Dirac synchronization is rhythmic and explosive

The study explores how topology, beyond traditional node-based dynamics, shapes synchronization when signals are defined both on nodes and links of a network. Prior work has examined higher-order interactions and edge (link) signals but often with global adaptive couplings. The central research questions are whether locally coupled topological signals (via a topological and local mechanism) can generate discontinuous (explosive) synchronization transitions and whether such coupling yields a stable hysteresis loop even on fully connected networks. The authors propose Dirac synchronization: a Kuramoto-like dynamics coupling node and link phases through time-dependent phase lags determined locally by the topological Dirac operator. They aim to characterize the phase diagram, predict the critical couplings for forward and backward transitions, and uncover a coherent non-stationary (rhythmic) phase potentially relevant to brain rhythms.

• Classical synchronization frameworks: Kuramoto and extensions describe continuous transitions in node-only dynamics, with phase lags (Sakaguchi–Kuramoto) yielding rich behaviors under specific conditions.
• Higher-order networks and simplicial complexes: Recent advances highlight topology’s role (Hodge Laplacians, Dirac operators) and synchronization of higher-dimensional topological signals (e.g., higher-order Kuramoto on links/simplices). Global adaptive couplings can produce abrupt transitions but are not local and lack stable hysteresis on fully connected graphs.
• Edge (link) signals: Increasingly studied in neuroscience (edge time series, synaptic oscillations) and in flux networks (transport, power grids), motivating dynamics that explicitly couple node and link signals.
• Explosive synchronization: Observed via various mechanisms (degree-frequency correlations, adaptation, multiplexing). A recent unifying framework exists, but the mechanism here is topologically induced and distinct, involving local coupling via the Dirac operator and producing a rhythmic non-stationary coherent phase. Overall, the literature suggests topology can qualitatively alter synchronization, but local, topologically grounded couplings between nodes and links with stable hysteresis remained an open question.

Model definition:

• Network: Consider a network G=(V,E) with N nodes and L links, using an oriented incidence matrix B (size N×L). The topological Dirac operator D couples node and link spaces, with D^2 yielding block Laplacians L[0]=BB^T (graph Laplacian on nodes) and L[1]=B^T B (1-Laplacian on links).
• Dirac synchronization dynamics: A Kuramoto-like system with local, topology-driven adaptive phase lags couples node phases θ (on nodes) and link phases φ (on links). The full coupled dynamics is Φ̇ = Ω − D sin(D Φ − K^{-1} C Φ), where Φ stacks node and link phases, K is block-diagonal with node degree matrix K[0] and link generalized degree matrix K[1] (2 on all link diagonals), and γ ensures non-trivial phase diagram and positive-definite linearization. This substitution introduces time-dependent, local phase lags proportional to the Laplacian.
• For analysis on fully connected networks: Node intrinsic frequencies ω_i ~ N(Ω,1), link frequencies ω_ij ~ N(0, 1/√(N−1)) so that the induced node-projected link frequencies are Gaussian with controlled variance.
• Projected dynamics on nodes: Define the projection ψ = B φ. The coupled equations become (for complete graphs with suitable normalization): θ̇ = ω − (1/N) B sin(B(θ + ψ/2)), ψ̇ = ω − (1/N) B sin(ψ − K_1 θ), with ω = B ω (projected link frequencies). Introduce phase combinations α_i = θ_i + ψ_i/2 and β_i = (θ_i − θ_j) − ψ_i/2 and corresponding complex order parameters X_α = (1/N) Σ e^{iα_i}, X_β = (1/N) Σ e^{iβ_i}.
• Linearization and interpretation: The linearized dynamics couples nodes and links via L and D K^{-1} L terms, indicating local diffusion-induced coupling across nodes, incident links, and neighboring nodes’ links. Analytical treatment:
• Continuity equation and generalized Ott–Antonsen (OA) ansatz: Formulate a 2D continuity equation for the density ρ(α,β|ω_1,ω_2). Extend OA by assuming Fourier coefficients factorize via complex functions a(ω_1,ω_2,t) and b(ω_1,ω_2,t). Obtain closed-form evolution equations for a and b, derive stationary solutions and constraints under which stationary coherent states can exist.
• Stationary phases: Incoherent phase (R_α=0; R_β determined by drift/freeze conditions) and a stationary coherent phase requiring |a_i|=|b_i|=1 for all nodes (impossible in the thermodynamic limit for unbounded frequency distributions; only possible at finite N when σ exceeds bounds set by the maximal frequencies).
• Non-stationary coherent (rhythmic) phase: Show that stationary coherent solutions generally do not exist for Gaussian frequencies as N→∞; the coherent phase is intrinsically non-stationary (order parameters rotate/oscillate). Provide an approximate theory classifying nodes into frozen, α-oscillating, β-oscillating, and drifting, and derive self-consistent expressions for R_α and R_β in the rhythmic phase (e.g., via integrals over regions |d|≤1).
• Stability analysis of the incoherent phase: Linearize around incoherence within the generalized OA framework to derive the onset of instability (critical coupling σ_c) by evaluating the Lyapunov exponent through self-consistency integrals. Numerical simulations:
• Networks: Fully connected, primarily N=20,000 for phase diagrams; additional sizes (N=1,250 up to 80,000) for finite-size scaling; N=500 for long-time rhythmic dynamics.
• Integration: 4th-order Runge–Kutta with Δt=0.005. Coupling σ is varied adiabatically in steps Δσ=0.03, with equilibration time T_max (e.g., 10; also 5).
• Measurements: Compute order parameters R_α and R_β (real parts and full complex trajectories), determine forward/backward transitions, assess hysteresis. Perform finite-size scaling to quantify deviation of observed threshold σ*(N) from theoretical σ_c. Measure oscillation frequency Ω_α in the rhythmic phase versus σ and N.

