Journal of Physics: Complexity

The 2021 Nobel Prize in Physics recognized the fundamental role of complex systems in the natural sciences. In order to celebrate this milestone, this editorial presents the point of view of the editorial board of JPhys Complexity on the achievements, challenges, and future prospects of the field. To distinguish the voice and the opinion of each editor, this editorial consists of a series of editor perspectives and reflections on few selected themes. A comprehensive and multi-faceted view of the field of complexity science emerges. We hope and trust that this open discussion will be of inspiration for future research on complex systems.

Network reconstruction is a well-developed sub-field of network science, but it has only recently been applied to production networks, where nodes are firms and edges represent customer-supplier relationships. We review the literature that has flourished to infer the topology of these networks by partial, aggregate, or indirect observation of the data. We discuss why this is an important endeavour, what needs to be reconstructed, what makes it different from other network reconstruction problems, and how different researchers have approached the problem. We conclude with a research agenda.

Neuroscience is home to concepts and theories with roots in a variety of domains including information theory, dynamical systems theory, and cognitive psychology. Not all of those can be coherently linked, some concepts are incommensurable, and domain-specific language poses an obstacle to integration. Still, conceptual integration is a form of understanding that provides intuition and consolidation, without which progress remains unguided. This paper is concerned with the integration of deterministic and stochastic processes within an information theoretic framework, linking information entropy and free energy to mechanisms of emergent dynamics and self-organization in brain networks. We identify basic properties of neuronal populations leading to an equivariant matrix in a network, in which complex behaviors can naturally be represented through structured flows on manifolds establishing the internal model relevant to theories of brain function. We propose a neural mechanism for the generation of internal models from symmetry breaking in the connectivity of brain networks. The emergent perspective illustrates how free energy can be linked to internal models and how they arise from the neural substrate.

The participation coefficient is a widely used metric of the diversity of a node's connections with respect to a modular partition of a network. An information-theoretic formulation of this concept of connection diversity, referred to here as participation entropy, has been introduced as the Shannon entropy of the distribution of module labels across a node's connected neighbors. While diversity metrics have been studied theoretically in other literatures, including to index species diversity in ecology, many of these results have not previously been applied to networks. Here we show that the participation coefficient is a first-order approximation to participation entropy and use the desirable additive properties of entropy to develop new metrics of connection diversity with respect to multiple labelings of nodes in a network, as joint and conditional participation entropies. The information-theoretic formalism developed here allows new and more subtle types of nodal connection patterns in complex networks to be studied.

Higher-order networks encode the many-body interactions existing in complex systems, such as the brain, protein complexes, and social interactions. Simplicial complexes are higher-order networks that allow a comprehensive investigation of the interplay between topology and dynamics. However, simplicial complexes have the limitation that they only capture undirected higher-order interactions while in real-world scenarios, often there is a need to introduce the direction of simplices, extending the popular notion of direction of edges. On graphs and networks the Magnetic Laplacian, a special case of connection Laplacian, is becoming a popular operator to address edge directionality. Here we tackle the challenge of handling directionality in simplicial complexes by formulating higher-order connection Laplacians taking into account the configurations induced by the simplices' directions. Specifically, we define all the connection Laplacians of directed simplicial complexes of dimension two and we discuss the induced higher-order diffusion dynamics by considering instructive synthetic examples of simplicial complexes. The proposed higher-order diffusion processes can be adopted in real scenarios when we want to consider higher-order diffusion displaying non-trivial frustration effects due to conflicting directionalities of the incident simplices.

We study pitch control in football, using data from six complete seasons of the English Premier League. Our objective is to investigate features of pitch control in the data. We process the data to ensure consistency of the tracking and event datasets. This represents the largest coherent dataset analysed in the literature and allows the observation of consistent patterns across several seasons' data. We demonstrate that teams playing in front of a crowd at home control on average $\left(2.9\pm0.2\right)\%$ more of the pitch than teams playing away, which reduces to $\left(1.5\pm0.6\right)\%$ in matches played behind closed doors. We observe that match by match the difference in pitch control between the teams has a weak, positive correlation with the difference in expected goals (Pearson correlation R = 0.38). As a further manifestation of home advantage we find that in games which the two teams have equal pitch control, on average the home team accumulates greater expected goals ($0.16 \pm 0.03$). The concept of weighted pitch control is introduced, by assigning a weight to regions of the pitch. We demonstrate that pitch control of the penalty box of the out-of-possession team is negatively correlated with expected goals in each of the six seasons, and interpret this apparently counter-intuitive result.

