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.
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ISSN: 2632-072X
JPhys Complexity is a new, interdisciplinary and fully open access journal publishing the most exciting and significant developments across all areas of complex systems and networks.
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Ginestra Bianconi et al 2023 J. Phys. Complex. 4 010201
A Baptista et al 2023 J. Phys. Complex. 4 042001
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.
Diogo L M Souza et al 2024 J. Phys. Complex. 5 025010
Spiral waves are spatial-temporal patterns that can emerge in different systems as heart tissues, chemical oscillators, ecological networks and the brain. These waves have been identified in the neocortex of turtles, rats, and humans, particularly during sleep-like states. Although their functions in cognitive activities remain until now poorly understood, these patterns are related to cortical activity modulation and contribute to cortical processing. In this work, ,we construct a neuronal network layer based on the spatial distribution of pyramidal neurons. Our main goal is to investigate how local connectivity and coupling strength are associated with the emergence of spiral waves. Therefore, we propose a trustworthy method capable of detecting different wave patterns, based on local and global phase order parameters. As a result, we find that the range of connection radius (R) plays a crucial role in the appearance of spiral waves. For R < 20 µm, only asynchronous activity is observed due to small number of connections. The coupling strength () greatly influences the pattern transitions for higher R, where spikes and bursts firing patterns can be observed in spiral and non-spiral waves. Finally, we show that for some values of R and bistable states of wave patterns are obtained.
Viktor Jirsa and Hiba Sheheitli 2022 J. Phys. Complex. 3 015007
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.
Clàudia Payrató-Borràs et al 2024 J. Phys. Complex. 5 025013
Mutualistic relationships, where species interact to obtain mutual benefits, constitute an essential component of natural ecosystems. The use of ecological networks to represent the species and their ecological interactions allows the study of structural and dynamic patterns common to different ecosystems. However, by neglecting the temporal dimension of mutualistic communities, relevant insights into the organization and functioning of natural ecosystems can be lost. Therefore, it is crucial to incorporate empirical phenology -the cycles of species' activity within a season- to fully understand the impact of temporal variability on network architecture. In this paper, by using empirical datasets together with a set of synthetic models, we propose a framework to characterize the phenology of plant-pollinator communities and assess how it reshapes their portrayal as a network. Analyses of three empirical cases reveal that non-trivial information is missed when representing the network of interactions as static, which leads to overestimating the value of fundamental structural features. We discuss the implications of our findings for mutualistic relationships and intra-guild competition for common resources. We show that recorded interactions and species' activity duration are pivotal factors in accurately replicating observed patterns within mutualistic communities. Furthermore, our exploration of synthetic models underscores the system-specific character of the mechanisms driving phenology, increasing our understanding of the complexities of natural ecosystems.
Matheus Henrique Junqueira Saldanha and Yoshito Hirata 2024 J. Phys. Complex. 5 025015
Seismicity is a complex phenomenon with a multitude of components involved. In order to perform forecasting, which has yet to be done sufficiently well, it is paramount to be in possession of information of all these components, and use this information effectively in a prediction model. In the literature, the influence of the Sun and the Moon in seismic activity on Earth has been discussed numerous times. In this paper we contribute to such discussion, giving continuity to a previous work. Most importantly, we instrument four earthquake catalogs from different regions, calculating the Moon tidal force at the region and time of each earthquake, which allows us to analyze the relation between the tidal forces and the earthquake magnitudes. At first, we find that the dynamical system governing Moon motion is unidirectionally coupled with seismic activity, indicating that the position of the Moon drives, to some extent, the earthquake generating process. Furthermore, we present an analysis that demonstrates a clear positive correlation between tidal force and earthquake magnitude. Finally, it is shown that the use of Moon tidal force data and sunspot number data can be used to improve next-day maximum magnitude forecasting, with the highest accuracy being achieved when using both kinds of data. We hope that our results encourage researchers to include data from Moon tidal forces and Sun activity in their earthquake forecasting models.
Luca Mungo et al 2024 J. Phys. Complex. 5 012001
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.
