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Explosive Phenomena in Complex Networks

Raissa M. D'Souza, Jesus Gómez-Gardeñes, Jan Nagler, Alex Arenas

arXiv:1907.09957v1physics.soc-phcond-mat.dis-nnnlin.AO

TL;DR

Large-scale connectivity and synchronization shape complex-network function and failure, but abrupt transitions and their mechanisms remain incompletely understood. This review organizes explosive phenomena by underlying mechanisms, connects models, surveys applications, and identifies directions for future research.

  • Problem

    Abrupt transitions in complex networks and the mechanisms underlying explosive phenomena remain not well understood despite extensive research.

  • Method

    The review organizes the literature into thematic categories, synthesizes connections between models, surveys application areas, and classifies explosive phenomena by their underlying mechanisms.

  • Results

    The literature reveals many mechanisms and models producing explosive percolation and explosive synchronization, along with exotic behaviors and universality classes.

  • Takeaways & Limitations

    Explosive phenomena provide opportunities to understand abrupt network transitions and model failure modes, interventions, and consequences in technological and economic systems.

Abstract

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The emergence of large-scale connectivity and synchronization are crucial to the structure, function and failure of many complex socio-technical networks. Thus, there is great interest in analyzing phase transitions to large-scale connectivity and to global synchronization, including how to enhance or delay the onset. These phenomena are traditionally studied as second-order phase transitions where, at the critical threshold, the order parameter increases rapidly but continuously. In 2009, an extremely abrupt transition was found for a network growth process where links compete for addition in attempt to delay percolation. This observation of "explosive percolation" was ultimately revealed to be a continuous transition in the thermodynamic limit, yet with very atypical finite-size scaling, and it started a surge of work on explosive phenomena and their consequences. Many related models are now shown to yield discontinuous percolation transitions and even hybrid transitions. Explosive percolation enables many other features such as multiple giant components, modular structures, discrete scale invariance and non-self-averaging, relating to properties found in many real phenomena such as explosive epidemics, electric breakdowns and the emergence of molecular life. Models of explosive synchronization provide an analytic framework for the dynamics of abrupt transitions and reveal the interplay between the distribution in natural frequencies and the network structure, with applications ranging from epileptic seizures to waking from anesthesia. Here we review the vast literature on explosive phenomena and synthesize the fundamental connections between models and survey the application areas. We attempt to classify explosive phenomena based on underlying mechanisms and to provide a coherent overview and perspective for future research to address the many vital questions that remained unanswered.

I. INTRODUCTION

Explosive phenomena challenge the traditional view that connectivity and synchronization emerge through smooth second-order transitions. This review organizes explosive percolation and synchronization around microscopic dynamics that delay macroscopic component formation and connects these mechanisms to real systems.

  • The review synthesizes mechanisms including cluster-size control, overtaking, suppression of the largest component, and correlated processes, while identifying open challenges.
  • Percolation and synchronization traditionally exhibit sharp but smooth second-order transitions, whereas explosive phenomena produce abrupt changes in connectivity or global coherence.
  • Large-scale connectivity supports transportation and communication but can also enable epidemics, making its onset context-dependent.
  • Explosive percolation was discovered in 2009 through small interventions that delayed large-scale connectivity and prompted extensive study of alternative models and mechanisms.
  • Explosive synchronization similarly links abrupt global coherence to network structure and natural-frequency organization, with reported connections to seizures and other real systems.
  • Competitive processes such as the Product Rule select between candidate edges to delay the growth of large components, producing atypical connectivity evolution relative to conventional coalescence.In ordinary random graph growth, multiplicative coalescence makes larger clusters more likely to merge and drives gelation.

1. Achlioptas Process (AP)

Achlioptas Processes add edges through competition among candidate choices, delaying percolation and producing transitions whose apparent abruptness depends strongly on model and system size. Rigorous and numerical studies show that fixed-choice APs on random graphs are continuous in the thermodynamic limit, often with unusual finite-size scaling and distinct universality classes.

