Table of contents

Stationarity of the constituents of the body and of its functionalities is a basic requirement for life, being equivalent to survival in first place. Assuming that the resting state activity of the brain serves essential functionalities, stationarity entails that the dynamics of the brain needs to be regulated on a time-averaged basis. The combination of recurrent and driving external inputs must therefore lead to a non-trivial stationary neural activity, a condition which is fulfiled for afferent signals of varying strengths only close to criticality. In this view, the benefits of working in the vicinity of a second-order phase transition, such as signal enhancements, are not the underlying evolutionary drivers, but side effects of the requirement to keep the brain functional in first place. It is hence more appropriate to use the term 'self-regulated' in this context, instead of 'self-organized'.

The increasing availability of data sources and analysis tools borrowed from computer science and physical science have sharply changed traditional methodologies of social sciences, leading to a new branch named computational socioeconomics, which studies various phenomena in socioeconomic development by using quantitative methods based on large-scale real-world data. Sited on recent publications, this perspective will introduce three representative methods: (i) natural data analyses, (ii) large-scale online experiments, and (iii) integration of big data and surveys. This perspective ends up with in-depth discussion on the limitations and challenges of the above-mentioned emerging methods.

A homeostatic mechanism that keeps the brain highly susceptible to stimuli and optimizes many of its functions—although this is a compelling theoretical argument in favor of the brain criticality hypothesis, the experimental evidence accumulated during the last two decades is still not entirely convincing, causing the idea to be seemingly unknown in the more clinically-oriented neuroscience community. In this perspective review, we will briefly review the theoretical framework underlying such bold hypothesis, and point to where theory and experiments agree and disagree, highlighting potential ways to try and bridge the gap between them. Finally, we will discuss how the stand point of statistical physics could yield practical applications in neuroscience and help with the interpretation of what is a healthy or unhealthy brain, regardless of being able to validate the critical brain hypothesis.

Duopolies are one of the simplest economic situations where interactions between firms determine market behavior. The standard model of a price-setting duopoly is the Bertrand model, which has the unique solution that both firms set their prices equal to their costs—a paradoxical result where both firms obtain zero profit, which is generally not observed in real market duopolies. Here we propose a new game theory model for a price-setting duopoly, which we show resolves the paradoxical behavior of the Bertrand model and provides a consistent general model for duopolies.

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.

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.

In dissipative systems without any driving or positive feedback all motion stops ultimately since the initial kinetic energy is dissipated away during time evolution. If chaos is present, it can only be of transient type. Traditional transient chaos is, however, supported by an infinity of unstable orbits. In the lack of these, chaos in undriven dissipative systems is of another type: it is termed doubly transient chaos as the strength of transient chaos is diminishing in time, and ceases asymptotically. Here we show that a clear view of such dynamics is provided by identifying KAM tori or chaotic regions of the dissipation-free case, and following their time evolution in the dissipative dynamics. The tori often smoothly deform first, but later they become disintegrated and dissolve in a kind of shrinking chaos. We identify different dynamical measures for the characterization of this process which illustrate that the strength of chaos is first diminishing, and after a while disappears, the motion enters the phase of ultimate stopping.

A chaotic saddle is a common nonattracting chaotic set well known for generating finite-time chaotic behavior in low and high-dimensional systems. In general, dynamical systems possessing chaotic saddles in their state-space exhibit irregular behavior with duration lengths following an exponential distribution. However, when these systems are coupled into networks the chaotic saddle plays a role in the long-term dynamics by trapping network trajectories for times that are indefinitely long. This process transforms the network's high-dimensional state-space by creating an alternative persistent desynchronized state coexisting with the completely synchronized one. Such coexistence threatens the synchronized state with vulnerability to external perturbations. We demonstrate the onset of this phenomenon in complex networks of discrete-time units in which the synchronization manifold is perturbed either in the initial instant of time or in arbitrary states of its asymptotic dynamics. The role of topological asymmetries of Erdös–Rényi and Barabási–Albert graphs are investigated. Besides, the required coupling strength for the occurrence of trapping in the chaotic saddle is unveiled.