Paolo Bova et al 2024 J. Phys. Complex. 5 025009
Auditors can play a vital role in ensuring that tech companies develop and deploy AI systems safely, taking into account not just immediate, but also systemic harms that may arise from the use of future AI capabilities. However, to support auditors in evaluating the capabilities and consequences of cutting-edge AI systems, governments may need to encourage a range of potential auditors to invest in new auditing tools and approaches. We use evolutionary game theory to model scenarios where the government wishes to incentivise auditing but cannot discriminate between high and low-quality auditing. We warn that it is alarmingly easy to stumble on 'Adversarial Incentives', which prevent a sustainable market for auditing AI systems from forming. Adversarial Incentives mainly reward auditors for catching unsafe behaviour. If AI companies learn to tailor their behaviour to the quality of audits, the lack of opportunities to catch unsafe behaviour will discourage auditors from innovating. Instead, we recommend that governments always reward auditors, except when they find evidence that those auditors failed to detect unsafe behaviour they should have. These 'Vigilant Incentives' could encourage auditors to find innovative ways to evaluate cutting-edge AI systems. Overall, our analysis provides useful insights for the design and implementation of efficient incentive strategies for encouraging a robust auditing ecosystem.
Lewis Higgins et al 2023 J. Phys. Complex. 4 025008
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 more of the pitch than teams playing away, which reduces to 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 (). 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.
Xinshan Jiao et al 2024 J. Phys. Complex. 5 025014
Link prediction aims to predict the potential existence of links between two unconnected nodes within a network based on the known topological characteristics. Evaluation metrics are used to assess the effectiveness of algorithms in link prediction. The discriminating ability of these evaluation metrics is vitally important for accurately evaluating link prediction algorithms. In this study, we propose an artificial network model, based on which one can adjust a single parameter to monotonically and continuously turn the prediction accuracy of the specifically designed link prediction algorithm. Building upon this foundation, we show a framework to depict the effectiveness of evaluating metrics by focusing on their discriminating ability. Specifically, a quantitative comparison in the abilities of correctly discerning varying prediction accuracies was conducted encompassing nine evaluation metrics: Precision, Recall, F1-Measure, Matthews correlation coefficient, balanced precision, the area under the receiver operating characteristic curve (AUC), the area under the precision-recall curve (AUPR), normalized discounted cumulative gain (NDCG), and the area under the magnified receiver operating characteristic. The results indicate that the discriminating abilities of the three metrics, AUC, AUPR, and NDCG, are significantly higher than those of other metrics.
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Matheus Henrique Junqueira Saldanha and Yoshito Hirata 2024 J. Phys. Complex. 5 025015
Seismicity is a complex phenomenon with a multitude of components involved. In order to perform forecasting, which has yet to be done sufficiently well, it is paramount to be in possession of information of all these components, and use this information effectively in a prediction model. In the literature, the influence of the Sun and the Moon in seismic activity on Earth has been discussed numerous times. In this paper we contribute to such discussion, giving continuity to a previous work. Most importantly, we instrument four earthquake catalogs from different regions, calculating the Moon tidal force at the region and time of each earthquake, which allows us to analyze the relation between the tidal forces and the earthquake magnitudes. At first, we find that the dynamical system governing Moon motion is unidirectionally coupled with seismic activity, indicating that the position of the Moon drives, to some extent, the earthquake generating process. Furthermore, we present an analysis that demonstrates a clear positive correlation between tidal force and earthquake magnitude. Finally, it is shown that the use of Moon tidal force data and sunspot number data can be used to improve next-day maximum magnitude forecasting, with the highest accuracy being achieved when using both kinds of data. We hope that our results encourage researchers to include data from Moon tidal forces and Sun activity in their earthquake forecasting models.
Xinshan Jiao et al 2024 J. Phys. Complex. 5 025014
Link prediction aims to predict the potential existence of links between two unconnected nodes within a network based on the known topological characteristics. Evaluation metrics are used to assess the effectiveness of algorithms in link prediction. The discriminating ability of these evaluation metrics is vitally important for accurately evaluating link prediction algorithms. In this study, we propose an artificial network model, based on which one can adjust a single parameter to monotonically and continuously turn the prediction accuracy of the specifically designed link prediction algorithm. Building upon this foundation, we show a framework to depict the effectiveness of evaluating metrics by focusing on their discriminating ability. Specifically, a quantitative comparison in the abilities of correctly discerning varying prediction accuracies was conducted encompassing nine evaluation metrics: Precision, Recall, F1-Measure, Matthews correlation coefficient, balanced precision, the area under the receiver operating characteristic curve (AUC), the area under the precision-recall curve (AUPR), normalized discounted cumulative gain (NDCG), and the area under the magnified receiver operating characteristic. The results indicate that the discriminating abilities of the three metrics, AUC, AUPR, and NDCG, are significantly higher than those of other metrics.