  • AP mechanism: An Achlioptas Process selects among candidate edges according to a criterion, such as minimizing or maximizing the resulting component sizes, rather than adding one uniformly random edge.The Product Rule retains the edge minimizing the product of the sizes of the components it would join.
  • AP mechanism: The Product Rule delays percolation, but its initially abrupt-looking transition was later shown to be continuous in the thermodynamic limit.Early simulations observed a narrow scaling window and a critical point near tc ≈ 0.888, while later proofs established continuity for fixed-choice APs on random graphs.
  • Finite-size behavior: For m-edge APs, the largest single-edge growth gap decays as ∆Cmax ∼ N^-β, with β = 0.065 for the Product Rule, so finite systems can still display large discrete jumps.The vanishing gap distinguishes the thermodynamic-limit transition from the pronounced abruptness observed in networks of practical size.
  • Transition mechanisms: The transition character depends on the growth mechanism: allowing direct growth of the largest component can yield continuity, whereas PGr = 0 necessarily produces discontinuity during the process.When growth occurs only by overtaking, merging smaller components can create a macroscopic component abruptly; powder-keg states provide a related mechanism for discontinuity.
  • Critical behavior: Numerical and rate-equation studies find continuous AP transitions with distinct universality classes and novel critical exponents across competitive percolation models.For the dCDGM model, the order parameter follows S ∼ (t − tc)^β, with β = 0.0555(1) ≈ 1/18.
  • Finite-size behavior: Finite-size order-parameter distributions can become increasingly double-peaked even though the distance between peaks shrinks to zero as N →∞.This coexistence of sharper finite-system peaks with vanishing asymptotic separation is one source of APs’ unusual scaling behavior.

3. AP with m-edge rules on lattices

Achlioptas processes on lattices and related competitive-growth models can delay or sharply alter percolation by restricting candidate edges or suppressing large-component growth. These mechanisms produce discontinuous, continuous-with-atypical-scaling, hybrid, and multi-giant-component behaviors, while their thermodynamic interpretation remains partly unresolved.

  • AP with m-edge rules on lattices: Fixed-m candidate edges can produce truly discontinuous explosive percolation transitions on lattices.For the SCA model, discontinuity occurs when d < 6 and m ≥ mc, with mc = d/(d − dBB).
  • AP with m-edge rules on lattices: For m = mc, the discontinuous transition occurs at an intermediate process time; for m > mc, spanning can encompass the entire system.The m →∞ random-graph limit likewise produces global percolation at the final step.
  • AP with k-vertex rules on random graphs: Random-vertex rules select k vertices and compare all k(k − 1)/2 possible edges, enabling criteria that keep component sizes similar.This selection can prohibit direct growth of the largest component, unlike some m-edge rules.
  • AP with k-vertex rules on random graphs: k-vertex rules necessarily begin with a continuous connectivity transition but may exhibit secondary discontinuous jumps, including a first jump arbitrarily close to it.These stochastic jumps generate “Devil’s staircase” behavior.
  • Thermodynamic formulations of EP: A key unresolved issue is reconciling thermodynamic results requiring traditional subcritical-to-supercritical scaling with evidence for different explosive-percolation scalings.The review identifies this reconciliation as an open question.
  • Thermodynamic formulations of EP: Thermodynamic formulations connect some discontinuous percolation transitions to first-order link-density transitions and hysteresis, although their relation to explosive-percolation mechanisms is not direct.A separate rate-equation approach models the evolving finite-cluster size distribution P(s, t).
  • Suppressing the largest component: Several mechanisms suppress large-component growth, including uniform cluster sizes, overtaking, and correlated processes, but models can differ in transition class and scaling.Four competitive models remain continuous asymptotically while displaying distinct universality behavior; other models show discontinuity or coexistence peaks.
  • Tuning edge rejection rate (The BFW model): The BFW model can yield two or more stable giant components, with α controlling their number and β controlling allowable direct growth of the largest component.For β < 1, overtaking dominates and the transition is discontinuous; for β > 1, unavoidable direct growth leads to continuity.

3. Cluster aggregation

Cluster-aggregation models describe explosive percolation through merger kinetics, with transition type controlled by aggregation rules and cluster-size homogeneity. Generalized models reveal continuous, discontinuous, ultra-slow, and non-self-averaging behaviors, while the classification remains limited and some theories have restricted topology.

