vanishing measure available. To test the pertinence of our approach, we refer to the following complex systems issues: (i) Basic questions, such as demonstration of paradigms equivalence, illustration of self-organization, thermodynamic viewpoint of diversity, biological or other. (ii) Derivation of empirical laws, e.g., ranked data distributions (Zipf law), biological regularities (Kleiber law), river and cosmological structures (Hack law). (iii) Complex systems methods, for example, evolutionary game theory, self-similar networks, central-limit theorem questions. (iv) Condensed-matter physics complex problems (and their analogs in other disciplines), like, critical fluctuations (catastrophes), glass formation (traffic jams), localization transition (foraging, collective motion).

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Advances in artificial intelligence (AI) offer the prospect of manipulating beliefs and behaviors on a population-wide level (1). Large language models (LLMs) and autonomous agents (2) let influence campaigns reach unprecedented scale and precision. Generative tools can expand propaganda output without sacrificing credibility (3) and inexpensively create falsehoods that are rated as more human-like than those written by humans (3, 4). Techniques meant to refine AI reasoning, such as chain-of-thought prompting, can be used to generate more convincing falsehoods. Enabled by these capabilities, a disruptive threat is emerging: swarms of collaborative, malicious AI agents. Fusing LLM reasoning with multiagent architectures (2), these systems are capable of coordinating autonomously, infiltrating communities, and fabricating consensus efficiently. By adaptively mimicking human social dynamics, they threaten democracy. Because the resulting harms stem from design, commercial incentives, and governance, we prioritize interventions at multiple leverage points, focusing on pragmatic mechanisms over voluntary compliance.

DANIEL THILO SCHROEDER, MEEYOUNG CHA, ANDREA BARONCHELLI, NICK BOSTROM, NICHOLAS A. CHRISTAKIS, DAVID GARCIA, AMIT GOLDENBERG, YARA KYRYCHENKO, KEVIN LEYTON-BROWN, NINA LUTZ, GARY MARCUS, FILIPPO MENCZER, GORDON PENNYCOOK, DAVID G. RAND, MARIA RESSA, FRANK SCHWEITZER, DAWN SONG, CHRISTOPHER SUMMERFIELD, AUDREY TANG, JAY J. VAN BAVEL, SANDER VAN DER LINDEN, AND JONAS R. KUNST

SCIENCE 22 Jan 2026 Vol 391, Issue 6783 pp. 354-357

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Ricard Solé, Luis F Seoane, Jordi Pla-Mauri, Michael Timothy Bennett, Michael E. Hochberg, Michael Levin
Cognitive processes are realized across an extraordinary range of natural, artificial, and hybrid systems, yet there is no unified framework for comparing their forms, limits, and unrealized possibilities. Here, we propose a cognition space approach that replaces narrow, substrate-dependent definitions with a comparative representation based on organizational and informational dimensions. Within this framework, cognition is treated as a graded capacity to sense, process, and act upon information, allowing systems as diverse as cells, brains, artificial agents, and human-AI collectives to be analyzed within a common conceptual landscape. We introduce and examine three cognition spaces -- basal aneural, neural, and human-AI hybrid -- and show that their occupation is highly uneven, with clusters of realized systems separated by large unoccupied regions. We argue that these voids are not accidental but reflect evolutionary contingencies, physical constraints, and design limitations. By focusing on the structure of cognition spaces rather than on categorical definitions, this approach clarifies the diversity of existing cognitive systems and highlights hybrid cognition as a promising frontier for exploring novel forms of complexity beyond those produced by biological evolution.

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Galen J. Wilkerson
The origin of life poses a problem of combinatorial feasibility: How can temporally supported functional organization arise in exponentially branching assembly spaces when unguided exploration behaves as a memoryless random walk? We show that nonlinear threshold-cascade dynamics in connected interaction networks provide a minimal, substrate-agnostic mechanism that can soften this obstruction. Below a critical connectivity threshold, cascades die out locally and structured input-output response mappings remain sparse and transient-a "functional desert" in which accumulation is dynamically unsupported. Near the critical percolation threshold, system-spanning cascades emerge, enabling discriminative functional responses. We illustrate this transition using a minimal toy model and generalize the argument to arbitrary networked systems. Also near criticality, cascades introduce finite-timescale structural and functional coherence, directional bias, and weak dynamical path-dependence into otherwise memoryless exploration, allowing biased accumulation. This connectivity-driven transition-functional percolation-requires only generic ingredients: interacting units, nonlinear thresholds, influence transmission, and non-zero coherence times. The mechanism does not explain specific biochemical pathways, but it identifies a necessary dynamical regime in which structured functional organization can emerge and be temporarily supported, providing a physical foundation for how combinatorial feasibility barriers can be crossed through network dynamics alone.