Clàudia Payrató-Borràs et al 2024 J. Phys. Complex. 5 025013
Mutualistic relationships, where species interact to obtain mutual benefits, constitute an essential component of natural ecosystems. The use of ecological networks to represent the species and their ecological interactions allows the study of structural and dynamic patterns common to different ecosystems. However, by neglecting the temporal dimension of mutualistic communities, relevant insights into the organization and functioning of natural ecosystems can be lost. Therefore, it is crucial to incorporate empirical phenology -the cycles of species' activity within a season- to fully understand the impact of temporal variability on network architecture. In this paper, by using empirical datasets together with a set of synthetic models, we propose a framework to characterize the phenology of plant-pollinator communities and assess how it reshapes their portrayal as a network. Analyses of three empirical cases reveal that non-trivial information is missed when representing the network of interactions as static, which leads to overestimating the value of fundamental structural features. We discuss the implications of our findings for mutualistic relationships and intra-guild competition for common resources. We show that recorded interactions and species' activity duration are pivotal factors in accurately replicating observed patterns within mutualistic communities. Furthermore, our exploration of synthetic models underscores the system-specific character of the mechanisms driving phenology, increasing our understanding of the complexities of natural ecosystems.
Daniele Vilone et al 2024 J. Phys. Complex. 5 025012
The COVID-19 pandemic, caused by the SARS-CoV-2 virus, suddenly erupted in China at the beginning of 2020 and soon spread worldwide. This has resulted in an outstanding increase on research about the virus itself and, more in general, epidemics in many scientific fields. In this work we focus on the dynamics of the epidemic spreading and how it can be affected by the individual variability in compliance with social norms, i.e. in the adoption of preventive social norms by population's members, which influences the infectivity rate throughout the population and through time. By means of theoretical considerations, we show how such heterogeneities of the infection rate make the population more resistant against the epidemic spreading. Finally, we depict possible empirical tests aimed to confirm our results.
A Provata 2024 J. Phys. Complex. 5 025011
When chaotic oscillators are coupled in complex networks a number of interesting synchronization phenomena emerge. Notable examples are the frequency and amplitude chimeras, chimera death states, solitary states as well as combinations of these. In a previous study (Provata 2020 J. Phys. Complex.1 025006), a toy model was introduced addressing possible mechanisms behind the formation of frequency chimera states. In the present study a variation of the toy model is proposed to address the formation of amplitude chimeras. The proposed oscillatory model is now equipped with an additional 3rd order equation modulating the amplitude of the network oscillators. This way, the single oscillators are constructed as bistable in amplitude and depending on the initial conditions their amplitude may result in one of the two stable fixed points. Numerical simulations demonstrate that when these oscillators are nonlocally coupled in networks, they organize in domains with alternating amplitudes (related to the two fixed points), naturally forming amplitude chimeras. A second extension of this model incorporates nonlinear terms merging amplitude together with frequency, and this extension allows for the spontaneous production of composite amplitude-and-frequency chimeras occurring simultaneously in the network. Moreover the extended model allows to understand the emergence of bump states via the continuous passage from chimera states, when both fixed point amplitudes are positive, to bump states when one of the two fixed points vanishes. The synchronization properties of the network are studied as a function of the system parameters for the case of amplitude chimeras, bump states and composite amplitude-and-frequency chimeras. The proposed mechanisms of creating domains with variable amplitudes and/or frequencies provide a generic scenario for understanding the formation of the complex synchronization phenomena observed in networks of coupled nonlinear and chaotic oscillators.
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Luca Mungo et al 2024 J. Phys. Complex. 5 012001
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.
A Baptista et al 2023 J. Phys. Complex. 4 042001
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.
Christopher S Dunham et al 2021 J. Phys. Complex. 2 042001
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.
Henrik Jeldtoft Jensen 2021 J. Phys. Complex. 2 032002
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.
Sindre W Haugland 2021 J. Phys. Complex. 2 032001
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.
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Caldarola et al
Economic Complexity (EC) methods have gained increasing popularity across fields and disciplines. In particular, the EC toolbox has proved particularly promising in the study of complex and interrelated phenomena, such as the transition towards a greener economy. Using the EC approach, scholars have been investigating the relationship between EC and sustainability, proposing to identify the distinguishing characteristics of green products and to assess the readiness of productive and technological structures for the sustainability transition. This article proposes to review and summarize the data, methods, and empirical literature that are relevant to the study of the sustainability transition from an EC perspective. 
We review three distinct but connected blocks of literature on EC and environmental sustainability. First, we survey the evidence linking measures of EC to indicators related to environmental sustainability. Second, we review articles that strive to assess the green competitiveness of productive systems. Third, we examine evidence on green technological development and its connection to non-green knowledge bases. Finally, we summarize the findings for each block and identify avenues for further research in this recent and growing body of empirical literature.