  • Cluster-aggregation framework: The Smoluchowski rate equation models cluster aggregation through a collision kernel K_ij proportional to the merger probability of clusters of sizes i and j.Solutions remain unknown for most kernels, limiting analytical insight.
  • Cluster-aggregation framework: For power-law kernels K_ij ∼(ij)^ω, the transition is continuous for ω > 0.5 and discontinuous for ω < 0.5.For ω < 0.5, gelation does not occur in finite time; above the threshold, mass is conserved and finite-time gelation is guaranteed.
  • Generalized aggregation: Using different exponents for mergers involving the largest cluster creates multiple coalescence time scales and can produce continuous, discontinuous, ultra-slow, and non-self-averaging transitions.Controlling the largest cluster is intended to delay percolation.
  • Transition classification: A mean-field two-species theory identifies homogeneity of the cluster kinetic rule with respect to cluster sizes as the key determinant of transition type.The theory examines the cluster-size distribution immediately before the transition and derives necessary conditions for genuinely discontinuous transitions.
  • Transition classification: The theory is expected to be non-exact for lattices, correlated or clustered structures, and networks with non-random locally non-loop-free topologies.Its broader applicability beyond mean-field settings therefore remains constrained.
  • Classification: Explosive-percolation transitions are classified into six behavioral classes, including type I anomalous continuous transitions with atypical finite-size scaling.Type I transitions have a significant finite-system order-parameter jump, largest-gap scaling ∆Cmax ∼N^-β with β ≪1, and finite critical slope.
  • Additional structures: Budget mechanisms can stabilize multiple giant components, with the asymptotic edge-rejection rate controlling their number and connecting explosive percolation to modular organization.This feature follows from breaking the multiplicative coalescence of classic percolation.

3. Type III: Discontinuity at the end of the process

Type III transitions place the discontinuous jump at the end of the growth process, often because percolation is delayed until full link density. Related models also exhibit hybrid critical behavior, slow convergence, and persistent realization-to-realization fluctuations.

  • Type III definition: In type III transitions, the giant component emerges only at the end of the process, corresponding to full link density p_c = 1 in the stated setting.In the kinetic picture, gelation does not occur before the process ends.
  • Type III examples: Examples include one-dimensional percolation, infinite-choice m-edge rules, power-law aggregation with α < 0.5, and SCA below d_c with sufficiently large m > m_c.These models show an order-parameter jump at the end of growth.
  • Slow convergence: For multi-time-scale aggregation, the transition point converges ultra-slowly while the order-parameter slope increases with system size N.The reported slope and gap scaling are illustrated for α = 0, β = 1 and N = 10^3, 10^5, and 10^7.
  • Hybrid transitions: Hybrid transitions combine a genuine discontinuity with second-order critical divergence and require a diverging order-parameter slope at p_c, hence β < 1.An exponent β ≥1 or finite supercritical slope does not constitute a hybrid transition.
  • Staircases: Models with stochastic staircases can show realization-dependent jumps, with the order parameter failing to converge to a function of p in the thermodynamic limit.For DS and mER, supercritical behavior varies strongly between realizations because growth requires the two largest clusters to have equal size.
  • Staircases: Multi-time-scale aggregation can remain stochastic in both ensembles and individual realizations even as N →∞.This differs from models whose staircase onset is stochastic but whose later jumps are described as deterministic incomplete devil’s staircases.
  • Finite-size effects: For the dCDGM model, 1/β = 1/0.0555 ≈18, so N = 10^18 can still show a roughly 10% discrete jump.The associated crossover length is therefore beyond most real-world networks.

3. Modular network structure in the real-world

Explosive-percolation mechanisms appear across physical, biological, economic, and social systems by suppressing large-component growth or delaying global connectivity. Related extensions show that transition abruptness depends on network growth rules, directionality, and the amount of information available for edge selection.

