Environmental, social and economic challenges antagonize sustainable development. Understanding the emergent collective dynamics of the underlying complex interconnected systems is key to predicting processes and preventing failures. This requires integrating data-driven methods, computational modeling and conceptual theoretical progress as well as specific insights from a variety of different fields. Here we review recent advances in analyzing collective dynamics of infrastructure systems, focusing on electric energy supply and sustainable mobility. Key phenomena include flow disruptions, supply fluctuations and non-equilibrium states emerging through distributed demand-highlighting the need for cross-disciplinary tools to guide sustainable system design.

Swarmalators constitute a paradigmatic model for understanding the collective dynamics of coupled moving agents, integrating both internal and spatial degrees of freedom. Empirical evidence from systems such as bird flocks and living matter highlights the relevance of topological, metric-free coupling, but their impact on swarmalator dynamics remains largely unknown to date. Here, we present and analyze a topological swarmalator model in which the units interact topologically, on Delaunay networks. We find intriguing self-organized collective dynamics, including patterns with local vortices and unprecedented spatiotemporal patterns absent in metric-based models. Identifying three order parameters to quantify synchrony, spatial order, and vortex formation, we map the phase diagram that classifies these diverse patterns. Notably, we uncover a first-order transition even if the phases of all units are frozen, a dynamics inverted relative to the classical Kuramoto model. These insights not only advance our theoretical understanding of locally coupled systems of moving agents, but also offer key guidelines for their control.

Diffusion processes and random walks describe the dynamics of systems subject to many unpredictable external influences. Here, we show how trajectories of individual ridepooling vehicles are characterized by a novel type of persistent random walk with a preplanned future route. We find three distinct timescales that reflect short- and long-term alignment and medium-term detours emerging in the routing dynamics, together inducing ballistic, subdiffusive, and diffusive motions of the vehicles. The diffusion constant links the timescales underlying the microscopic dynamics of the routes to macroscopic system parameters such as fleet size and effective demand. Observables quantifying the service quality of ridepooling collapse to a single scaling function of the diffusion constant, directly connecting microscopic timescales to macroscopic properties of system operations.

The emergence of synchrony essentially underlies the functionality of many systems across physics, biology and engineering. In all established synchronization phase transitions so far, a stable synchronous state is connected to a stable incoherent state: For continuous transitions, stable synchrony directly connects to stable incoherence at a critical point, whereas for discontinuous transitions, stable synchrony is connected to stable incoherence via an additional unstable branch. Here we present a novel type of transition between synchrony and incoherence where the synchronous state does not connect to the state of incoherence. We uncover such transitions in the complexified Kuramoto model with their variables and coupling strength parameter analytically continued. Deriving a self-consistency equation for a quaternion order parameter that we propose helps to mathematically pin down the mechanisms underlying this transition type. Local numerical analysis suggests that the transition is linked to a Hopf bifurcation destabilizing synchrony, in contrast to branching point bifurcations established for the transition between synchrony and incoherence so far.

Across natural and human-made systems, transition points mark sudden changes of order and are thus key to understanding overarching system features. Motivated by recent experimental observations, we here uncover an intriguing class of transitions in coupled oscillators, extreme synchronization transitions, from asynchronous disordered states to synchronous states with almost completely ordered phases. Whereas such a transition appears like discontinuous or explosive phase transitions, it exhibits markedly distinct features. First, the transition occurs already in finite systems of N units and so constitutes an intriguing bifurcation of multi-dimensional systems rather than a genuine phase transition that emerges in the thermodynamic limit N → ∞ only. Second, the synchronization order parameter jumps from moderate values of the order of N-1/2 to values extremely close to 1, its theoretical maximum, immediately upon crossing a critical coupling strength. We analytically explain the mechanisms underlying such extreme transitions in coupled complexified Kuramoto oscillators. Extreme transitions may similarly occur across other systems of coupled oscillators as well as in certain percolation processes. In applications, their occurrence impacts our ability of ensuring or preventing strong forms of ordering, for instance in biological and engineered systems.

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.