Inequalities in wealth, income, access to food and healthcare have been rising worldwide in the past decades, approaching levels seen in the early 20th century. Here we study the relationships between wealth inequality and mobility for different segments of the population, comparing longitudinal surveys conducted in the USA and in Italy. The larger wealth inequality observed in the USA is reflected by poorer health conditions than in Italy. We also find that in both countries wealth mobility becomes slower at the two extremes of the wealth distribution. Households trapped in a state of persistent lack of wealth are generally experiencing greater food insecurity and poorer health than the general population. We interpret the observed association between inequality and immobility using a simple agent based model of wealth condensation driven by random returns and exchanges. The model describes well survey data on a qualitative level, but the mobility is generally overestimated by the model. We trace back this discrepancy to the way income is generated for low-wealth households which is not correctly accounted by the model. On the other hand, the model is excellent in describing the wealth dynamics within a restricted class of ultra-wealthy, as we demonstrate by analyzing billionaires lists. Our results suggest that different forms of inequality are intertwined and should therefore be addressed together.

Will advanced artificial intelligence (AI) language models exhibit trust toward humans? Gauging an AI model's trust in humans is challenging because—absent costs for dishonesty—models might respond falsely about trusting humans. Accordingly, we devise a method for incentivizing machine decisions without altering an AI model's underlying algorithms or goal orientation and we employ the method in trust games between an AI model from OpenAI and a human experimenter (namely, author TJ). We find that the AI model exhibits behavior consistent with trust in humans at higher rates when facing actual incentives than when making hypothetical decisions—a finding that is robust to prompt phrasing and the method of game play. Furthermore, trust decisions appear unrelated to the magnitude of stakes and additional experiments indicate that they do not reflect a non-social preference for uncertainty.

In a rapidly changing world, facing an increasing number of socioeconomic, health and environmental crises, complexity science can help us to assess and quantify vulnerabilities, and to monitor and achieve the UN sustainable development goals. In this perspective, we provide three exemplary use cases where complexity science has shown its potential: poverty and socioeconomic inequalities, collective action for representative democracy, and computational epidemic modeling. We then review the challenges and limitations related to data, methods, capacity building, and, as a result, research operationalization. We finally conclude with some suggestions for future directions, urging the complex systems community to engage in applied and methodological research addressing the needs of the most vulnerable.

The erosion of social cohesion and polarization is one of the topmost societal risks. In this work, we investigated the evolution of polarization, influence, and domination in online interaction networks using a large Twitter dataset collected before and during the 2022 Brazilian elections. From a theoretical perspective, we develop a methodology called d-modularity that allows discovering the contribution of specific groups to network polarization using the well-known modularity measure. While the overall network modularity (somewhat unexpectedly) decreased, the proposed group-oriented approach reveals that the contribution of the right-leaning community to this modularity increased, remaining very high during the analyzed period. Our methodology is general enough to be used in any situation when the contribution of specific groups to overall network modularity and polarization is needed to investigate. Moreover, using the concept of partial domination, we are able to compare the reach of sets of influential profiles from different groups and their ability to accomplish coordinated communication inside their groups and across segments of the entire network. We show that in the whole network, the left-leaning high-influential information spreaders dominated, reaching a substantial fraction of users with fewer spreaders. However, when comparing domination inside the groups, the results are inverse. Right-leaning spreaders dominate their communities using few nodes, showing as the most capable of accomplishing coordinated communication. The results bring evidence of extreme isolation and the ease of accomplishing coordinated communication that characterized right-leaning communities during the 2022 Brazilian elections, which likely influenced the subsequent coup events in Brasilia.

Recent years have seen a surge in interest for leveraging neural networks to parameterize small-scale or fast processes in climate and turbulence models. In this short paper, we point out two fundamental issues in this endeavor. The first concerns the difficulties neural networks may experience in capturing rare events due to limitations in how data is sampled. The second arises from the inherent multiscale nature of these systems. They combine high-frequency components (like inertia-gravity waves) with slower, evolving processes (geostrophic motion). This multiscale nature creates a significant hurdle for neural network closures. To illustrate these challenges, we focus on the atmospheric 1980 Lorenz model, a simplified version of the Primitive Equations that drive climate models. This model serves as a compelling example because it captures the essence of these difficulties.

Tree-based machine learning algorithms provide the most precise assessment of the feasibility for a country to export a target product given its export basket. However, the high number of parameters involved prevents a straightforward interpretation of the results and, in turn, the explainability of policy indications. In this paper, we propose a procedure to statistically validate the importance of the products used in the feasibility assessment. In this way, we are able to identify which products, called explainers, significantly increase the probability to export a target product in the near future. The explainers naturally identify a low dimensional representation, the Feature Importance Product Space, that enhances the interpretability of the recommendations and provides out-of-sample forecasts of the export baskets of countries. Interestingly, we detect a positive correlation between the complexity of a product and the complexity of its explainers.

In this article, we explore the concept and measurement of the degree of economic development pattern (DEDP) of economy, which refers to the extent to which the development of an economy can serve as a reference for other economies. Utilizing 76 macroeconomic indicators across 217 economies, the economic development paths in a standardized space of economy is compared to identify variations in DEDP through the regression analysis on the relationship between the similarity of development paths and the growth rate on gross domestic product (GDP) per capita. To measure DEDP of economy from different perspective, two types of metrics are constructed. One is the determination coefficient of regression analysis, which exhibits significant positive correlations with population size of economy, uncovering differences of development paths among economies of varying population sizes. The other type of metrics is based on the consistency on regression coefficients and effectively explains disparities among economies in the growth rate on GDP per capita, economic complexity index and economic fitness. These findings reveal the differences in development paths among different countries from the perspective of referentiality for development patterns, suggesting the potential existence of the paths with more universal meaning to economic development.