  • Modular mechanisms: Suppressing the largest component produces modular structures relevant to real-world systems and underlies several explosive-percolation models.The review identifies delayed macroscopic-component formation as a shared mechanism across explosive phenomena.
  • Real-world applications: Exact mapping of a rational-agent trading optimization problem to a local model links cost-minimizing demand satisfaction with explosive percolation in global trade networks.Nodes satisfy demand locally when feasible, with broader connectivity emerging only when local satisfaction is insufficient.
  • Real-world applications: Uniform nanotube-bundle sizes motivate explosive-percolation models for insulator-to-conductor transitions, while conducting particles on insulating substrates exhibit genuine discontinuous electric breakdowns.These examples connect explosive transitions to experimentally motivated physical processes.
  • Real-world applications: Suppressing mergers between substantially different component sizes generates multiple discontinuous jumps with randomly distributed sizes and locations, resembling crackling noise.The model enhances mergers whose size ratio approaches a fixed target ratio f.
  • Real-world applications: Information-sharing models delay global outbreaks by keeping unverifiable information localized, then produce an abrupt global burst at a critical stage.The process uses lattice epidemic spreading combined with a smallest-group selection rule implemented through the Sum Rule.
  • Growth dynamics: Node arrival can mitigate abruptness while delaying percolation, and the dCDGM growing-graph variant establishes significant delay using order-parameter scaling and fourth-order-cumulant crossings.These results extend explosive-percolation analysis beyond fixed-node graphs.
  • Network extensions: Directed competition processes retain many undirected explosive features less strongly, with the directed analogue changing critical scaling relative to the undirected AP.The directed rule minimizes the product of the tail node’s in-component and the head node’s out-component.
  • Network extensions: With global information, m = N, DPR becomes an infinite-choice model, while degree-dependent growth with α > 0 preferentially connects low-degree nodes and asymptotically approaches an explosive-percolation transition.These cases show how edge-choice scope and degree bias alter transition behavior.

4. Explosive Ising from an AP

Explosive phenomena arise when microscopic rules delay macroscopic connectivity or synchronization, but their apparent transition type can depend on dynamics, control parameters, and order-parameter choices. The review links these mechanisms to discontinuous transitions, multiple components, and synchronization behavior.

  • Explosive Ising from an AP: Adaptive cluster dynamics can turn the continuous two-dimensional Ising transition into a discontinuous transition.An AP process is applied to a modification of Swendsen–Wang cluster dynamics.
  • Mechanisms of explosive growth: Preventing direct growth of the largest component forces growth by overtaking and produces a discontinuous giant-component emergence.The minimum discontinuous jump in relative size is ΔCmax ≥1/3.
  • Mechanisms of explosive growth: Suppressing largest-component growth and using increasingly non-local evolution rules are recurring mechanisms for producing explosive percolation.Examples include edge-rejection budgets, size-biased component creation, and rules that keep clusters similar in size.
  • The role of order parameter: The observed transition can depend on the order parameter: summing all macroscopic components may make some apparently explosive transitions globally continuous.This choice can also affect interpretations of discontinuities and non-self-averaging behavior.
  • Reversibility and control parameters: Explosive models are usually irreversible, and changing from event count to physical time can change the apparent phase-transition type.A diffusion-limited aggregation model is discontinuous versus aggregation events but not versus the physical time of its governing equation.
  • Explosive synchronization: In the classical Kuramoto model, the order parameter measures phase coherence, and its mean-field onset obeys r ∼(K − Kc)^β with β = 1/2.The valid scaling interval shrinks as δK ∼N^-α with α ≈1.5, while a uniform frequency distribution yields a first-order transition and other listed distributions yield second-order transitions.

B. The Kuramoto model in networks

The Kuramoto model extends synchronization analysis from all-to-all coupling to complex network topologies, where the critical coupling depends strongly on degree-distribution moments and network structure. Mean-field theory provides analytical thresholds under assumptions, while topology, frequency weighting, random fields, and degree heterogeneity alter the transition.

  • Network formulation: Reformulating the Kuramoto model with an adjacency matrix allows synchronization analysis on complex network topologies, including uniform coupling between connected nodes.The all-to-all model is recovered when every pair is connected and coupling scales as K/N.
  • Numerical network studies: Early numerical studies found synchronization on Watts–Strogatz and Barabási–Albert networks, with small rewiring fractions sufficient for collective synchronization in ring-derived networks.For fixed average degree, the all-to-all critical coupling is recovered only as the mean degree increases, not simply by taking rewiring probability toward one.
  • Mean-field threshold: The critical coupling λc scales with ⟨k⟩/⟨k^2⟩, so it vanishes for scale-free networks with γ ≤ 3 but remains finite for γ > 3.This result is mathematically correct for locally tree-like networks and can work well for some clustered networks.
  • Mean-field threshold: Mean-field analysis predicts that synchronization thresholds can share the same functional form across synchronization, percolation, and epidemic-spreading processes.The implication that topology determines critical properties across dynamics remains under numerical scrutiny.
  • Extensions: Degree-dependent frequency weighting can restore a finite critical coupling for scale-free networks with 2 < γ < 3, contrasting with the unweighted case where λc = 0.The result follows from extending the mean-field formalism to degree-dependent coupling cases.
  • Extensions: Random fields can produce a tricritical point separating second-order behavior from first-order behavior in the Kuramoto model.For scale-free networks with 2 < γ < 5, the transition is second-order at any field magnitude, except at γ = 3, where it is infinite-order at λc = 0.