The COVID-19 pandemic and the resulting economic recession negatively affected many people's physical, social, and psychological health and has been shown to change population-level mobility, but little attention has been given to park visitations as an indicator. Estimating the frequency of park visitations from aggregated mobility data of all the parks in Washington State (USA), we study trends in park use one year prior to and two years during the COVID-19 pandemic. Our findings indicate that the gravity model is a robust model for the park visitation behavior in different spatial resolutions of city level and state level and different socio-economical classes. Incorporating network structure, our detailed analysis highlights that high-income level residents changed their recreational behavior by visiting their local parks more and a broader recreational options outside of their local census area; whereas the low-income residents changed their visitation behavior by reducing their recreational choices.

How nonlinear systems dynamically respond to external perturbations essentially determines their function. Weak perturbations induce response dynamics near a stable operating point, often approximately characterized by linear response theory. However, stronger driving signals may induce genuinely nonlinear responses, including tipping transitions to qualitatively different dynamical states. Here, we analyze how inter-unit coupling impacts responses to periodic perturbations. We find that already in minimal systems of two identical and linearly coupled units, coupling impacts the dynamical responses in a distinct way. Any non-zero coupling strength extends the regime of non-tipping local responses relative to uncoupled units. Intriguingly, finite coupling may be more effective than infinitely strong coupling in keeping responses from tipping. Interestingly, already weak coupling may create novel response modes in strongly driven systems, implying the existence of multiple tipping points instead of only one. These results persist for systems of non-identical units, systems with nonlinear coupling, and larger networks of coupled units.

We present the finite-size Kuramoto model analytically continued from real to complex variables and analyze its collective dynamics. For strong coupling, synchrony appears through locked states that constitute attractors, as for the real-variable system. However, synchrony persists in the form of complex locked states for coupling strengths K below the transition K^{(pl)} to classical phase locking. Stable complex locked states indicate a locked subpopulation of zero mean frequency in the real-variable model and their imaginary parts help identifying which units comprise that subpopulation. We uncover a second transition at K^{'}<K^{(pl)} below which complex locked states become linearly unstable yet still exist for arbitrarily small coupling strengths.

The Global South, encompassing more than 80% of the world population, heavily relies on informal paratransit services with ad-hoc routes. Yet, it remains unclear how efficiently such informal public transport services organize and operate. Here, we analyze and compare the structural efficiency of more than 7000 formal and informal bus service routes in 36 cities across 22 countries globally. Intriguingly, informal transport self-organizes in ways at or above efficiency levels of centralized services. They exhibit fewer detours, more uniform paths, and comparable interconnectivities, all while remaining profitable without the major subsidies common in the Global North. These insights challenge the global perception of informal transport as an inferior alternative to centrally organized services. More generally, analyzing large-scale microscopic transport data and condensing them into informative macroscopic observables may qualitatively improve system understanding and reveal specific options to create more accessible, efficient, and sustainable public transport solutions. Informal transportation services constitute the primary form of public transport in the Global South. Here, the authors analyze the structure of route networks in cities across the globe, showing how informal routes self-organize into consistent line services that often outperform centralized services in the Global North, exhibiting fewer detours and comparable interconnectivity.

Ridepooling combines trips of multiple passengers in the same vehicle and may thereby provide a more sustainable option than transport by private cars. The efficiency and sustainability of ridepooling is typically quantified by key performance indicators such as the average vehicle occupancy or the total distance driven by all ridepooling vehicles relative to individual transport. However, even if the average occupancy is high and rides are shared, ridepooling services may increase the total distance driven due to additional detours and deadheading. Moreover, these key performance indicators are difficult to predict without large-scale simulations or actual ridepooling operation. Here, we propose a dimensionless parameter to estimate the sustainability of ridepooling by quantifying the load on a ridepooling service, relating characteristic timescales of demand and supply. The load bounds the relative distance driven and uniquely marks the break-even point above which the total distance driven by all vehicles of a ridepooling service falls below that of motorized individual transport. Detailed event-based simulations and a comparison with empirical observations from a ridepooling pilot project in a rural area of Germany validate the theoretical prediction. Importantly, the load follows directly from a small set of aggregate parameters of the service setting and is thus predictable a priori. The load may thus complement standard key performance indicators and simplify planning, operation and evaluation of ridepooling services.

