To celebrate the launch of Frontiers in Complex Systems, Frontiers has organized a series of special issue Research Topics to highlight the latest advances in this field. “Insights In Complex Systems Theory” solicits brief, forward-looking contributions from the members of the journal's accomplished Editorial Board, focused on new insights, novel developments, current challenges, latest discoveries, major accomplishments and future perspectives in the field. Authors are encouraged to identify the greatest challenges in their disciplines, and how to address these challenges. The goal of this special edition Research Topic is to provide a thorough overview of the field, whilst inspiring, informing and directing researchers in the field.

University of Catania

Sapienza University of Rome

Istanbul University

Frontiers in Complex Systems

Complex Systems Theory

Insights In Complex Systems Theory

BackgroundHistorically, although researchers in the science of complex systems proposed the idea of the edge of chaos and/or self-organized criticality as the essential feature of complex organization, they were not able to generalize this concept. Complex organization is regarded at the edge of chaos between the order phase and the chaos phase and a very rare case. Additionally, in cellular automata, the critical property is class IV, which is also rarely found. Therefore, there can be overestimation for natural selection. More recently, developments in cognitive and brain science have led to the free energy principle based on Bayesian inference, while quantum cognition has been established to explain various cognitive phenomena. Since Bayesian inference results in the perspective of a steady state, it can be described in Boolean logic. Considering that quantum logic consists of multiple Boolean logic in terms of lattice theory, the perspective of the free energy principle is the perspective of order, and the perspective of quantum logic might be the perspective of multiple worlds, which is strongly relevant for the edge of chaos.</sec>ProblemThe next question arises whether the perspective derived from quantum logic can be generalized for the complex behavior consisting of both order and chaos, instead of the edge of chaos or self-organized criticality, to reveal the property of critical behavior such as a power-law distribution.</sec>SolutionIn this study, we define quantum logic automata, which entail quantum logic (orthomodular lattice) in terms of lattice theory and have the features of a dynamical system. Because quantum logic automata are applied to a binary sequence, one can estimate the behavior of those automata with respect to patterns and a time series. Here, we show that most of a group of quantum logic automata display class IV-like behavior, in which oscillatory traveling waves collide with each other, leading to complex behavior; moreover, a time series of binary sequences displays <mml:math id="m1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>f</mml:mi></mml:mrow></mml:math></inline-formula> noise. Therefore, one can see that quantum logic automata generalize and expand the idea of the edge of chaos.</sec>

Background: Historically, although researchers in the science of complex systems proposed the idea of the edge of chaos and/or self-organized criticality as the essential feature of complex organization, they were not able to generalize this concept. Complex organization is regarded at the edge of chaos between the order phase and the chaos phase and a very rare case. Additionally, in cellular automata, the critical property is class IV, which is also rarely found. Therefore, there can be overestimation for natural selection. More recently, developments in cognitive and brain science have led to the free energy principle based on Bayesian inference, while quantum cognition has been established to explain various cognitive phenomena. Since Bayesian inference results in the perspective of a steady state, it can be described in Boolean logic. Considering that quantum logic consists of multiple Boolean logic in terms of lattice theory, the perspective of the free energy principle is the perspective of order, and the perspective of quantum logic might be the perspective of multiple worlds, which is strongly relevant for the edge of chaos.Problem: The next question arises whether the perspective derived from quantum logic can be generalized for the complex behavior consisting of both order and chaos, instead of the edge of chaos or self-organized criticality, to reveal the property of critical behavior such as a power-law distribution.Solution: In this study, we define quantum logic automata, which entail quantum logic (orthomodular lattice) in terms of lattice theory and have the features of a dynamical system. Because quantum logic automata are applied to a binary sequence, one can estimate the behavior of those automata with respect to patterns and a time series. Here, we show that most of a group of quantum logic automata display class IV-like behavior, in which oscillatory traveling waves collide with each other, leading to complex behavior; moreover, a time series of binary sequences displays <math id="m1"><mrow><mn>1</mn><mo>/</mo><mi>f</mi></mrow></math> noise. Therefore, one can see that quantum logic automata generalize and expand the idea of the edge of chaos.