C. Explosive Synchronization in networks

Explosive synchronization arises when oscillator dynamics interact with network structure in ways that delay global coherence. Degree–frequency correlations generate abrupt transitions and hysteresis in scale-free networks, while analytical treatments characterize thresholds and their dependence on network correlations.

  • Scope: The review cautions that most abrupt explosive-synchronization phenomena lack rigorous proofs establishing genuine discontinuities.Accordingly, it uses “abrupt” rather than asserting discontinuity when evidence is primarily numerical.
  • Origins: Explosive synchronization was introduced through degree–frequency correlations, linking each oscillator’s natural frequency to its number of network connections.The framework targets abrupt changes in the global synchronization order parameter and can produce hysteresis.
  • Numerical evidence: Continuation simulations show smooth, coincident forward and backward branches in homogeneous networks but abrupt transitions and hysteresis in scale-free networks.In the scale-free case, effective frequencies remain near their natural values until λ ∼ 1.42, then collapse to the mean frequency Ω = ⟨k⟩ = 6.
  • Analytical characterization: The degree-correlated framework derives a critical coupling from the non-trivial synchronization solution near r → 0+, with extensions for partial degree–frequency correlations.When the correlation applies to all oscillators, the generalized expression recovers the previously derived threshold.
  • Analytical characterization: Under an annealed-network approximation, scale-free networks have first-order transitions for 2 < γ < 3, second-order transitions for γ > 3, and a hybrid transition at γ = 3.At γ = 3, the order parameter follows r − rc ∝ (λ − λc)^2/3 near onset, while γ < 3 supports coexistence and hysteresis.

2. The star network

Star networks provide a minimal heterogeneous topology for analytically studying explosive synchronization and hysteresis. Their hub–leaf structure yields explicit synchronization branches, while assortativity and modularity substantially reshape hysteresis and transition locations.

  • 2. The star network: Star networks reduce scale-free-like degree heterogeneity to one hub and K leaves, enabling analytical treatment of correlated degree–frequency synchronization.They were used in the original analysis establishing explosive synchronization in the correlated framework.
  • 2. The star network: For large K, the backward critical coupling approaches one and the backward synchronization order parameter approaches K/(K + 1).The latter expression coincides with an alternative derivation when K ≫ 1.
  • 2. The star network: The star’s backward and forward branches are stable synchronization solutions whose existence bounds exactly recover the limits of the hysteresis cycle.A third steady state is unstable, while the stable solutions correspond to the backward and forward branches.
  • Structural properties: Assortative scale-free networks exhibit much larger hysteresis cycles than disassortative networks at moderate assortativity, with degree heterogeneity further magnifying irreversibility.The network degree distribution is preserved while degree correlations are varied through targeted rewiring.
  • Structural properties: Increasing modularity lowers the forward critical coupling, while tuning intermodule mixing in two coupled stars produces a double sharp transition with adjustable synchronization gaps.The gaps and transition positions depend on the fraction of leaves shared by the two hubs.

4. More general CDF scenarios

Generalizations of correlated degree–frequency models show that explosive synchronization depends sensitively on the correlation function, noise, time delay, and frustration. These modifications can suppress, stimulate, or transform abrupt transitions into hierarchical or multistage synchronization.