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Erik Hoel

Patterns, Volume 7, Issue 1101472January 09, 2026

Complex systems can be described at myriad different scales, and their causal workings often have a multiscale structure (e.g., a computer can be described at the microscale of its hardware circuitry, the mesoscale of its machine code, and the macroscale of its operating system). While scientists study and model systems across the full hierarchy of their scales, from microphysics to macroeconomics, there is debate about what the macroscales of systems can possibly add beyond mere compression. To resolve this long-standing issue, here, a new theory of emergence is introduced that can distinguish which scales irreducibly contribute to a system’s causal workings. The theory’s application is demonstrated in coarse grains of Markov chains, revealing a novel measure of emergent complexity: how widely distributed a system’s causal contributions are across its hierarchy of scales.

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Didier Sornette, Sandro Claudio Lera, Ke Wu
Recent reports of large language models (LLMs) exhibiting behaviors such as deception, threats, or blackmail are often interpreted as evidence of alignment failure or emergent malign agency. We argue that this interpretation rests on a conceptual error. LLMs do not reason morally; they statistically internalize the record of human social interaction, including laws, contracts, negotiations, conflicts, and coercive arrangements. Behaviors commonly labeled as unethical or anomalous are therefore better understood as structural generalizations of interaction regimes that arise under extreme asymmetries of power, information, or constraint. Drawing on relational models theory, we show that practices such as blackmail are not categorical deviations from normal social behavior, but limiting cases within the same continuum that includes market pricing, authority relations, and ultimatum bargaining. The surprise elicited by such outputs reflects an anthropomorphic expectation that intelligence should reproduce only socially sanctioned behavior, rather than the full statistical landscape of behaviors humans themselves enact. Because human morality is plural, context-dependent, and historically contingent, the notion of a universally moral artificial intelligence is ill-defined. We therefore reframe concerns about artificial general intelligence (AGI). The primary risk is not adversarial intent, but AGI's role as an endogenous amplifier of human intelligence, power, and contradiction. By eliminating longstanding cognitive and institutional frictions, AGI compresses timescales and removes the historical margin of error that has allowed inconsistent values and governance regimes to persist without collapse. Alignment failure is thus structural, not accidental, and requires governance approaches that address amplification, complexity, and regime stability rather than model-level intent alone.

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Sara Najem, Amer E. Mouawad
Determining whether two graphs are structurally identical is a fundamental problem with applications spanning mathematics, computer science, chemistry, and network science. Despite decades of study, graph isomorphism remains a challenging algorithmic task, particularly for highly symmetric structures. Here we introduce a new algorithmic approach based on ideas from spectral graph theory and geometry that constructs candidate correspondences between vertices using their curvatures. Any correspondence produced by the algorithm is explicitly verified, ensuring that non-isomorphic graphs are never incorrectly identified as isomorphic. Although the method does not yet guarantee success on all isomorphic inputs, we find that it correctly resolves every instance tested in deterministic polynomial time, including a broad collection of graphs known to be difficult for classical spectral techniques. These results demonstrate that enriched spectral methods can be far more powerful than previously understood, and suggest a promising direction for the practical resolution of the complexity of the graph isomorphism problem.

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Galen J. Wilkerson
Understanding how network structure constrains and enables information processing is a central problem in the statistical mechanics of interacting systems. Here we study random networks across the structural percolation transition and analyze how connectivity governs realizable input-output transformations under cascade dynamics. Using Erdos-Renyi networks as a minimal ensemble, we examine structural, functional, and information-theoretic observables as functions of mean degree. We find that the emergence of the giant connected component coincides with a sharp transition in realizable information processing: complex input-output response functions become accessible, functional diversity increases rapidly, output entropy rises, and directed information flow, quantified by transfer entropy, extends beyond local neighborhoods. We term this coincidence of structural, functional, and informational transitions functional percolation, referring to a sharp expansion of the space of realizable input-output functions at the percolation threshold. Near criticality, networks exhibit a Pareto-optimal tradeoff between functional complexity and diversity, suggesting that percolation criticality may provide a general organizing principle of information processing capacity in systems with local interactions and propagating influences.