Corberi et al
The voter model is an extremely simple yet nontrivial prototypical model of ordering dynamics, which has been studied in great detail. Recently, a great deal of activity has focused on long-range statistical physics models, where interactions take place among faraway sites, with a probability slowly decaying with distance. In this paper, we study analytically the one-dimensional long-range voter model, where an agent takes the opinion of another at distance r with probability ∝ r−α. The model displays rich and diverse features as α is changed. For α > 3 the behavior is similar to the one of the nearest-neighbor version, with the formation of ordered domains whose typical size grows as R(t) ∝ t1/2 until consensus (a fully ordered configuration) is reached. The correlation function C(r,t) between two agents at distance r obeys dynamical scaling with sizeable corrections at large distances r > r∗(t), slowly fading away in time. For 2 < α ≤ 3 violations of scaling appear, due to the simultaneous presence of two lengh-scales, the size of domains growing as t(α−2)/(α−1), and the distance L(t) ∝ t1/(α−1) over which correlations extend. For α ≤ 2 the system reaches a partially ordered stationary state, characterised by an algebraic correlator, whose lifetime diverges in the thermodynamic limit of infinitely many agents, so that consensus is not reached. For a finite system escape towards the fully ordered configuration is finally promoted by development of large distance correlations. In a system of N sites, global consensus is achieved after a time T ∝ N2 for α>3,T ∝Nα−1 for2<α≤3,andT ∝N forα≤2.
Silva et al
The post-World War II decades experienced rapid growth in international trade, but recently a trend of weakening globalization has been consolidating. We construct an International Trade Network (ITN) using bilateral trade (2010 to 2022) to assess how interconnectedness has evolved in the face of recent developments. Our analysis reveals that while network connectivity initially improved, there was a shift towards a negative trend from 2018, coinciding with an increasingly unfavorable environment for international trade. We also document significant changes in the roles of countries within the ITN. While the USA remains the primary hub and China solidifies its second position, key countries like Germany, France, Great Britain, and Japan have notably lost relevance, whereas nations like India and the Republic of Korea are gaining prominence. Finally, employing an econometric model, we show that countries with large economies, significant manufacturing sector, lower inward foreign direct investment stock, and economic and geopolitical stability tend to occupy more central positions in the ITN.
Yap et al
This work involves an investigation of the mechanics of the herding behaviour using a non-linear timescale, with the aim to generalize the herding model which helps to explain frequently occurring complex behaviour in the real world, such as the financial markets. A herding model with fractional order of derivatives was developed. This model involves the use of derivatives of order α where 0<α ≤1. We have found the generalized result that the number of groups of agents with size k increases linearly with time as nk={p(2p-1)(2-α)/[p(1-α)+1]}Γ(α+(2-α)/(1-p){Γ(k)/[Γ(k-1+α+(2-α)/(1-p))}t for α ∈ (0,1], where p is a growth parameter. The result reduces to that in a previous herding model with derivative order of 1 for α=1. The results corresponding to various values of α and p are presented. The group size distribution at long time is found to decay as a generalized power law, with an exponent depending on both α and p, thereby demonstrating that the scale invariance property of a complex system holds regardless of the order of the derivatives. The physical interpretation of fractional differentiation and fractional integration is also explored based on the results of this work.
Krawciw et al
Complex network theory has focused on properties of networks with real-valued edge weights. However, in signal transfer networks, such as those representing the transfer of light across an interferometer, complex-valued edge weights are needed to represent the manipulation of the signal in both magnitude and phase. These complex-valued edge weights introduce interference into the signal transfer, but it is unknown how such interference affects network properties such as small-worldness. To address this gap, we have introduced a small-world interferometer network model with complex-valued edge weights and generalized existing network measures to define the interferometric clustering coefficient, the apparent path length, and the interferometric small-world coefficient. Using high-performance computing resources, we generated a large set of small-world interferometers over a wide range of parameters in system size, nearest-neighbor count, and edge-weight phase and computed their interferometric network measures. We found that the interferometric small-world coefficient depends significantly on the amount of phase on complex-valued edge weights: for small edge-
weight phases, constructive interference led to a higher interferometric small-world coefficient; while larger edge-weight phases induced destructive interference which led to a lower interferometric small-world coefficient. Thus, for the small-world interferometer model, interferometric measures are necessary to capture the effect of interference on signal transfer. This model is an example of the type of problem that necessitates interferometric measures, and applies to any wave-based network including quantum networks.