Explosive Synchronization in Networks

Updated 26 February 2026

Explosive Synchronization is a discontinuous phase transition in oscillator networks, characterized by an abrupt macroscopic jump in synchrony when a critical coupling strength is exceeded.
Mathematical frameworks like the generalized Kuramoto model reveal that degree–frequency correlations and adaptive, multilayer couplings drive the emergence of bistability and hysteresis.
Empirical and simulation studies in neural, power-grid, and combustion systems demonstrate both controlled and catastrophic transitions, highlighting ES's practical implications.

Explosive synchronization is a genuine first-order (discontinuous) phase transition in networks of coupled oscillators, distinguished by an abrupt macroscopic jump in collective synchrony as a control parameter (typically coupling strength) is varied. Unlike the classical Kuramoto transition, which is continuous, explosive synchronization (ES) exhibits bistability and hysteresis: coherent and incoherent states coexist over a broad parameter interval. The origin of ES has been traced to positive correlations between oscillator natural frequency and structural features such as node degree or local coupling; however, recent research has uncovered broader classes of mechanisms—including adaptive and multilayer coupling, as well as topological and dynamical factors—that robustly induce first-order synchronization transitions.

1. Mathematical Foundations and Dynamical Models

The canonical framework for explosive synchronization is the generalized Kuramoto model, describing a network of $N$ phase oscillators with phases $\theta_i(t)\in[0,2\pi)$ : $\frac{d\theta_i}{dt} = \omega_i + \frac{\lambda_i}{k_i} \sum_{j=1}^N A_{ij} \sin(\theta_j - \theta_i)$ where $\omega_i$ is the intrinsic frequency, $A_{ij}$ encodes the network structure, $k_i = \sum_j A_{ij}$ is the degree, and $\lambda_i$ is the local coupling. Two key variants distinguish regular synchronization (RS) from ES:

RS (classical Kuramoto): $\lambda_i = \lambda$ (constant), $A_{ij}$ describing a homogeneous or complete network.
ES: $\lambda_i = \lambda\,|\omega_i|$ (frequency-coupling proportionality), and/or the imposition of a degree–frequency correlation ( $\omega_i \propto k_i$ ), generally in a heterogeneous (e.g., scale-free or Erdős–Rényi) network.

The global phase coherence is captured by the complex order parameter: $R e^{i\Psi} = \frac{1}{N}\sum_{j=1}^N e^{i\theta_j}, \quad 0 \leq R \leq 1$ with $R\approx 0$ indicating incoherence and $R\approx 1$ perfect synchrony (Choi et al., 2018, Gomez-Gardenes et al., 2011).

2. Microscopic Mechanisms and Network Correlations

Explosive synchronization generically requires a mechanism that suppresses the gradual coalescence of coherent clusters and instead enforces a collective, system-wide transition. The archetypal mechanism is a positive monotonic correlation between oscillator natural frequency and structural centrality:

Degree–frequency correlation: $\omega_i = k_i$ on heterogeneous networks (e.g., scale-free), as in Gómez-Gardeñes et al. (Gomez-Gardenes et al., 2011).
Frequency–weighted coupling: $\lambda_i = \lambda\,|\omega_i|$ (Choi et al., 2018).

These correlations delay the entrainment of high-frequency (or hub) nodes, delaying global order until a critical coupling $\lambda_c$ induces an abrupt, system-wide phase locking. Analytical reductions on star graphs yield closed-form expressions for the critical points and jump amplitudes: $\lambda_c = \frac{K-1}{K+1}, \quad r_c = \frac{K}{K+1}$ where $K$ is the hub degree (Gomez-Gardenes et al., 2011, Vlasov et al., 2014). Simulations confirm these results in scale-free networks, and breaking the correlation (assigning frequencies randomly) restores a continuous transition.

Experimental and numerical studies in FitzHugh–Nagumo relaxation oscillators with degree–frequency correlation reproduce ES with pronounced hysteresis in Barabási–Albert networks, in contrast to Erdős–Rényi graphs where synchronization remains continuous irrespective of frequency dispersion (Chen et al., 2012).

Disorder in frequencies can itself induce ES in mildly heterogeneous networks, with the width of the bistable region controlled by the disorder strength (Skardal et al., 2014).

3. Bifurcation Structure, Hysteresis, and Self-Consistency

The key dynamical hallmark of ES is multistability: for an interval $\lambda_c^{\rm backward} < \lambda < \lambda_c^{\rm forward}$ , both incoherent and coherent solutions coexist and are locally stable. The transition is associated with saddle-node bifurcations in a reduced order parameter space, as described by ensemble order-parameter (EOP) equations from Ott–Antonsen/Watanabe–Strogatz theory (Xu et al., 2014, Vlasov et al., 2014, Varshney et al., 2024).

Forward and backward sweeps in $\lambda$ reveal:

Upon increasing $\lambda$ , $R$ remains low until $\lambda_c^{\rm forward}$ , where it jumps abruptly to $O(1)$ .
Upon decreasing $\lambda$ , desynchronization occurs at a lower $\lambda_c^{\rm backward}$ , closing a wide hysteresis loop.
Analytical determination of critical points and sizes of hysteresis are available in explicit low-dimensional reductions (Vlasov et al., 2014).

In ring topologies with non-monotonic frequency arrangements, similar explosive or hybrid (mixed) transitions can occur, with the weighting exponent $\alpha$ in the coupling kernel controlling the nature and width of the hysteresis (Chen et al., 2017).