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Marlene C. L. Batzke, Peter Steiglechner, Jan Lorenz, Bruce Edmonds, František Kalvas

Political Psychology 

Political polarization represents a rising issue in many countries, making it more and more important to understand its relation to cognitive-motivational and social influence mechanisms. Yet, the link between micro-level mechanisms and macro-level phenomena remains unclear. We investigated the consequences of individuals striving for cognitive coherence in their belief systems on political polarization in society in an agent-based model. In this, we formalized how cognitive coherence affects how individuals update their beliefs following social influence and self-reflection processes. We derive agents' political beliefs as well as their subjective belief systems, defining what determines coherence for different individuals, from European Social Survey data via correlational class analysis. The simulation shows that agents polarize in their beliefs when they have a strong strive for cognitive coherence, and especially when they have structurally different belief systems. In a mathematical analysis, we not only explain the main findings but also underscore the necessity of simulations for understanding the complex dynamics of socially embedded phenomena such as political polarization.

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The Effects of Remote Working on Scientific Collaboration and Impact

Sara Venturini, Satyaki Sikdar, Martina Mazzarello, Francesco Rinaldi, Francesco Tudisco, Paolo Santi, Santo Fortunato, Carlo Ratti
The COVID-19 pandemic shifted academic collaboration from in-person to remote interactions. This study explores, for the first time, the effects on scientific collaborations and impact of such a shift, comparing research output before, during, and after the pandemic. Using large-scale bibliometric data, we track the evolution of collaboration networks and the resulting impact of research over time. Our findings are twofold: first, the geographic distribution of collaborations significantly shifted, with a notable increase in cross-border partnerships after 2020, indicating a reduction in the constraints of geographic proximity. Second, despite the expansion of collaboration networks, there was a concerning decline in citation impact, suggesting that the absence of spontaneous in-person interactions-which traditionally foster deep discussions and idea exchange-negatively affected research quality. As hybrid work models in academia gain traction, this study highlights the need for universities and research organizations to carefully consider the balance between remote and in-person engagement.

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Marco A. Rosas Pulido, Roberto Murcio, Omar R. Vázquez, Carlos Gershenson
Intricate interactions among firms, institutions, and spatial structures shape urban economic systems. In this study, we propose a framework based on three structural dimensions -- abundance, diversity, and longevity (ADL) of economic units -- as proxies of urban economic complexity and resilience. Using a decade of georeferenced firm-level data from Mexico City, we analyze the relationships among ADL variables using regression, spatial correlation, and time-series clustering. Our results reveal nonlinear dynamics across urban space, with powerlaw behavior in central zones and logarithmic saturation in peripheral areas, suggesting differentiated growth regimes. Notably, firm longevity modulates the relationship between abundance and diversity, particularly in periurban transition zones. These spatial patterns point to an emerging polycentric restructuring within a traditionally monocentric metropolis. By integrating economic complexity theory with spatial analysis, our approach provides a scalable method to assess the adaptive capacity of urban economies. This has implications for understanding informality, designing inclusive urban policies, and navigating structural transitions in rapidly urbanizing regions.

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Nick Holton, Marianne Cottin, Adam Wright, Michael Mannino, Dayanne S. Antonio, Marcelo Bigliassi

Antifragility challenges conventional thinking by proposing that adversity is not merely to be survived but actively used to promote growth. This scoping review synthesizes 18 emerging research studies focused on antifragility in human systems across disciplines, distinguishing antifragility from resilience and robustness and highlighting key empirical gaps, particularly in psychological research. During the screening process, articles were categorized as human or non-human systems. Non-human systems (n = 29; e.g., robotics, logistics, information systems, urban planning, artificial intelligence) were excluded from synthesis to align with the review’s focus on human domains (e.g., psychology, leadership, coaching, health). Drawing from biology, psychology, and organizational studies, the review summarizes applications in mental health, performance, and quality of life. Findings emphasize the proactive nature of antifragility, in which stressors are intentionally engaged to strengthen capabilities. Biological concepts like hormesis and psychological frameworks such as post-traumatic growth align with mechanisms relevant to growth through adversity. Yet empirical studies remain scarce, underscoring the need for robust measurement tools and longitudinal designs. Future directions include refining antifragility as a state, trait, or process, developing dose-specific models, and exploring biopsychosocial correlates. Embracing antifragility could transform how individuals and systems confront challenge, not by resisting breakdown, but by evolving beyond it.

