Malte Schröder

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.

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.

The Kuramoto model and its generalizations have been broadly employed to characterize and mechanistically understand various collective dynamical phenomena, especially the emergence of synchrony among coupled oscillators. Despite almost five decades of research, many questions remain open, in particular, for finite-size systems. Here, we generalize recent work [Thümler et al., Phys. Rev. Lett. 130, 187201 (2023)] on the finite-size Kuramoto model with its state variables analytically continued to the complex domain and also complexify its system parameters. Intriguingly, systems of two units with purely imaginary coupling do not actively synchronize even for arbitrarily large magnitudes of the coupling strengths, |K|→∞, but exhibit conservative dynamics with asynchronous rotations or librations for all |K|. For generic complex coupling, both traditional phase-locked states and asynchronous states generalize to complex locked states, fixed points off the real subspace that exist even for arbitrarily weak coupling. We analyze a new collective mode of rotations exhibiting finite, yet arbitrarily large rotation numbers. Numerical simulations for large networks indicate a novel form of discontinuous phase transition. We close by pointing to a range of exciting questions for future research.

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.

Understanding the structure and formation of networks is a central topic in complexity science. Economic networks are formed by decisions of individual agents and thus not properly described by established random graph models. In this article, we establish a model for the emergence of trade networks that is based on rational decisions of individual agents. The model incorporates key drivers for the emergence of trade, comparative advantage and economic scale effects, but also the heterogeneity of agents and the transportation or transaction costs. Numerical simulations show three macroscopically different regimes of the emerging trade networks. Depending on the specific transportation costs and the heterogeneity of individual preferences, we find centralized production with a star-like trade network, distributed production with all-to-all trading or local production and no trade. Using methods from statistical mechanics, we provide an analytic theory of the transitions between these regimes and estimates for critical parameters values.

Understanding the structure and formation of networks is a central topic in complexity science. Economic networks are formed by decisions of individual agents and thus not properly described by established random graph models. In this article, we establish a model for the emergence of trade networks that is based on rational decisions of individual agents. The model incorporates key drivers for the emergence of trade, comparative advantage and economic scale effects, but also the heterogeneity of agents and the transportation or transaction costs. Numerical simulations show three macroscopically different regimes of the emerging trade networks. Depending on the specific transportation costs and the heterogeneity of individual preferences, we find centralized production with a star-like trade network, distributed production with all-to-all trading or local production and no trade. Using methods from statistical mechanics, we provide an analytic theory of the transitions between these regimes and estimates for critical parameters values.

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.

Ride-pooling (or ride-sharing) services combine trips of multiple customers along similar routes into a single vehicle. The collective dynamics of the fleet of ride-pooling vehicles fundamentally underlies the efficiency of these services. In simplified models, the common features of these dynamics give rise to scaling laws of the efficiency that are valid across a wide range of street networks and demand settings. However, it is unclear how constraints of the vehicle fleet impact such scaling laws. Here, we map the collective dynamics of capacity-constrained ride-pooling fleets to services with unlimited passenger capacity and identify an effective fleet size of available vehicles as the relevant scaling parameter characterizing the dynamics. Exploiting this mapping, we generalize the scaling laws of ride-pooling efficiency to capacity-constrained fleets. We approximate the scaling function with a queueing theoretical analysis of the dynamics in a minimal model system, thereby enabling mean-field predictions of required fleet sizes in more complex settings. These results may help to transfer insights from existing ride-pooling services to new settings or service locations.