Fluctuations and damages crucially determine the operation and stability of networked systems across disciplines, from electrical powergrids, to vascular networks or neuronal networks. Local changes in the underlying dynamics may affect the whole network and, in the worst case, cause a total collapse of the system through a cascading failure. It has been demonstrated that certain subgraphs can reduce failure spreading drastically, or even inhibit it completely. However, this shielding effect is poorly understood for non-linear dynamical models. Here, we study the effect of perturbations in networks of oscillators coupled via the Kuramoto model. We demonstrate how the network structure can be optimised for suppressing specific, targeted fluctuations at a desired operational state while letting others pass. We illustrate our approach by demonstrating that a significant reduction in time-dependent fluctuations may be achieved by optimising the edge weights. Finally, we demonstrate how to apply the developed method to real-world supply networks such as power grids. Our findings reveal that a targeted shielding of specific solutions in multistable systems is possible which may be applied to make supply networks more robust.

In various application areas, networked data is collected by measuring interactions involving some specific set of core nodes. This results in a network dataset containing the core nodes along with a potentially much larger set of fringe nodes that all have at least one interaction with a core node. In many settings, this type of data arises for structures that are richer than graphs, because they involve the interactions of larger sets; for example, the core nodes might be a set of individuals under surveillance, where we observe the attendees of meetings involving at least one of the core individuals. We model such scenarios using hypergraphs, and we study the problem of core recovery: if we observe the hypergraph but not the labels of core and fringe nodes, can we recover the 'planted' set of core nodes in the hypergraph? We provide a theoretical framework for analyzing the recovery of such a set of core nodes and use our theory to develop a practical and scalable algorithm for core recovery. The crux of our analysis and algorithm is that the core nodes are a hitting set of the hypergraph, meaning that every hyperedge has at least one node in the set of core nodes. We demonstrate the efficacy of our algorithm on a number of real-world datasets, outperforming competitive baselines derived from network centrality and core-periphery measures.

Mutation is an unavoidable and indispensable phenomenon in both biological and social systems undergoing evolution through replication-selection processes. Here we show that mutation in a generation-wise nonoverlapping population with two-player-two-strategy symmetric game gives rise to coexisting stable population states, one of which can even be chaotic; the chaotic state prevents the cooperators in the population from going extinct. Specifically, we use replicator maps with additive and multiplicative mutations, and rigorously find all possible two dimensional payoff matrices for which physically allowed solutions can be achieved in the equations. Subsequently, we discover the various possibilities of bistable outcomes—e.g., coexistences of fixed point and periodic orbit, periodic orbit and chaos, and chaos and fixed point—in the resulting replicator-mutator maps.

We use methods from computational algebraic topology to study functional brain networks in which nodes represent brain regions and weighted edges encode the similarity of functional magnetic resonance imaging (fMRI) time series from each region. With these tools, which allow one to characterize topological invariants such as loops in high-dimensional data, we are able to gain understanding of low-dimensional structures in networks in a way that complements traditional approaches that are based on pairwise interactions. In the present paper, we use persistent homology to analyze networks that we construct from task-based fMRI data from schizophrenia patients, healthy controls, and healthy siblings of schizophrenia patients. We thereby explore the persistence of topological structures such as loops at different scales in these networks. We use persistence landscapes and persistence images to represent the output of our persistent-homology calculations, and we study the persistence landscapes and persistence images using k-means clustering and community detection. Based on our analysis of persistence landscapes, we find that the members of the sibling cohort have topological features (specifically, their one-dimensional loops) that are distinct from the other two cohorts. From the persistence images, we are able to distinguish all three subject groups and to determine the brain regions in the loops (with four or more edges) that allow us to make these distinctions.

The 1 + 1 dimensional Kuramoto–sine-Gordon system consists of a set of N nonlinear coupled equations for N scalar fields θ_i, which constitute the nodes of a complex system. These scalar fields interact by means of Kuramoto nonlinearities over a network of connections determined by N(N − 1)/2 symmetric coupling coefficients a_ij. This system, regarded as a chirally invariant quantum field theory, describes a single decoupled massless field together with N − 1 scalar boson excitations of nonzero mass depending on a_ij, which propagate and interact over the network. For N = 2 the equations decouple into separate sine-Gordon and wave equations. The system allows an extensive array of soliton configurations which interpolate between the various minima of the 2π-periodic potential, including sine-Gordon solitons in both static and time-dependent form, as well as double sine-Gordon solitons which can be imbedded into the system for any N. The precise form of the stable soliton depends critically on the coupling coefficients a_ij. We investigate specific configurations for N = 3 by classifying all possible potentials, and use the symmetries of the system to construct static solitons in both exact and numerical form.

