[Paper Review] Adaptive Dynamical Networks

Rico Berner, Thilo Groß+3 more|arXiv (Cornell University)|Apr 12, 2023

Neural Networks and Applications5 citations

TL;DR

This review presents adaptive dynamical networks as a unifying framework for modeling systems where network structure and dynamics co-evolve, enabling the study of complex phenomena like synchronization, plasticity, and resilience. It synthesizes mathematical methods, applications across neuroscience, epidemiology, and climate science, and identifies open challenges in multiscale modeling and mean-field approximations.

ABSTRACT

It is a fundamental challenge to understand how the function of a network is related to its structural organization. Adaptive dynamical networks represent a broad class of systems that can change their connectivity over time depending on their dynamical state. The most important feature of such systems is that their function depends on their structure and vice versa. While the properties of static networks have been extensively investigated in the past, the study of adaptive networks is much more challenging. Moreover, adaptive dynamical networks are of tremendous importance for various application fields, in particular, for the models for neuronal synaptic plasticity, adaptive networks in chemical, epidemic, biological, transport, and social systems, to name a few. In this review, we provide a detailed description of adaptive dynamical networks, show their applications in various areas of research, highlight their dynamical features and describe the arising dynamical phenomena, and give an overview of the available mathematical methods developed for understanding adaptive dynamical networks.

Motivation & Objective

• To establish adaptive dynamical networks as a fundamental modeling paradigm for complex systems where structure and function co-evolve.
• To systematize the classification of adaptation mechanisms, including event-based and continuous rules, and their time-scale separation.
• To unify diverse applications—from neuronal plasticity and opinion formation to power grids and epidemic models—under a common dynamical framework.
• To identify and analyze key dynamical phenomena such as explosive synchronization, multistability, and chimera states in adaptive networks.
• To advance mathematical tools, including mean-field theories and multiscale decomposition, for analyzing large-scale adaptive systems.

Proposed method

• Classifying adaptive networks by adaptation type: event-based (e.g., spike-timing-dependent plasticity) versus continuous adaptation rules.
• Applying slow-fast time-scale decomposition to derive conditions for complex dynamical states in small networks.
• Using mean-field approaches, including Vlasov-Fokker-Planck and moment equations, to describe collective behavior in large ensembles.
• Employing continuum limit and bifurcation analysis to study pattern formation and stability in adaptive systems.
• Integrating network control theory with adaptive dynamics, including edge snapping and speed-gradient methods for synchronization.
• Extending models to multilayer, hypernetworks, and simplicial complexes to capture higher-order interactions and structural complexity.

Experimental results

Research questions

• RQ1 How do event-based adaptation rules in coupled oscillators approximate continuous dynamics, and what are the conditions for such approximations to hold?
• RQ2 What dynamical phenomena—such as explosive synchronization or chimera states—emerge in adaptive networks with co-evolving topology and dynamics?
• RQ3 How can mean-field theories be generalized to describe the collective behavior of large adaptive dynamical networks?
• RQ4 What are the implications of time-scale separation in adaptive networks for the emergence of chaos or recurrent synchronization?
• RQ5 How can adaptive network models be extended to multilayer or higher-order structures to capture complex interdependencies in real-world systems?

Key findings

• Event-based adaptation rules, such as those in spike-timing-dependent plasticity, can be effectively approximated by continuous dynamics, enabling analytical tractability.
• Adaptive networks exhibit rich dynamical phenomena including explosive synchronization with hysteresis, solitary states, and multistability, often absent in static networks.
• Multiscale decomposition reveals conditions under which complex states like cluster synchronization emerge in adaptive networks with distinct time scales.
• Mean-field approaches, though recently developed for adaptive networks, remain limited and represent a key frontier for future analysis.
• Adaptive dynamical networks can model real-world systems such as neuronal circuits, epidemic spread, and power grids, capturing feedbacks between function and structure.
• The interplay between network adaptivity and inertia in phase oscillator models leads to non-trivial dynamical consequences, including altered synchronization thresholds.