  • 4. More general CDF scenarios: For ωi = αk_i^β with β < 0, the abrupt transition is suppressed, the onset is enhanced, and synchronization proceeds hierarchically.High-degree nodes with lower natural frequencies lead the cluster, while lower-degree nodes adapt as coupling increases.
  • 4. More general CDF scenarios: Randomly assigning frequency signs creates a symmetric distribution but eliminates explosive synchronization, replacing it with a two-stage transition involving two synchronized clusters.The first stage begins at λ1 and is associated with a bimodal frequency distribution induced by the degree mapping.
  • 4. More general CDF scenarios: Adding bounded frequency noise can stimulate explosive transitions in otherwise homogeneous networks, including the C. elegans neural network, scale-free graphs with γ > 3, and stretched-exponential networks.These cases are illustrated through synchronization diagrams in the cited studies.
  • 4. More general CDF scenarios: Time delay produces a non-trivial dependence of transition type: in a scale-free network, explosive transitions occur only at τ = 0 and τ = 1.0 among the tested delays.Other tested delays yield smooth transitions with different order-parameter scaling.
  • 4. More general CDF scenarios: Increasing frustration in scale-free networks enhances synchronization onset while transforming explosive synchronization into the usual second-order transition.The effect is reported for the Sakaguchi–Kuramoto model under the correlated framework.

E. The correlated frequency-coupling (CFC) framework

The CFC framework correlates oscillators’ natural frequencies with coupling strength, producing explosive synchronization across frequency distributions and, theoretically, independently of network topology. Its synchronized state consists of two phase-locked clusters, but predictions can fail on sparse complex networks and under altered frequency conditions.

  • Mechanism: CFC correlates each oscillator’s natural frequency with its coupling strength, allowing different frequency distributions on the same underlying network.This removes the CDF requirement that the frequency distribution equal the degree distribution.
  • Explosive transition: For symmetric unimodal frequency distributions centered at Ω = 0, CFC produces explosive synchronization even in all-to-all networks.The result is reported for fully connected, ER, and SF topologies.
  • Microscopic structure: The synchronized regime comprises two coexisting clusters of fully phase-locked oscillators that progressively approach each other as coupling increases.Numerical results confirm this microscopic cluster picture.
  • Critical behavior: The backward phase-locked regime exists for λ > 2, while the forward critical coupling depends on the frequency distribution g(ω).The forward and backward branches therefore have distinct critical behavior.
  • Limitations: The continuum theory is topology-independent, but its assumptions are too strong for complex networks with small average degree and can mispredict the backward threshold.The discrepancy is observed for ER and SF networks in Fig. 26(c)–(d).
  • Dependence on frequency structure: Explosive synchronization can become second order when |ω_i| is replaced by ω_i or when the frequency-distribution center shifts sufficiently from Ω = 0.The microscopic synchronization patterns may nevertheless remain unlike those of the traditional Kuramoto model.

4. Including frequency mismatch

Frequency mismatch can induce explosive synchronization by weakening links between similarly paced oscillators and strengthening links between dissimilar ones. The resulting transition depends on mismatch, network heterogeneity, and whether coupling is static, adaptive, or dynamically modulated.

  • Mismatch-weighted coupling: Weighting links by frequency mismatch weakens coupling between similar-frequency neighbors and strengthens coupling between neighbors with different frequencies.This implements an effective decoupling of oscillators that would otherwise synchronize readily.
  • Network dependence: For homogeneous, ER, and random-regular graphs, sufficiently large mismatch-weighting exponent α produces explosive synchronization across several frequency distributions.The explosive transition is progressively lost as networks become more heterogeneous and approach the SF limit.
  • Synchronization seeds: The modified effective adjacency matrix identifies synchronization seeds through the eigenvector associated with its largest eigenvalue.The resulting ranking defines a synchronization-centrality measure.
  • Mechanism: In networks exhibiting explosive synchronization, CFC homogenizes synchronization centrality and suppresses structural loci that would otherwise seed synchronization.This provides a network-level interpretation of the suppressive mechanism.
  • FGC framework: The FGC framework constructs networks so neighboring nodes satisfy prescribed frequency mismatches, rather than modifying the oscillator equations.The construction starts from isolated nodes, assigns frequencies, and adds links under the mismatch constraint.
  • FGC results: For mismatch γ > γ_c, FGC networks display explosive synchronization and develop V-shaped frequency–degree correlations absent at γ = 0.The mismatch threshold and hysteresis region are reported in Fig. 28.
  • Adaptive framework: In adaptive node coupling, a critical fraction f_c exists such that f > f_c changes the second-order transition into an abrupt one.The adaptive scheme is effective in homogeneous ER networks, while its behavior in SF networks remains under study.
  • Finite-size visibility: For any nonzero f, theory predicts explosive synchronization, but small f produces a bistable interval too narrow to detect in finite networks.This explains the contrast between the theoretical result and numerical insets.