We investigate the ground states of spin models defined on networks that we imprint (e.g., non-complex random networks like Erdos–Renyi, or complex networks like Watts–Strogatz, and Barabasi–Albert), and their response to decohering processes which we model with network attacks. We quantify the complexity of these ground states, and their response to the attacks, by calculating distributions of network measures of an emergent network whose link weights are the pairwise mutual information between spins. We focus on attacks which projectively measure spins. We find that the emergent networks in the ground state do not satisfy the usual criteria for complexity, and their average properties are captured well by a single dimensionless parameter in the Hamiltonian. While the response of classical networks to attacks is well-studied, where classical complex networks are known to be more robust to random attacks than random networks, we find counter-intuitive results for our quantum networks. We find that the ground states for Hamiltonians defined on different classes of imprinted networks respond similarly to all our attacks, and the attacks rescale the average properties of the emergent network by a constant factor. Mean field theory explains these results for relatively dense networks, but we also find the simple rescaling behavior away from the regime of validity of mean field theory. Our calculations indicate that complex spin networks are not more robust to projective measurement attacks, and presumably also other quantum attacks, than non-complex spin networks, in contrast to the classical case. Understanding the response of the spin networks to decoherence and attacks will have applications in understanding the physics of open quantum systems, and in designing robust complex quantum systems—possibly even a robust quantum internet in the long run—that is maximally resistant to decoherence.

Network representations often cannot fully account for the structural richness of complex systems spanning multiple levels of organisation. Recently proposed high-order information-theoretic signals are well-suited to capture synergistic phenomena that transcend pairwise interactions; however, the exponential-growth of their cardinality severely hinders their applicability. In this work, we combine methods from harmonic analysis and combinatorial topology to construct efficient representations of high-order information-theoretic signals. The core of our method is the diagonalisation of a discrete version of the Laplace–de Rham operator, that geometrically encodes structural properties of the system. We capitalise on these ideas by developing a complete workflow for the construction of hyperharmonic representations of high-order signals, which is applicable to a wide range of scenarios.

We analyze maximum entropy random graph ensembles with constrained degrees, drawn from arbitrary degree distributions, and a tuneable number of three-cycles (triangles). We find that such ensembles generally exhibit two transitions, a clustering and a shattering transition, separating three distinct regimes. At the clustering transition, the graphs change from typically having only isolated cycles to forming cycle clusters. At the shattering transition the graphs break up into many small cliques to achieve the desired three-cycle density. The locations of both transitions depend nontrivially on the system size. We derive a general formula for the three-cycle density in the regime of isolated cycles, for graphs with degree distributions that have finite first and second moments. For bounded degree distributions we present further analytical results on cycle densities and phase transition locations, which, while non-rigorous, are all validated via MCMC sampling simulations. We show that the shattering transition is of an entropic nature, occurring for all three-cycle density values, provided the system is large enough.

Complex network theory has shown success in understanding the emergent and collective behavior of complex systems Newman 2010 Networks: An Introduction (Oxford: Oxford University Press). Many real-world complex systems were recently discovered to be more accurately modeled as multiplex networks Bianconi 2018 Multilayer Networks: Structure and Function (Oxford: Oxford University Press); Boccaletti et al 2014 Phys. Rep.544 1–122; Lee et al 2015 Eur. Phys. J. B88 48; Kivelä et al 2014 J. Complex Netw.2 203–71; De Domenico et al 2013 Phys. Rev. X3 041022—in which each interaction type is mapped to its own network layer; e.g. multi-layer transportation networks, coupled social networks, metabolic and regulatory networks, etc. A salient physical phenomena emerging from multiplexity is super-diffusion: exhibited by an accelerated diffusion admitted by the multi-layer structure as compared to any single layer. Theoretically super-diffusion was only known to be predicted using the spectral gap of the full Laplacian of a multiplex network and its interacting layers. Here we turn to machine learning (ML) which has developed techniques to recognize, classify, and characterize complex sets of data. We show that modern ML architectures, such as fully connected and convolutional neural networks (CNN), can classify and predict the presence of super-diffusion in multiplex networks with 94.12% accuracy. Such predictions can be done in situ, without the need to determine spectral properties of a network.