Spiking neural network models characterize the emergent collective dynamics of circuits of biological neurons and help engineer neuro-inspired solutions across fields. Most dynamical systems’ models of spiking neural networks typically exhibit one of two major types of interactions: First, the response of a neuron’s state variable to incoming pulse signals (spikes) may be additive and independent of its current state. Second, the response may depend on the current neuron’s state and multiply a function of the state variable. Here we reveal that deterministic spiking neural network models with additive coupling are equivalent to models with multiplicative coupling for simultaneously modified intrinsic neuron time evolution. As a consequence, the same collective dynamics can be attained by state-dependent multiplicative and constant (state-independent) additive coupling. Such a mapping enables the transfer of theoretical results between spiking neural network models with different types of interaction mechanisms and at the same time extends the option space for hardware implementation or modeling. By allowing to choose the coupling type or neuron type that is the simplest one to implement in a given practical situation where a specific dynamic or functionality is required, it potentially allows simpler or more effective engineering applications.

Complex and networked dynamical systems characterize the time evolution of most of the natural and human-made world. The dimension of their state space, i.e., the number of (active) variables in such systems, arguably constitutes their most fundamental property yet is hard to access in general. Recent work [Haehne et al., Phys. Rev. Lett. 122, 158301 (2019)] introduced a method of inferring the state space dimension of a multi-dimensional networked system from repeatedly measuring time series of only some fraction of observed variables, while all other variables are hidden. Here, we show how time series observations of one single variable are mathematically sufficient for dimension inference. We reveal how successful inference in practice depends on numerical constraints of data evaluation and on experimental choices, in particular the sampling intervals and the total duration of observations. We illustrate robust inference for systems of up to N=10 to N=100 variables by evaluating time series observations of a single variable. We discuss how the faithfulness of the inference depends on the quality and quantity of collected data and formulate some general rules of thumb on how to approach the measurement of a given system.

Networked dynamical systems, i.e., systems of dynamical units coupled via nontrivial interaction topologies, constitute models of broad classes of complex systems, ranging from gene regulatory and metabolic circuits in our cells to pandemics spreading across continents. Most of such systems are driven by irregular and distributed fluctuating input signals from the environment. Yet how networked dynamical systems collectively respond to such fluctuations depends on the location and type of driving signal, the interaction topology and several other factors and remains largely unknown to date. As a key example, modern electric power grids are undergoing a rapid and systematic transformation towards more sustainable systems, signified by high penetrations of renewable energy sources. These in turn introduce significant fluctuations in power input and thereby pose immediate challenges to the stable operation of power grid systems. How power grid systems dynamically respond to fluctuating power feed-in as well as other temporal changes is critical for ensuring a reliable operation of power grids yet not well understood. In this work, we systematically introduce a linear response theory (LRT) for fluctuation-driven networked dynamical systems. The derivations presented not only provide approximate analytical descriptions of the dynamical responses of networks, but more importantly, also allow to extract key qualitative features about spatio-temporally distributed response patterns. Specifically, we provide a general formulation of a LRT for perturbed networked dynamical systems, explicate how dynamic network response patterns arise from the solution of the linearised response dynamics, and emphasise the role of LRT in predicting and comprehending power grid responses on different temporal and spatial scales and to various types of disturbances. Understanding such patterns from a general, mathematical perspective enables to estimate network responses quickly and intuitively, and to develop guiding principles for, e.g., power grid operation, control and design.

The computation of rank ordering plays a fundamental role in cognitive tasks and offers a basic building block for computing arbitrary digital functions. Spiking neural networks have been demonstrated to be capable of identifying the largest k out of N analog input signals through their collective nonlinear dynamics. By finding partial rank orderings, they perform k-winners-take-all computations. Yet, for any given study so far, the value of k is fixed, often to k equal one. Here we present a concept for spiking neural networks that are capable of (re)configurable computation by choosing k via one global system parameter. The spiking network acts via pulse-suppression induced by inhibitory pulse-couplings. Couplings are proportional to each units' state variable (neuron voltage), constituting an uncommon but straightforward type of leaky integrate-and-fire neural network. The result of a computation is encoded as a stable periodic orbit with k units spiking at some frequency and others at lower frequency or not at all. Orbit stability makes the resulting analog-to-digital computation robust to sufficiently small variations of both, parameters and signals. Moreover, the computation is completed quickly within a few spike emissions per neuron. These results indicate how reconfigurable k -winners-take-all computations may be implemented and effectively exploited in simple hardware relying only on basic dynamical units and spike interactions resembling simple current leakages to a common ground.