Quantum logic automata generalizing the edge of chaos in complex systems

The global socio-ecological system in the era of the Capitalocene—the world system created by the use of fossil fuels to provide energy for the development of a growth-oriented capitalist logic in all areas of production and consumption—is facing a set of interwoven sub-system crises that come together to make it extremely unlikely that the global system can continue in its present form. The whole system is in a state of crisis—a system state that cannot continue to exist and in which the system must either return to a previous system state—be resilient in the common usage of that word to mean “bouncing back”—or be transformed into a new relatively long-lasting but qualitatively different state. The most evident whole system crisis is, of course, a product of the impending climate transformation contingent upon global warming, but there are related crises of increasing social inequality, demographic structures, healthcare systems, fiscal and public expenditure processes, and urban systems in an urbanized world. These are all interwoven to constitute a polycrisis across the global socio-ecological world system. They are also manifested at all geographical levels and, in particular, at the level of city regions, which, in a predominantly urbanized world, are crucial levels for administration and action. The complex realist frame of reference can be used to inform the development of scenarios for the available alternative system states in the path-dependent possibility space. We have to start from where we are to get to where we want to go. Scenarios are not only descriptions of possible futures but also include a specification of the actions—the drivers—that shape the creation of specific kinds of futures in those available to us. The construction of scenarios should be done through a process of action research, involving a dialog among system scientists, key actors in governance systems, and civil society. The co-production of knowledge as a guide to action is essential.

The global socio-ecological system in the era of the Capitalocene&#x2014;the world system created by the use of fossil fuels to provide energy for the development of a growth-oriented capitalist logic in all areas of production and consumption&#x2014;is facing a set of interwoven sub-system crises that come together to make it extremely unlikely that the global system can continue in its present form. The whole system is in a state of crisis&#x2014;a system state that cannot continue to exist and in which the system must either return to a previous system state&#x2014;be resilient in the common usage of that word to mean &#x201c;bouncing back&#x201d;&#x2014;or be transformed into a new relatively long-lasting but qualitatively different state. The most evident whole system crisis is, of course, a product of the impending climate transformation contingent upon global warming, but there are related crises of increasing social inequality, demographic structures, healthcare systems, fiscal and public expenditure processes, and urban systems in an urbanized world. These are all interwoven to constitute a polycrisis across the global socio-ecological world system. They are also manifested at all geographical levels and, in particular, at the level of city regions, which, in a predominantly urbanized world, are crucial levels for administration and action. The complex realist frame of reference can be used to inform the development of scenarios for the available alternative system states in the path-dependent possibility space. We have to start from where we are to get to where we want to go. Scenarios are not only descriptions of possible futures but also include a specification of the actions&#x2014;the drivers&#x2014;that shape the creation of specific kinds of futures in those available to us. The construction of scenarios should be done through a process of action research, involving a dialog among system scientists, key actors in governance systems, and civil society. The co-production of knowledge as a guide to action is essential.

Scenarios—using the complexity frame of reference to inform the construction of available futures in the possibility space

The process of training an artificial neural network involves iteratively adapting its parameters so as to minimize the error of the network’s prediction, when confronted with a learning task. This iterative change can be naturally interpreted as a trajectory in network space–a time series of networks–and thus the training algorithm (e.g., gradient descent optimization of a suitable loss function) can be interpreted as a dynamical system in graph space. In order to illustrate this interpretation, here we study the dynamical properties of this process by analyzing through this lens the network trajectories of a shallow neural network, and its evolution through learning a simple classification task. We systematically consider different ranges of the learning rate and explore both the dynamical and orbital stability of the resulting network trajectories, finding hints of regular and chaotic behavior depending on the learning rate regime. Our findings are put in contrast to common wisdom on convergence properties of neural networks and dynamical systems theory. This work also contributes to the cross-fertilization of ideas between dynamical systems theory, network theory and machine learning.

The process of training an artificial neural network involves iteratively adapting its parameters so as to minimize the error of the network&#x2019;s prediction, when confronted with a learning task. This iterative change can be naturally interpreted as a trajectory in network space&#x2013;a time series of networks&#x2013;and thus the training algorithm (e.g., gradient descent optimization of a suitable loss function) can be interpreted as a dynamical system in graph space. In order to illustrate this interpretation, here we study the dynamical properties of this process by analyzing through this lens the network trajectories of a shallow neural network, and its evolution through learning a simple classification task. We systematically consider different ranges of the learning rate and explore both the dynamical and orbital stability of the resulting network trajectories, finding hints of regular and chaotic behavior depending on the learning rate regime. Our findings are put in contrast to common wisdom on convergence properties of neural networks and dynamical systems theory. This work also contributes to the cross-fertilization of ideas between dynamical systems theory, network theory and machine learning.