The complexity underlying real-world systems implies that standard statistical hypothesis testing methods may not be adequate for these peculiar applications. Specifically, we show that the likelihood-ratio (LR) test's null-distribution needs to be modified to accommodate the complexity found in multi-edge network data. When working with independent observations, the p-values of LR tests are approximated using a χ² distribution. However, such an approximation should not be used when dealing with multi-edge network data. This type of data is characterized by multiple correlations and competitions that make the standard approximation unsuitable. We provide a solution to the problem by providing a better approximation of the LR test null-distribution through a beta distribution. Finally, we empirically show that even for a small multi-edge network, the standard χ² approximation provides erroneous results, while the proposed beta approximation yields the correct p-value estimation.

Doubly transient chaos was recently characterized as the general form of chaos in undriven dissipative systems. Here we study this type of complex behavior in the advective dynamics of decaying incompressible open flows. Using a decaying version of the blinking vortex-sink map as a prototype, we show that the resulting dynamics is markedly distinct from the one of mechanical systems addressed in previous works. In particular, the asymptotic codimension of the set of initial conditions of non-escaping particles is zero rather than one and the time-dependent escape rates either undergo an exponential decay rather than growth (for moderate and fast energy dissipation) or display a complex, possibly nonmonotonic behavior (for slow energy dissipation).

An emerging paradigm for predicting the state evolution of chaotic systems is machine learning with reservoir computing, the core of which is a dynamical network of artificial neurons. Through training with measured time series, a reservoir machine can be harnessed to replicate the evolution of the target chaotic system for some amount of time, typically about half dozen Lyapunov times. Recently, we developed a reservoir computing framework with an additional parameter channel for predicting system collapse and chaotic transients associated with crisis. It was found that the crisis point after which transient chaos emerges can be accurately predicted. The idea of adding a parameter channel to reservoir computing has also been used by others to predict bifurcation points and distinct asymptotic behaviors. In this paper, we address three issues associated with machine-generated transient chaos. First, we report the results from a detailed study of the statistical behaviors of transient chaos generated by our parameter-aware reservoir computing machine. When multiple time series from a small number of distinct values of the bifurcation parameter, all in the regime of attracting chaos, are deployed to train the reservoir machine, it can generate the correct dynamical behavior in the regime of transient chaos of the target system in the sense that the basic statistical features of the machine generated transient chaos agree with those of the real system. Second, we demonstrate that our machine learning framework can reproduce intermittency of the target system. Third, we consider a system for which the known methods of sparse optimization fail to predict crisis and demonstrate that our reservoir computing scheme can solve this problem. These findings have potential applications in anticipating system collapse as induced by, e.g., a parameter drift that places the system in a transient regime.

Multistability is a common phenomenon which naturally occurs in complex networks. If coexisting attractors are numerous and their basins of attraction are complexly interwoven, the long-term response to a perturbation can be highly uncertain. We examine the uncertainty in the outcome of perturbations to the synchronous state in a Kuramoto-like representation of the British power grid. Based on local basin landscapes which correspond to single-node perturbations, we demonstrate that the uncertainty shows strong spatial variability. While perturbations at many nodes only allow for a few outcomes, other local landscapes show extreme complexity with more than a hundred basins. Particularly complex domains in the latter can be related to unstable invariant chaotic sets of saddle type. Most importantly, we show that the characteristic dynamics on these chaotic saddles can be associated with certain topological structures of the network. We find that one particular tree-like substructure allows for the chaotic response to perturbations at nodes in the north of Great Britain. The interplay with other peripheral motifs increases the uncertainty in the system response even further.

The spatiotemporal dynamics of excitable media may exhibit chaotic transients. We investigate this transient chaos in the 2D Fenton–Karma model describing the propagation of electrical excitation waves in cardiac tissue and compute the average duration of chaotic transients in dependence on model parameter values. Furthermore, other characteristics like the dominant frequency, the size of the excitable gap, pseudo ECGs, the number of phase singularities and parameters characterizing the action potential duration restitution curve are determined and it is shown that these quantities can be used to predict the average transient time using polynomial regression.

COVID-19 is not a universal killer. We study the spread of COVID-19 at the county level for the United States up until the 15th of August, 2020. We show that the prevalence of the disease and the death rate are correlated with the local socio-economic conditions often going beyond local population density distributions, especially in rural areas. We correlate the COVID-19 prevalence and death rate with data from the US Census Bureau and point out how the spreading patterns of the disease show asymmetries in urban and rural areas separately and are preferentially affecting the counties where a large fraction of the population is non-white. Our findings can be used for more targeted policy building and deployment of resources for future occurrence of a pandemic due to SARS-CoV-2. Our methodology, based on interpretable machine learning and game theory, can be extended to study the spread of other diseases.