H. Other paradigmatic dynamical models

Explosive transitions extend beyond first-order phase Kuramoto dynamics to inertial, amplitude–phase, and other oscillator models. Their manifestations include cluster synchronization, oscillation death, remote synchronization, and topology- or parameter-dependent abrupt coherence.

  • Second-order Kuramoto model: For inertial Kuramoto networks, the central question is how explosive-transition frameworks alter the model’s existing second-order synchronization transition.The analysis focuses on coupling structural and dynamical features through the CDF framework.
  • Second-order Kuramoto model: CDF inertia produces cluster synchronization as a cascade of sudden synchronizations among different degree groups rather than a single explosive transition.The phenomenon is analyzed using degree-dependent oscillator densities and pendulum dynamics.
  • Second-order Kuramoto model: Degree classes map to damped driven pendula with different parameters, whose bistability and hysteresis explain abrupt cluster transitions.Increasing and decreasing coupling follow different transitions between synchronized fixed points and incoherent limit cycles.
  • Oscillators with amplitude dynamics: The Stuart–Landau model connects amplitude–phase dynamics to Kuramoto dynamics when α is large relative to coupling, and uses separate phase and global-amplitude coherence measures.rθ measures phase coherence, whereas ru also includes amplitude coherence.
  • Oscillators with amplitude dynamics: CDF in Stuart–Landau networks does not produce explosive synchronization in the studied topologies but does produce remote synchronization.Remote synchronization is phase coherence without directly connected or phase-coherent intermediary units.
  • Oscillators with amplitude dynamics: CFC in Stuart–Landau networks produces explosive transitions corresponding to sudden oscillation death, with bistability in both phase and global synchronization.The synchronized regime self-organizes into two equal-amplitude clusters with different phases.
  • Other dynamical models: Other amplitude–phase models show topology-independent transition points, while average degree strongly affects whether abrupt transitions are amplitude death or oscillation death.A related model exhibits explosive synchronization only on SF networks above a critical parameter δ_c ≃ 0.2.

4. Rossler chaotic oscillators

Chaotic oscillator experiments and models show that synchronization can be either smooth or explosive depending on dynamical parameters. Across explosive-synchronization frameworks, microscopic rules delay macroscopic coherence by isolating small clusters, but their relation to real systems and explosive percolation remains incomplete.

  • Rössler chaotic oscillators: In coupled Rössler-like oscillators on SF networks, the synchronization transition can be explosive or continuous depending on the dynamical parameter R.For R ≳ 88, explosive synchronization of chaotic states appears.
  • Implementations in the lab: Experiments show a smooth transition at R_exp = 55 but sharp growth and hysteresis at R_exp = 65 and 70.This was the first experimental realization of the theoretical explosive-synchronization predictions.
  • Spontaneous synchronization in real systems: Explosive synchronization has been reported as a possible model for epileptic-seizure onset following network rewiring that increases local clustering.The proposed mechanism links structural changes to abrupt collective dynamics.
  • Spontaneous synchronization in real systems: A cochlear model exhibits an abrupt synchronization transition that may explain frequency selectivity in acoustical signal transduction.The application concerns the auditory portion of the inner ear.
  • Scope boundary: The review emphasizes that empirical links between explosive synchronization and real-system observations remain at an early stage.More work is needed to relate observed phenomena accurately to the proposed mechanisms.
  • Connection between EP and ES: Across EP and ES, microscopic rules delay formation of a macroscopic connected or synchronized component, but an exact mapping between the two phenomena is not established.The review identifies suppressive rules and isolated frequency-similar clusters as partial points of contact.
  • Connection between EP and ES: Before the forward threshold, synchronized clusters form along the frequency-ordered matrix diagonal and remain isolated from disparate-frequency nodes.This structure reflects suppression of synchronized links near the center of the frequency distribution.
  • Connection between EP and ES: Both CDF and CFC produce synchronized clusters that disappear progressively while preserving constituent phase distributions, indicating self-similarity.The observation concerns clusters at the backward transition.

V. OTHER MODELS WITH EXPLOSIVE BEHAVIOR

Explosive behavior appears across congestion, information routing, epidemic spreading, and related network processes. These models produce abrupt transitions through traffic-aware routing, dynamical interactions, resource limitations, or synergistic contagion effects.