Dynamical stability and chaos in artificial neural network trajectories along training

This paper makes the very simple, perhaps straightforward point that adaptiveness cannot be taken for granted when analysing a complex system. The paradigm of Complex Adaptive Systems (CAS) theory makes it clear that a key feature of complex systems is the ability to adapt to changes in their environment. This is, indeed, relevant to many systems (e.g., living and social systems) since change is embedded in the way in which systems evolve over time. At the same time, adaptiveness is a strong assumption to make, since it prioritises change over stability and it can be a straight jacket, especially when it comes to studying complexity in the context of human social organising. By using a Case Study, this paper highlights the limits of a focus on adaptiveness and pushes for a more “neutral” perspective that allows researchers to appreciate a wider set of mechanisms, norms, and behaviours pertaining to complex social systems.

This paper makes the very simple, perhaps straightforward point that adaptiveness cannot be taken for granted when analysing a complex system. The paradigm of Complex Adaptive Systems (CAS) theory makes it clear that a key feature of complex systems is the ability to adapt to changes in their environment. This is, indeed, relevant to many systems (e.g., living and social systems) since change is embedded in the way in which systems evolve over time. At the same time, adaptiveness is a strong assumption to make, since it prioritises change over stability and it can be a straight jacket, especially when it comes to studying complexity in the context of human social organising. By using a Case Study, this paper highlights the limits of a focus on adaptiveness and pushes for a more “neutral” perspective that allows researchers to appreciate a wider set of mechanisms, norms, and behaviours pertaining to complex social systems.

What about adaptiveness? The case of organisational resilience and cognition

Introduction: Collective human distrust—and its associated mis/disinformation—is one of the most complex phenomena of our time, given that approximately 70% of the global population is now online. Current examples include distrust of medical expertise, climate change science, democratic election outcomes—and even distrust of fact-checked events in the current Israel-Hamas and Ukraine-Russia conflicts.Methods: Here we adopt the perspective of the system being a complex dynamical network, in order to address these questions. We analyze a Facebook network of interconnected in-built communities (Facebook Page communities) totaling roughly 100 million users who, prior to the pandemic, were just focused on distrust of vaccines.Results: Mapping out this dynamical network from 2019 to 2023, we show that it has quickly self-healed in the wake of Facebook’s mitigation campaigns which include shutdowns. This confirms and extends our earlier finding that Facebook’s ramp-ups during COVID-19 were ineffective (e.g., November 2020). We also show that the post-pandemic network has expanded its topics and has developed a dynamic interplay between global and local discourses across local and global geographic scales.Discussion: Hence current interventions that target specific topics and geographical scales will be ineffective. Instead, our findings show that future interventions need to resonate across multiple topics and across multiple geographical scales. Unlike many recent studies, our findings do not rely on third-party black-box tools whose accuracy for rigorous scientific research is unproven, hence raising doubts about such studies’ conclusions–nor is our network built using fleeting hyperlink mentions which have questionable relevance.

Introduction: Collective human distrust—and its associated mis/disinformation—is one of the most complex phenomena of our time, given that approximately 70% of the global population is now online. Current examples include distrust of medical expertise, climate change science, democratic election outcomes—and even distrust of fact-checked events in the current Israel-Hamas and Ukraine-Russia conflicts.Methods: Here we adopt the perspective of the system being a complex dynamical network, in order to address these questions. We analyze a Facebook network of interconnected in-built communities (Facebook Page communities) totaling roughly 100 million users who, prior to the pandemic, were just focused on distrust of vaccines.Results: Mapping out this dynamical network from 2019 to 2023, we show that it has quickly self-healed in the wake of Facebook’s mitigation campaigns which include shutdowns. This confirms and extends our earlier finding that Facebook’s ramp-ups during COVID-19 were ineffective (e.g., November 2020). We also show that the post-pandemic network has expanded its topics and has developed a dynamic interplay between global and local discourses across local and global geographic scales.Discussion: Hence current interventions that target specific topics and geographical scales will be ineffective. Instead, our findings show that future interventions need to resonate across multiple topics and across multiple geographical scales. Unlike many recent studies, our findings do not rely on third-party black-box tools whose accuracy for rigorous scientific research is unproven, hence raising doubts about such studies’ conclusions–nor is our network built using fleeting hyperlink mentions which have questionable relevance.