Discovering and quantifying the drivers of language change is a major challenge. Hypotheses about causal factors proliferate, but are difficult to rigorously test. Here we ask a simple question: can 20th century changes in English be explained as a consequence of spatial diffusion, or have other processes created bias in favour of certain linguistic forms? Using two of the most comprehensive spatial datasets available, which measure the state of English at the beginning and end of the 20th century, we calibrate a simple spatial model so that, initialised with the early state, it evolves into the later. Our calibrations reveal that while some changes can be explained by diffusion alone, others are clearly the result of substantial asymmetries between variants. We discuss the origins of these asymmetries and, as a by-product, we generate a full spatio–temporal prediction for the spatial evolution of English features over the 20th century, and a prediction of the future.

Complex networks represent the natural backbone to study epidemic processes in populations of interacting individuals. Such a modeling framework, however, is naturally limited to pairwise interactions, making it less suitable to properly describe social contagion, where individuals acquire new norms or ideas after simultaneous exposure to multiple sources of infections. Simplicial contagion has been proposed as an alternative framework where simplices are used to encode group interactions of any order. The presence of these higher-order interactions leads to explosive epidemic transitions and bistability. In particular, critical mass effects can emerge even for infectivity values below the standard pairwise epidemic threshold, where the size of the initial seed of infectious nodes determines whether the system would eventually fall in the endemic or the healthy state. Here we extend simplicial contagion to time-varying networks, where pairwise and higher-order simplices can be created or destroyed over time. By following a microscopic Markov chain approach, we find that the same seed of infectious nodes might or might not lead to an endemic stationary state, depending on the temporal properties of the underlying network structure, and show that persistent temporal interactions anticipate the onset of the endemic state in finite-size systems. We characterize this behavior on higher-order networks with a prescribed temporal correlation between consecutive interactions and on heterogeneous simplicial complexes, showing that temporality again limits the effect of higher-order spreading, but in a less pronounced way than for homogeneous structures. Our work suggests the importance of incorporating temporality, a realistic feature of many real-world systems, into the investigation of dynamical processes beyond pairwise interactions.

A dynamical variable driven by the combination of a deterministic multiplicative process with stochastic reset events develops, at long times, a stationary power-law distribution. Here, we analyze how such distribution changes when several variables of the same kind interact with each other through diffusion-like coupling. While for weak coupling the variables are still distributed following power-law functions, their distributions are severely distorted as interactions become stronger, with sudden appearance of cutoffs and divergent singularities. We explore these effects both analytically and numerically, for coupled ensembles of identical and non-identical variables. The most relevant consequences of ensemble heterogeneity are assessed, and preliminary results for spatially distributed ensembles are presented.

Understanding and predicting uncertain things are the central themes of scientific evolution. Human beings revolve around these fears of uncertainties concerning various aspects like a global pandemic, health, finances, to name but a few. Dealing with this unavoidable part of life is far tougher due to the chaotic nature of these unpredictable activities. In the present article, we consider a global network of identical chaotic maps, which splits into two different clusters, despite the interaction between all nodes are uniform. The stability analysis of the spatially homogeneous chaotic solutions provides a critical coupling strength, before which we anticipate such partial synchronization. The distance between these two chaotic synchronized populations often deviates more than eight times of standard deviation from its long-term average. The probability density function of these highly deviated values fits well with the generalized extreme value distribution. Meanwhile, the distribution of recurrence time intervals between extreme events resembles the Weibull distribution. The existing literature helps us to characterize such events as extreme events using the significant height. These extremely high fluctuations are less frequent in terms of their occurrence. We determine numerically a range of coupling strength for these extremely large but recurrent events. On-off intermittency is the responsible mechanism underlying the formation of such extreme events. Besides understanding the generation of such extreme events and their statistical signature, we furnish forecasting these events using the powerful deep learning algorithms of an artificial recurrent neural network. This long short-term memory (LSTM) can offer handy one-step forecasting of these chaotic intermittent bursts. We also ensure the robustness of this forecasting model with two hundred hidden cells in each LSTM layer.

