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A Universal Route to Explosive Phenomena
Christian Kuehn, Christian Bick
TL;DR
Critical transitions in complex systems can be continuous or abrupt, and added model features have been associated with shifts to first-order transitions. The paper develops a mathematical argument for why such shifts are generically expected under two-parameter variation and illustrates it in epidemic, synchronization, and percolation models.
Problem
Whether shifts from continuous to abrupt critical transitions share a common dynamical origin is a question raised by recurring examples across complex systems.
Source: Introduction · pp. 1–2Method
The paper uses a bifurcation-theory argument for generic two-parameter variations of classical models and illustrates it with three physical examples.
Source: Introduction · p. 2, Normal forms for critical transitions · p. 3, Change of criticality for a generic… · p. 5Results
A generic two-parameter variation of a classical model with a second-order transition must lead to a change in criticality to first-order, illustrated in epidemic, synchronization, and percolation models.
Source: Introduction · p. 2, Transitions in adaptive epidemics · pp. 6–7, Synchronization in phase oscillator… · p. 8Takeaways & Limitations
The framework may help identify perturbations that induce or prevent abrupt critical transitions in physical systems.
Source: Introduction · p. 2The analysis assumes the transition corresponds to an equilibrium point of the macroscopic equations, although the full-system transition state need not be stationary.
Source: Normal forms for critical transitions · p. 2
Abstract
from arXiv · showhide
Critical transitions are observed in many complex systems. This includes the onset of synchronization in a network of coupled oscillators or the emergence an epidemic state within a population. "Explosive" first-order transitions have caught particular attention in a variety of systems when classical models are generalized by incorporating additional effects. Here we give a mathematical argument that the emergence of such first-order transitions is not surprising but rather a universally expected effect: Varying a classical model along a generic two-parameter family must lead to a change of the criticality. To illustrate our framework, we give three explicit examples of the effect in distinct physical systems: a model of adaptive epidemic dynamics, for a generalization of the Kuramoto model, and for a percolation transition.
Introduction
Complex systems can undergo continuous or discontinuous critical transitions, and added model features have repeatedly been associated with shifts from the former to the latter. The paper argues that such shifts are generically expected when a classical model is varied along a two-parameter family.
Introduction
- In prior examples, adding an effect and varying its associated parameter changes a previously second-order transition into a first-order one.Examples include synchronization and epidemic models with added features.Source: Introduction · p. 2
- A generic two-parameter variation of a classical model with a second-order transition must lead to a change in criticality to first-order.The authors present this as a common dynamical origin for abrupt transitions across different complex systems.Source: Introduction · p. 2
- The paper illustrates the framework with adaptive epidemic dynamics, Kuramoto synchronization with non-additive higher-order interactions, and percolation.The proposed insight may help identify perturbations that induce or prevent abrupt transitions.Source: Introduction · p. 2
Results
A generic second parameter can change a continuous transition into a discontinuous one by changing the sign of the leading nonlinear term. The paper connects this route to adaptive epidemics, higher-order oscillator interactions, and Potts-model percolation.
Normal forms for critical transitions
- Near a critical transition, a high-dimensional system can be reduced to one-dimensional center-manifold dynamics, where equilibrium branching corresponds to Landau phase transitions.Source: Normal forms for critical transitions · pp. 2–3
Change of criticality for a generic variation of an additional parameter
- A generic second parameter can change transition criticality by changing the sign of the leading nonlinear coefficient; symmetries or nongeneric parameter dependence can prevent that change.The sign change generically switches between second- and first-order transitions.Source: Change of criticality for a generic… · pp. 4–5
Discontinuous critical transitions from a universal perspective
- The framework relates three seemingly distinct examples—adaptive epidemics, synchronization, and percolation—to a common dynamical route for changing transition criticality.Source: Discontinuous critical transitions… · p. 6
Transitions in adaptive epidemics
- For the epidemic model, re-wiring changes the transition from second-order to first-order at q = 21/19.The re-wiring parameter appears in the reduced center-manifold dynamics as a generic second unfolding parameter.Source: Transitions in adaptive epidemics · p. 7
Synchronization in phase oscillator networks
- In the generalized Kuramoto model, the transition at K2 = 2 is second-order for K3 = 0 and becomes first-order for K3 > 2.Source: Synchronization in phase oscillator… · pp. 8–9
Discontinuous percolation transitions
- For the q-state Potts model, the percolation transition of fixed point P*=0 of H occurs at critical bond density p-hat=1/2; varying p, it is continuous for 1≤q≤2 and discontinuous for q>2.At q=2, the change corresponds to a switch from supercritical to subcritical transcritical bifurcation.Source: Discontinuous percolation transitions · p. 10
Discussion
The framework links abrupt critical transitions in percolation, synchronization, and epidemic spreading through a common dynamical mechanism. The authors expect it to apply to additional systems, including neural networks and chemical reaction networks.
Discussion
- The formalism attributes abrupt critical transitions to a shared underlying dynamical mechanism.The authors present three physical examples to illustrate this route.Source: Discussion · p. 10
- The framework explicitly links percolation, synchronization, and epidemic spreading through a common dynamical framework.Source: Discussion · p. 11
- The authors expect the framework to extend to neural networks, laser dynamics, flocking, and chemical reaction networks.They suggest it may help explain discontinuous firing transitions in neural networks.Source: Discussion · p. 11
Methods
The results are based on theoretical considerations.
Methods
- The results are based on theoretical considerations.Source: Methods · p. 11