4. Generalizations: Weighted, Adaptive, and Multilayer Networks

ES is not restricted to classical frequency–degree or coupling correlations:

Weighted networks: Detuning-based link weighting (e.g., $\Omega_{ij} = |\omega_i-\omega_j|^\alpha$ ) on homogeneous graphs can induce ES, tunable by the exponent $\alpha$ . In heterogeneous graphs, augmenting with edge betweenness as a global topological factor restores ES (Leyva et al., 2013).
Adaptive and multilayer networks: Adaptive rules, where a node’s coupling adapts to local phase coherence, or systems with interlayer dependency links, admit ES without any explicit frequency–coupling correlation. The critical window for phase-locking shrinks quadratically with the global order parameter ( $|\omega| \leq \lambda R^2 k$ ), thus preventing the growth of large coherent clusters until an abrupt collective collapse (Zhang et al., 2014). The critical fraction of adaptively coupled nodes required for ES is $f_c\sim 0.7$ (Zhang et al., 2014).
Multilayer/Multiplex architectures: Cross-layer interactions can induce ES in layers that would otherwise synchronize continuously. Examples include positive–negative (excitatory–inhibitory) multiplexing (Jalan et al., 2019), or coupling between layers with and without intrinsic ES (e.g., second-order/inertial Kuramoto and first-order Kuramoto), extending bistability and ES across both (Jalan et al., 2020, Khanra et al., 2018, Laptyeva et al., 2023, Verma et al., 2022).

5. Experimental Evidence, Physical Mechanisms, and Real-World Relevance

Empirical evidence for explosive synchronization spans turbulent reactive flows (thermoacoustic combustion), multiplexed neural–glial networks, and electronic Rössler circuits:

In a turbulent combustor, the transition to synchrony in local flame oscillators and the global acoustic field is abrupt and exhibits clear hysteresis as the coupling (global feedback) parameter is ramped, confirmed via cross-correlations and order-parameter trajectories (Joseph et al., 2023).
In cortical neuron–glial multiplex models, higher-order (triadic) coupling facilitates ES across coupled layers even when individual layers cannot synchronize independently (Laptyeva et al., 2023).
In neuromorphic computing frameworks, tuning networks to operate at the edge of ES yields orders-of-magnitude gains in supervised learning and stability, exploiting the high-dimensional transients near the critical point (Choi et al., 2018).
Biological implications include the pathological onset of large-scale neural synchrony associated with epilepsy or chronic pain (Laptyeva et al., 2023, Miranda et al., 2023).

Self-organized ES, or "synchronization bombs," can be triggered by the addition or deletion of a single network link in a bistable regime, connecting the phenomenology of ES with explosive percolation and providing decentralized mechanisms for abrupt transitions in engineered and biological systems (Arola-Fernández et al., 2022).

6. Topological and Dynamical Suppression, Control, and Design

While the presence of ES can be catastrophic for technological and biological systems (e.g., power-grid failures or neural seizures), certain network features can suppress explosiveness:

Introduction of cycles (even a single one) into a star-like topology destroys the high spectral degeneracy necessary for bistability, restoring a continuous (second-order) transition (Miranda et al., 2023).
Degree-assortative mixing in networks, or disruption of degree–frequency correlations, narrows or eliminates the ES-hysteresis window (Chen et al., 2012).
Careful tuning of weighting parameters, noise, coupling ranges, and inhibitory interaction strength can modulate or eradicate ES-induced bistability (Verma et al., 2022, Leyva et al., 2013, Jalan et al., 2019).

Analytical design principles and decentralized local rules—such as maximizing synchrony gain per link addition or suppressing pairing among large-frequency-mismatch nodes—provide programmable control of ES onset, hysteresis width, and robustness (Arola-Fernández et al., 2022, Leyva et al., 2013).

7. Broader Implications and Directions

The study of explosive synchronization underpins understanding of abrupt dynamical transitions in complex systems—ranging from brain networks and power grids to adaptive social and ecological networks. Its emergence from generic micro-macro correlations, as well as adaptive, temporal, and multilayer interaction rules, demonstrates that such first-order phenomena are not rare, but rather robust and tunable. This has prompted ongoing research into early-warning indicators (lag-time distributions, local order-parameter variance (Leyva et al., 2020)), design of resilient networks, and novel computing paradigms exploiting criticality (Choi et al., 2018, Mbonwouo et al., 10 Sep 2025).

Comprehensive theoretical analyses, supported by experiments, confirm the wide generality of ES across oscillator types (Kuramoto, Stuart–Landau, FitzHugh–Nagumo, Rössler, van der Pol), network architectures, and control protocols, providing a unified framework for predicting, controlling, and exploiting catastrophic dynamical transitions in real-world systems.

Key References:

(Choi et al., 2018) Critical Neuromorphic Computing based on Explosive Synchronization
(Gomez-Gardenes et al., 2011) Explosive Synchronization Transitions in Scale-free Networks
(Vlasov et al., 2014) Explosive Synchronization is Discontinuous
(Chen et al., 2012) Explosive synchronization transitions in complex neural network
(Xu et al., 2014) Explosive or Continuous: Incoherent state determines the route to synchronization
(Leyva et al., 2013) Explosive synchronization in weighted complex networks
(Zhang et al., 2014) Explosive synchronization in adaptive and multilayer networks
(Skardal et al., 2014) Disorder induces explosive synchronization
(Jalan et al., 2019) Inhibition induced explosive synchronization in multiplex networks
(Laptyeva et al., 2023) Explosive synchronization in multiplex neuron-glial networks
(Mbonwouo et al., 10 Sep 2025) Energy-dynamics interplay in temporal networks triggers explosive synchronization
(Miranda et al., 2023) Topologically-induced suppression of explosive synchronization
(Chen et al., 2017, Varshney et al., 2024, Arola-Fernández et al., 2022, Jalan et al., 2020, Khanra et al., 2018, Verma et al., 2022, Joseph et al., 2023, Leyva et al., 2020)