  • Explosive congestion: Traffic-aware routing can change the congestion onset from a continuous to a first-order-like transition.The routing parameter h controls deviation from shortest paths according to queue congestion; h = 1 denotes no traffic awareness.
  • Explosive congestion: Congestion models relate explosive transitions to bootstrap percolation, where overloaded nodes activate when active neighbors exceed a threshold.Finite-buffer models exhibit a similar explosive transition to congestion.
  • Information routing: Changing unit properties in dynamical information routing can make the transition switch-like as a control parameter varies.For Kuramoto-like dynamics, the transition depends on individual properties, network topology, and external inputs.
  • Explosive epidemics: Markovian equations and Monte Carlo simulations both predict endemic-state coexistence and hysteresis for synergistic SIS dynamics.The phase diagram shows discontinuous transitions between endemic infected and infection-free states.
  • Explosive epidemics: Multiplicative synergy produces discontinuous epidemic transitions, with the effect determined by the synergistic parameter and the number of infected neighbors.The model rescales infectivity using synergy strength and infected-neighbor count.
  • Explosive epidemics: Resource-limited recovery and cooperative contagion can generate abrupt epidemic transitions instead of the usual continuous epidemic onset.These mechanisms include healing resources generated by healthy individuals and contagion interactions that produce effects greater than individual contributions.

C. Explosive phenomena on multilayer complex networks

Multilayer and interdependent networks support abrupt structural, percolation, and synchronization transitions. These transitions arise from interlayer coupling, dependency-induced cascades, and interactions between dynamical processes across network layers.

  • Structural transitions: Increasing interconnectivity sharply merges independent network layers into a system behaving as a single-level network.The transition separates structurally decoupled layers from a regime where the layers become indistinguishable.
  • Structural transitions: A supra-Laplacian spectral analysis identifies two algebraic-connectivity regimes separated by a discontinuity in the first derivative at p∗.The upper regime is bounded by the algebraic connectivity of the weighted layer superposition.
  • Structural transitions: Below p∗, the layers are structurally independent; above p∗, interlayer connections dominate and make them structurally indistinguishable.The characteristic-vector coordinates distinguish the two regimes.
  • Percolation and cascading failure: Interdependent-network failures exhibit sharp transitions in the existence of mutual giant components, including hybrid phase transitions.Optimal coupling studies show that interconnectivity can help avoid catastrophic failures.
  • Percolation and cascading failure: Mutual-component membership requires every interdependent replica to belong to the component, unlike ordinary monolayer connectivity.This definition underlies the sharp contrast between multilayer and traditional percolation.
  • Synchronization transitions: Multiplex coupling can produce abrupt synchronization through biased random walkers, with bistability appearing in the synchronization phase diagram.The model couples Kuramoto oscillators and walker dynamics across layers; a related duplex model reports a discontinuous backward transition under replica frustration.

VI. CONCLUSIONS AND OPEN CHALLENGES

The review organizes explosive percolation and synchronization around microscopic mechanisms that suppress macroscopic component formation. It also identifies unresolved theoretical, empirical, and control questions for future work.

  • Conclusions: The review synthesizes mechanisms, models, behaviors, and applications of explosive phenomena across complex networks.It aims to organize a vigorous literature and identify fruitful directions for future research.
  • Implications: Explosive percolation demonstrates that repeated small interventions can have a massive impact on network connectivity.The review presents this as one of the field’s discoveries since activity began in 2009.
  • Conclusions: Explosive percolation and synchronization share microscopic rules that delay formation of a macroscopic connected or synchronized component.Percolation typically uses explicit suppression that creates a powder-keg cluster distribution, whereas synchronization emerges from frequencies and connectivity.
  • Open challenges: A key open problem in explosive percolation is quantifying the tradeoff between global and local information required by the dynamics.For explosive synchronization, unresolved questions include how local order develops and whether chimera states exist.
  • Open challenges: Future research should identify natural physical and sociotechnical systems where suppression of large cluster mergers is intrinsic to the dynamics.Such systems could improve understanding and control of networked phenomena.
  • Implications: Abrupt transitions may underlie important natural system phenomena and functions, motivating further study of their mechanisms.The review anticipates increased research on dynamical processes unfolding on complex networks.
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