Complexity of the online distrust ecosystem and its evolution

Defining complexity is hard and far from unique—like defining beauty, intelligence, creativity, and many other such abstract concepts. In contrast, describing concrete complex systems is a sensibly simpler task. We focus here on such an issue from the perspective of entropic functionals, either additive or nonadditive. Indeed, for the systems currently referred to as simple, the statistical mechanics and associated (additive) entropy is that of Boltzmann–Gibbs, formulated 150 years ago. This formalism constitutes a pillar of contemporary theoretical physics and is typically grounded on strong chaos, mixing, ergodicity, and similar hypotheses, which typically emerge for systems with short-range space–time generic correlations. It fails, however, for the so-called complex systems, where generic long-range space–time correlations prevail, typically grounded on weak chaos. Many such nontrivial systems are satisfactorily handled within a generalization of the Boltzmann–Gibbs theory, namely, nonextensive statistical mechanics, introduced in 1988 and grounded on nonadditive entropies. Illustrations are presented in terms of D-dimensional simplexes such as nodes (D = 0), bonds (D = 1), plaquettes (D = 2), polyhedra (D = 3, …), and higher-order ones. A regularly updated bibliography is available at <ext-link ext-link-type="uri" xlink:href="http://tsallis.cat.cbpf.br/biblio.htm" xmlns:xlink="http://www.w3.org/1999/xlink">http://tsallis.cat.cbpf.br/biblio.htm.</ext-link>

Defining complexity is hard and far from unique—like defining beauty, intelligence, creativity, and many other such abstract concepts. In contrast, describing concrete complex systems is a sensibly simpler task. We focus here on such an issue from the perspective of entropic functionals, either additive or nonadditive. Indeed, for the systems currently referred to as simple, the statistical mechanics and associated (additive) entropy is that of Boltzmann–Gibbs, formulated 150 years ago. This formalism constitutes a pillar of contemporary theoretical physics and is typically grounded on strong chaos, mixing, ergodicity, and similar hypotheses, which typically emerge for systems with short-range space–time generic correlations. It fails, however, for the so-called complex systems, where generic long-range space–time correlations prevail, typically grounded on weak chaos. Many such nontrivial systems are satisfactorily handled within a generalization of the Boltzmann–Gibbs theory, namely, nonextensive statistical mechanics, introduced in 1988 and grounded on nonadditive entropies. Illustrations are presented in terms of D-dimensional simplexes such as nodes (D = 0), bonds (D = 1), plaquettes (D = 2), polyhedra (D = 3, …), and higher-order ones. A regularly updated bibliography is available at <a href="http://tsallis.cat.cbpf.br/biblio.htm">http://tsallis.cat.cbpf.br/biblio.htm.</a>

When may a system be referred to as complex?—an entropic perspective

Complexity science and machine learning are two complementary approaches to discovering and encoding regularities in irreducibly high dimensional phenomena. Whereas complexity science represents a coarse-grained paradigm of understanding, machine learning is a fine-grained paradigm of prediction. Both approaches seek to solve the “Wigner-Reversal” or the unreasonable ineffectiveness of mathematics in the adaptive domain where broken symmetries and broken ergodicity dominate. In order to integrate these paradigms I introduce the idea of “Meta-Ockham” which 1) moves minimality from the description of a model for a phenomenon to a description of a process for generating a model and 2) describes low dimensional features–schema–in these models. Reinforcement learning and natural selection are both parsimonious in this revised sense of minimal processes that parameterize arbitrarily high-dimensional inductive models containing latent, low-dimensional, regularities. I describe these models as “super-Humean” and discuss the scientic value of analyzing their latent dimensions as encoding functional schema. 

Complexity science and machine learning are two complementary approaches to discovering and encoding regularities in irreducibly high dimensional phenomena. Whereas complexity science represents a coarse-grained paradigm of understanding, machine learning is a fine-grained paradigm of prediction. Both approaches seek to solve the &#x201c;Wigner-Reversal&#x201d; or the unreasonable ineffectiveness of mathematics in the adaptive domain where broken symmetries and broken ergodicity dominate. In order to integrate these paradigms I introduce the idea of &#x201c;Meta-Ockham&#x201d; which 1) moves minimality from the description of a model for a phenomenon to a description of a process for generating a model and 2) describes low dimensional features&#x2013;schema&#x2013;in these models. Reinforcement learning and natural selection are both parsimonious in this revised sense of minimal processes that parameterize arbitrarily high-dimensional inductive models containing latent, low-dimensional, regularities. I describe these models as &#x201c;super-Humean&#x201d; and discuss the scientic value of analyzing their latent dimensions as encoding functional schema.