Special Session 31: Data-Driven Modeling and Control of Complex Systems

Dynamical Phase Transitions in Nonequilibrium Networks

Jiazhen Liu Fudan University Peoples Rep of China Co-Author(s):

Abstract: Dynamical phase transitions (DPTs) characterize critical changes in system behavior occurring at finite times, providing a lens to study nonequilibrium phenomena beyond conventional equilibrium physics. While extensively studied in quantum systems, DPTs have remained largely unexplored in classical settings. Recent experiments on complex systems, from social networks to financial markets, have revealed abrupt dynamical changes analogous to quantum DPTs, motivating the search for a theoretical understanding. Here, we present a minimal model for nonequilibrium networks, demonstrating that nonlinear interactions among network edges naturally give rise to DPTs. Specifically, we show that network degree diverges at a finite critical time, following a universal hyperbolic scaling, consistent with empirical observations. Our analytical results predict that key network properties, including degree distributions and clustering coefficients, exhibit critical scaling as criticality approaches. These findings establish a theoretical foundation for understanding emergent nonequilibrium criticality across diverse complex systems.

High-dimensional Density Estimation

Lirong Qu School of Mathematical Sciences, Beijing Normal University Peoples Rep of China Co-Author(s):

Abstract: With the aim of inferring probability density functions from limited observed samples, density estimation via discrepancy based sequential partition (DSP) has been proposed to learn an adaptive piecewise constant approximation defined on a binary sequential partition of the underlying domain, where the star discrepancy is adopted to measure the uniformity of particle distribution. However, the calculation of the star discrepancy is NP-hard. We use heuristic algorithms such as SA, TA, and PT to accelerate the computation of the star discrepancy, and find that TA exhibits the best convergence speed. Furthermore, star discrepancy does not satisfy the reflection invariance and rotation invariance either. To this end, we use the mixture discrepancy and the comparison of moments as a replacement of the star discrepancy, leading to the density estimation via mixture discrepancy based sequential partition (DSP-mix) and density estimation via moments based sequential partition (MSP), respectively. Both DSP-mix and MSP are computationally tractable and exhibit the reflection and rotation invariance. Numerical experiments in reconstructing the $d$-D mixture of Gaussians and Betas with $d = 2-30$ demonstrate that both DSP-mix and MSP run approximately ten times faster than DSP while maintaining the same accuracy.

Modeling environmental feedback in opinion dynamics: Pattern formation and phase transitions

Rui Wang School of Mathematical Sciences, Beijing Normal University Peoples Rep of China Co-Author(s):

Abstract: The public opinions in modern information environments are shaped by the interplay between social conformity and environmental exposure. Interactions among individuals promote opinion alignment, while environmental heterogeneity and selective information exposure may reinforce the alignments with their own interests, known as the information cocoon. This work aims at a mathematical description of the underlying mechanisms by combining the opinion dynamics and bounded-confidence interactions with environmentally biased movement. Starting from the particle-level Hegselmann-Krause model and random movements of agents biased by the environment, a nonlocal PDE system under the mean-field scaling is derived, where opinion transport is driven by consensus dynamics and modulated by an attention-dependent cross-diffusion mechanism. The stability of spatially homogeneous states is analyzed, and the threshold conditions that characterize the phase transition between dispersed and aggregated opinion distributions are proposed. The theoretical results are supported by our numerical results produced by a structure-preserving third-order numerical scheme. Moreover, it demonstrates that the formation of spatial opinion clusters enhances the group consensus, and the environmental information flow may modify the timing and the pathway through which consensus is triggered. These observations provide a quantitative framework for understanding how social conformity and environmental exposure jointly shape the organization of public opinion.

Predicting first-order phase transitions in coupled oscillator systems with machine learning

Zhonghua Zhang Northwestern polytechnical university Peoples Rep of China Co-Author(s):    Liang Wang, Stefano Boccaletti, Ludovico Minati, Wei Xu

Abstract: First-order phase transitions such as explosive death can cause abrupt and irreversible suppression of collective oscillations, making their critical prediction important for preventing dynamical collapse in complex systems. In this talk, I will present two recent machine-learning approaches for predicting such transitions from oscillatory data alone. For low-dimensional coupled oscillators, we adopt a parameter-aware next-generation reservoir computing framework that embeds control parameters into nonlinear feature construction and enables accurate identification of critical points, hysteresis, and bifurcation structures in unseen regimes. For oscillator networks, we further introduce a parameter-aware parallel reservoir computing framework, where node-wise reservoirs preserve scalability while retaining sensitivity to parameter variations. The method accurately predicts explosive amplitude death and oscillation death in networks with different topologies, including nearest-neighbor ring, small-world, and random regular networks, while also reconstructing the corresponding spatiotemporal dynamics. Together, these results provide an efficient and scalable data-driven paradigm for forecasting first-order phase transitions in nonlinear dynamical systems and complex networks.