We define the topological Dirac equation describing the evolution of a topological wave function on networks or on simplicial complexes. On networks, the topological wave function describes the dynamics of topological signals or cochains, i.e. dynamical signals defined both on nodes and on links. On simplicial complexes the wave function is also defined on higher-dimensional simplices. Therefore the topological wave function satisfies a relaxed condition of locality as it acquires the same value along simplices of dimension larger than zero. The topological Dirac equation defines eigenstates whose dispersion relation is determined by the spectral properties of the Dirac operator defined on networks and generalized network structures including simplicial complexes and multiplex networks. On simplicial complexes the Dirac equation leads to multiple energy bands. On multiplex networks the topological Dirac equation can be generalized to distinguish between different mutlilinks leading to a natural definition of rotations of the topological spinor. The topological Dirac equation is here initially formulated on a spatial network or simplicial complex for describing the evolution of the topological wave function in continuous time. This framework is also extended to treat the topological Dirac equation on 1 + d lattices describing a discrete space-time with one temporal dimension and d spatial dimensions with d ∈ {1, 2, 3}. It is found that in this framework space-like and time-like links are only distinguished by the choice of the directional Dirac operator and are otherwise structurally indistinguishable. This work includes also the discussion of numerical results obtained by implementing the topological Dirac equation on simplicial complex models and on real simple and multiplex network data.

Criticality has been conjectured as an integral part of neuronal network dynamics. Operating at a critical threshold requires precise parameter tuning and a corresponding mechanism remains an open question. Recent studies have suggested that topological features observed in brain networks give rise to a Griffiths phase, leading to power-law scaling in brain activity dynamics and the operational benefits of criticality in an extended parameter region. Motivated by growing evidence of neural correlates of different states of consciousness, we investigate how topological changes affect the expression of a Griffiths phase. We analyze the activity decay in modular networks using a susceptible-infected-susceptible propagation model and find that we can control the extension of the Griffiths phase by altering intra- and intermodular connectivity. We find that by adjusting system parameters, we can counteract changes in critical behavior and maintain a stable critical region despite changes in network topology. Our results give insight into how structural network properties affect the emergence of a Griffiths phase and how its features are linked to established topological network metrics. We discuss how those findings could contribute to an understanding of the changes in functional brain networks.

We analyze the citation time-series of manuscripts in three different fields of science; physics, social science and technology. The evolution of the time-series of the yearly number of citations, namely the citation trajectories, diffuse anomalously, their variance scales with time ∝t^2H, where H ≠ 1/2. We provide detailed analysis of the various factors that lead to the anomalous behavior: non-stationarity, long-ranged correlations and a fat-tailed increment distribution. The papers exhibit a high degree of heterogeneity across the various fields, as the statistics of the highest cited papers is fundamentally different from that of the lower ones. The citation data is shown to be highly correlated and non-stationary; as all the papers except the small percentage of them with high number of citations, die out in time.

We propose and demonstrate a nonlinear control method that can be applied to unknown, complex systems where the controller is based on a type of artificial neural network known as a reservoir computer. In contrast to many modern neural-network-based control techniques, which are robust to system uncertainties but require a model nonetheless, our technique requires no prior knowledge of the system and is thus model-free. Further, our approach does not require an initial system identification step, resulting in a relatively simple and efficient learning process. Reservoir computers are well-suited to the control problem because they require small training data sets and remarkably low training times. By iteratively training and adding layers of reservoir computers to the controller, a precise and efficient control law is identified quickly. With examples on both numerical and high-speed experimental systems, we demonstrate that our approach is capable of controlling highly complex dynamical systems that display deterministic chaos to nontrivial target trajectories.

Characterizing the efficiency of movements is important for a better management of the cities. More specifically, the connection between the efficiency and uncertainty (entropy) production of a transport process is not established yet. In this study, we consider the movements of selfish drivers from their homes (origins) to work places (destinations) to see how interactions and randomness in the movements affect a measure of efficiency and entropy production (uncertainty in the destination time intervals) in this process. We employ realistic models of population distributions and mobility laws to simulate the movement process, where interactions are modelled by dependence of the local travel times on the local flows. We observe that some level of information (the travel times) sharing enhances a measure of predictability in the process without any coordination. Moreover, the larger cities display smaller efficiencies, for the same model parameters and population density, which limits the size of an efficient city. We find that entropy production is a good order parameter to distinguish the low- and high-congestion phases. In the former phase, the entropy production grows monotonically with the probability of random moves, whereas it displays a minimum in the congested phase; that is randomness in the movements can reduce the uncertainty in the destination time intervals. The findings highlight the role of entropy production in the study of efficiency and predictability of similar processes in a complex system like the city.