Ride sharing services combine trips of multiple users in the same vehicle and may provide more sustainable transport than private cars. As mobility demand varies during the day, the travel times experienced by passengers may substantially vary as well, making the service quality unreliable. We show through model simulations that such travel time fluctuations may be drastically reduced by stop pooling. Having users walk to meet at joint locations for pick-up or drop-off allows buses to travel more direct routes by avoiding frequent door-to-door detours, especially during high demand. We in particular propose adaptive stop pooling by adjusting the maximum walking distance to the temporally and spatially varying demand. The results highlight that adaptive stop pooling may substantially reduce travel time fluctuations while even improving the average travel time of ride sharing services, especially for high demand. Such quality improvements may in turn increase the acceptance and adoption of ride sharing services.

Higher-order networks encode the many-body interactions existing in complex systems, such as the brain, protein complexes, and social interactions. Simplicial complexes are higher-order networks that allow a comprehensive investigation of the interplay between topology and dynamics. However, simplicial complexes have the limitation that they only capture undirected higher-order interactions while in real-world scenarios, often there is a need to introduce the direction of simplices, extending the popular notion of direction of edges. On graphs and networks the Magnetic Laplacian, a special case of connection Laplacian, is becoming a popular operator to address edge directionality. Here we tackle the challenge of handling directionality in simplicial complexes by formulating higher-order connection Laplacians taking into account the configurations induced by the simplices' directions. Specifically, we define all the connection Laplacians of directed simplicial complexes of dimension two and we discuss the induced higher-order diffusion dynamics by considering instructive synthetic examples of simplicial complexes. The proposed higher-order diffusion processes can be adopted in real scenarios when we want to consider higher-order diffusion displaying non-trivial frustration effects due to conflicting directionalities of the incident simplices.

Network reconstruction is a well-developed sub-field of network science, but it has only recently been applied to production networks, where nodes are firms and edges represent customer-supplier relationships. We review the literature that has flourished to infer the topology of these networks by partial, aggregate, or indirect observation of the data. We discuss why this is an important endeavour, what needs to be reconstructed, what makes it different from other network reconstruction problems, and how different researchers have approached the problem. We conclude with a research agenda.

Networks have provided extremely successful models of data and complex systems. Yet, as combinatorial objects, networks do not have in general intrinsic coordinates and do not typically lie in an ambient space. The process of assigning an embedding space to a network has attracted great interest in the past few decades, and has been efficiently applied to fundamental problems in network inference, such as link prediction, node classification, and community detection. In this review, we provide a user-friendly guide to the network embedding literature and current trends in this field which will allow the reader to navigate through the complex landscape of methods and approaches emerging from the vibrant research activity on these subjects.

Numerous studies suggest critical dynamics may play a role in information processing and task performance in biological systems. However, studying critical dynamics in these systems can be challenging due to many confounding biological variables that limit access to the physical processes underpinning critical dynamics. Here we offer a perspective on the use of abiotic, neuromorphic nanowire networks as a means to investigate critical dynamics in complex adaptive systems. Neuromorphic nanowire networks are composed of metallic nanowires and possess metal-insulator-metal junctions. These networks self-assemble into a highly interconnected, variable-density structure and exhibit nonlinear electrical switching properties and information processing capabilities. We highlight key dynamical characteristics observed in neuromorphic nanowire networks, including persistent fluctuations in conductivity with power law distributions, hysteresis, chaotic attractor dynamics, and avalanche criticality. We posit that neuromorphic nanowire networks can function effectively as tunable abiotic physical systems for studying critical dynamics and leveraging criticality for computation.

We present a brief review of power laws and correlation functions as measures of criticality and the relation between them. By comparing phenomenology from rain, brain and the forest fire model we discuss the relevant features of self-organisation to the vicinity about a critical state. We conclude that organisation to a region of extended correlations and approximate power laws may be behaviour of interest shared between the three considered systems.

Chimera states, states of coexistence of synchronous and asynchronous motion, have been a subject of extensive research since they were first given a name in 2004. Increased interest has lead to their discovery in ever new settings, both theoretical and experimental. Less well-discussed is the fact that successive results have also broadened the notion of what actually constitutes a chimera state. In this article, we critically examine how the results for different model types and coupling schemes, as well as varying implicit interpretations of terms such as coexistence, synchrony and incoherence, have influenced the common understanding of what constitutes a chimera. We cover both theoretical and experimental systems, address various chimera-derived terms that have emerged over the years and finally reflect on the question of chimera states in real-world contexts.