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The adoption of battery electric vehicles (BEVs) may significantly reduce greenhouse gas emissions caused by road transport. However, there is wide disagreement as to how soon battery electric vehicles will play a major role in overall transportation. Focusing on battery electric passenger cars, we analyze BEV adoption across 17 individual countries, Europe, and the World, and consistently find exponential growth trends. Modeling-based estimates of future adoption given past trends suggest system-wide adoption substantially faster than typical economic analyses have proposed so far. For instance, we estimate the majority of passenger cars in Europe to be electric by about 2031. Within regions, the predicted times of mass adoption are largely insensitive to model details. Despite significant differences in current electric fleet sizes across regions, their growth rates consistently indicate fast doubling times of approximately 15 months, hinting at radical economic and infrastructural consequences in the near future.

How spiking neuronal networks encode memories in their different time and spatial scales constitute a fundamental topic in neuroscience and neuro-inspired engineering. Much attention has been paid to large networks and long-term memory, for example in models of associative memory. Smaller circuit motifs may play an important complementary role on shorter time scales, where broader network effects may be of less relevance. Yet, compact computational models of spiking neural networks that exhibit short-term volatile memory and actively hold information until their energy source is switched off, seem not fully understood. Here we propose that small spiking neural circuit motifs may act as volatile memory components. A minimal motif consists of only two interconnected neurons – one self-connected excitatory neuron and one inhibitory neuron – and realizes a single-bit volatile memory. An excitatory, delayed self-connection promotes a bistable circuit in which a self-sustained periodic orbit generating spike trains co-exists with the quiescent state of no neuron spiking. Transient external inputs may straightforwardly induce switching between those states. Moreover, the inhibitory neuron may act as an autonomous turn-off switch. It integrates incoming excitatory pulses until a threshold is reached after which the inhibitory neuron emits a spike that then inhibits further spikes in the excitatory neuron, terminating the memory. Our results show how external bits of information (excitatory signal), can be actively held in memory for a pre-defined amount of time. We show that such memory operations are robust against parameter variations and exemplify how sequences of multidimensional input signals may control the dynamics of a many-bits memory circuit in a desired way.

Secure electric energy supply and thus stable operation of power grids fundamentally relies on their capability to cope with fluctuations. Here, we study how active voltage dynamics impacts the collective response dynamics of networked power grids. We find that the systems driven by ongoing fluctuating inputs exhibit a bulk, a resonance, and a localized grid frequency response regime, as for static voltages. However, active voltage dynamics generically weakens the degree of localization in the grid, thereby intensifying and spatially extending the high-frequency responses. An analytic approximation scheme that takes into account shortest signal propagation paths among the voltage, phase angle and frequency variables result in an asymptotic lowest-order expansion that helps understanding the boosted high-frequency responses. These results moreover offer a generic tool to systematically investigate fluctuation response patterns in power grid models with and without active voltage dynamics.

Standard epidemic models exhibit one continuous, second order phase transition to macroscopic outbreaks. However, interventions to control outbreaks may fundamentally alter epidemic dynamics. Here we reveal how such interventions modify the type of phase transition. In particular, we uncover three distinct types of explosive phase transitions for epidemic dynamics with capacity-limited interventions. Depending on the capacity limit, interventions may (i) leave the standard second order phase transition unchanged but exponentially suppress the probability of large outbreaks, (ii) induce a first-order discontinuous transition to macroscopic outbreaks, or (iii) cause a secondary explosive yet continuous third-order transition. These insights highlight inherent limitations in predicting and containing epidemic outbreaks. More generally our study offers a cornerstone example of a third order explosive phase transition in complex systems.