Complexities, Volume 2, Issue 1 (March 2026) – 7 articles

Shapley values are the most widely used point-valued solution concept for cooperative games and have recently garnered attention for their applicability in explainable machine learning. Due to the complexity of Shapley value computation, users mostly resort to Monte Carlo approximations for large problems. We take a detailed look at an approximation method grounded in multilinear extensions proposed in 2021 under the name “Owen sampling”. We point out why Owen sampling is biased and propose unbiased alternatives based on combining multilinear extensions with stratified sampling and importance sampling. Finally, we discuss empirical results of the presented algorithms for various cooperative games, including real-world explainability scenarios. Full article

Emergent properties are properties of a whole that cannot be reduced to the properties of its parts. Properties of a system can be defined as relations between a particular input given to a system and its corresponding output. From this perspective, whole systems formed by coupling component systems have properties different from the properties of their components. Wholes tend to arise spontaneously through a process of self-organization, in which components randomly interact until they settle in a stable configuration that in general cannot be predicted from the properties of the components. This configuration constrains the relations between the components, thus defining emergent “laws” that downwardly cause the further behavior of the components. Thus, emergent wholes and their properties arise in a simple and natural manner. Full article

Many complex systems, particularly glasses and disordered materials, exhibit energy landscapes with exponentially many metastable states. Such landscape structure strongly influences equilibrium behavior but is not explicitly represented in standard thermodynamic state spaces. We develop a constrained equilibrium framework in which configurational complexity, defined as the logarithmic density of metastable basins, is treated as an additional macroscopic coordinate. Starting from maximum entropy with simultaneous constraints on energy and complexity, we obtain a generalized Gibbs ensemble characterized by a conjugate bias parameter. Standard thermodynamic structure remains intact, with extended relations arising as constrained equilibrium identities. A mean-field glassy example with explicit complexity function demonstrates how complexity bias shifts the saddle-point structure of the partition function and modifies equilibrium response functions. The geometric formulation further provides a diagnostic of landscape reorganization within an enlarged state space. This framework offers a systematic equilibrium description of how energy-landscape structure influences thermodynamic behavior in systems with rugged configuration spaces. Full article

Agent-based models of financial markets often rely on a small set of hand-crafted trading rules, making it difficult to relate model heterogeneity to information that is observable in market data. We take a different standpoint and treat the design of heterogeneity as a representation problem under limited observations. In our framework, each agent’s decision rule is implemented as a neural-network mapping from recent price histories to order decisions, trained on historical index or stock price series. To describe and manipulate heterogeneity without pre-assigning mechanism labels, we introduce Fit Quality (FQ), an ex post effect-defined index summarizing how strongly each learned rule fits the price patterns it was trained on, and we use FQ solely as a coordinate for organizing agent populations and constructing controlled changes in agent composition, rather than as a measure of forecasting skill or economic performance. Using this representation, we examine whether simulations can reproduce several stylized features of return series. We also perform simple ablation experiments to assess how far the observed properties depend on the data-trained decision rules rather than on the market mechanism alone. Taken together, the framework is intended as a step toward more data-linked, representation-conscious agent-based models, in which alternative ways of organizing heterogeneity can be compared within a common market environment. Full article

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). Full article

Urban and peri-urban agriculture (UPA) has emerged as a critical strategy to address multidimensional urban challenges, including food insecurity, environmental degradation, and social inequality. Despite its potential benefits, UPA occupies a marginal position in municipal governance frameworks. Understanding how public policies and social influence mechanisms shape consumer behavior and producer viability requires a systems-thinking approach capable of capturing complex socio-economic-ecological interactions. Therefore, we developed an agent-based model (ABM) following the ODD + D protocol to simulate urban agriculture market dynamics, incorporating producer and consumer agents within a spatially explicit grid environment representing the urban landscape. We implemented three policy interventions and conducted six complementary experiments. Education campaigns achieved the highest local market share, demonstrating strict Pareto dominance over all subsidy-based strategies. Production subsidies yielded equivalent outcomes but at a fiscal cost, reducing producer income inequality (Gini). Stress tests revealed moderate resilience to production shocks. The findings demonstrate the power of agent-based modeling to uncover policy dynamics in complex urban food systems, providing actionable evidence for sustainable urban governance. Full article

This study investigated the impact of post-activation performance enhancement (PAPE) on the parameters of the 3 min all-out test (3MT) in non-motorized tethered running, applying the concept of complex networks for integrative analysis. Ten recreational runners underwent anthropometric assessments, a one-repetition maximum test (1RM), a running ramp test, and 3MT trials under both PAPE and CONTROL conditions across five separate sessions. The conditioning activity consisted of two sets of six back squats at 60% 1RM. For each scenario, complex network graphs were constructed and analyzed using Degree, Eigenvector, PageRank, and Betweenness centrality metrics. In the PAPE condition, anthropometric parameters and parameters related to aerobic efficiency exhibited greater centrality, ranking among the top five nodes. Paired Student’s t-tests (p ≤ 0.05) revealed significant differences between conditions for end power (EP-W) (CONTROL: 407.83 ± 119.30 vs. PAPE: 539.33 ± 177.10 (effect size d = −0.84)) and end power relativized by body mass (rEP-W·kg⁻¹) (CONTROL: 5.38 ± 1.70 vs. PAPE: 6.91 ± 2.00 (effect size d = −0.76)), as well as for the absolute and relative values of peak output power, mean output power, peak force, and mean force. These findings suggest that PAPE alters the configuration of complex networks, increasing network density, and may enhance neuromuscular function and running economy. Moreover, PAPE appears to modulate both aerobic and anaerobic contributions to performance. These results highlight the importance of network-based approaches for advancing exercise science and providing individualized strategies for training and performance optimization. Full article