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Giulia Fischetti, Anna Mancini, Giulio Cimini, Jessica T. Davis, Abby Leung, Alessandro Vespignani, Guido Caldarelli
Accurate representations of the World Air Transportation Network (WAN) are fundamental inputs to models of global mobility, epidemic risk, and infrastructure planning. However, high-resolution, real-time data on the WAN are largely commercial and proprietary, therefore often inaccessible to the research community. Here we introduce a generative model of the WAN that treats air travel as a stochastic process within a maximum-entropy framework. The model uses airport-level passenger flows to probabilistically generate connections while preserving traffic volumes across geographic regions. The resulting reconstructed networks reproduce key structural properties of the WAN and enable simulations of dynamic spreading that closely match those obtained using the real network. Our approach provides a scalable, interpretable, and computationally efficient framework for forecasting and policy design in global mobility systems.

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Ori Livson, Siddharth Pritam, Mikhail Prokopenko
Preference cycles are prevalent in problems of decision-making, and are contradictory when preferences are assumed to be transitive. This contradiction underlies Condorcet's Paradox, a pioneering result of Social Choice Theory, wherein intuitive and seemingly desirable constraints on decision-making necessarily lead to contradictory preference cycles. Topological methods have since broadened Social Choice Theory and elucidated existing results. However, characterisations of preference cycles in Topological Social Choice Theory are lacking. In this paper, we address this gap by introducing a framework for topologically modelling preference cycles that generalises Baryshnikov's existing topological model of strict, ordinal preferences on 3 alternatives. In our framework, the contradiction underlying Condorcet's Paradox topologically corresponds to the non-orientability of a surface homeomorphic to either the Klein Bottle or Real Projective Plane, depending on how preference cycles are represented. These findings allow us to reduce Arrow's Impossibility Theorem to a statement about the orientability of a surface. Furthermore, these results contribute to existing wide-ranging interest in the relationship between non-orientability, impossibility phenomena in Economics, and logical paradoxes more broadly.

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María Fernanda Sánchez-Puig, Carlos Gershenson, Carlos Pineda

The large digital archives of the American Physical Society (APS) offer an opportunity to quantitatively analyze the structure and evolution of scientific communication. In this paper, we perform a comparative analysis of the language used in eight APS journals (Phys. Rev. A, B, C, D, E, Lett., X, Rev. Mod. Phys.) using methods from statistical linguistics. We study word rank distributions (from monograms to hexagrams), finding that they are consistent with Zipf’s law. We also analyze rank diversity over time, which follows a characteristic sigmoid shape. To quantify the linguistic similarity between journals, we use the rank-biased overlap (RBO) distance, comparing the journals not only to each other, but also to corpora from Google Books and Twitter. This analysis reveals that the most significant differences emerge when focusing on content words rather than the full vocabulary. By identifying the unique and common content words for each specialized journal, we develop an article classifier that predicts a paper’s journal of origin based on its unique word distribution. This classifier uses a proposed “importance factor” to weigh the significance of each word. Finally, we analyze the frequency of mention of prominent physicists and compare it to their cultural recognitions ranked in the Pantheon dataset, finding a low correlation that highlights the context-dependent nature of scientific fame. These results demonstrate that scientific language itself can serve as a quantitative window into the organization and evolution of science.

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Andrea Roli, Sudip Patra, Stuart Kauffman

Interface Focus (2025) 15 (6): 20250038

We discuss the creation of information in the evolution of the biosphere by elaborating on the interplay between affordances and constraints. We maintain that information is created when affordances are seized and, therefore, at the same time, meaning is generated and a new space of possibilities is created.

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