• Explosive synchronization with hysteresis from local topological coupling: • Forward transition is discontinuous at σ_c = 2.14623… (onset of instability of the incoherent phase). • Backward transition is continuous at σ_b ≈ 1.66229… (bifurcation where the synchronized branch merges with incoherence). • Stable hysteresis loop on fully connected networks.
• Emergent rhythmic (non-stationary coherent) phase: • For σ_c < σ < σ_ξ, the coherent phase is non-stationary: complex order parameters X_α and X_β oscillate; X_α often rotates at low frequency with nearly constant magnitude, while X_β exhibits nontrivial phase and amplitude dynamics. • As N→∞, the stationary coherent phase disappears (requires bounded frequencies); thus the only coherent phase in the thermodynamic limit is rhythmic (non-stationary). • The emergent oscillation frequency Ω_α decreases with increasing σ and with increasing N, approaching zero at the onset of stationarity; the rhythmic phase extends further for larger N.
• Correct order parameters are mixed (R_α and R_β): Traditional uncoupled order parameters (node-only or link-projection-only) remain near zero; mixed order parameters reveal the transitions, reflecting the local node–link coupling.
• Numerical–theoretical agreement: • Bifurcation diagrams from simulations (N=20,000; RK4, Δt=0.005; Δσ=0.03; T_max=10) match theoretical predictions in both stationary and rhythmic regimes. • Finite-size effects shift observed forward-transition threshold to σ_c(N) < 2.14623…; deviation scales approximately as |σ*(N) − σ_c| ≈ C N^ξ with ξ ≈ 0.177 for T_max=5 (and ξ ≈ 0.1065 for T_max=10), consistent with finite-N fluctuations enabling earlier transitions.
• Mechanism distinct from prior frameworks: The discontinuous transition arises from a topological, local Dirac coupling and cannot be reduced to existing universal routes to explosive transitions based on global adaptation or other known mechanisms.

The work demonstrates that coupling node and link topological signals locally via the Dirac operator produces an explosive synchronization transition with a robust hysteresis loop, directly addressing the open question of whether local, topologically grounded coupling can yield discontinuous synchronization and stable hysteresis even on fully connected networks. The identification of mixed order parameters (R_α, R_β) reveals the intrinsically interdependent dynamics of node and link phases, contrasting with models using separate order parameters. The emergence of a coherent non-stationary rhythmic phase over a wide σ range, and its dominance in the thermodynamic limit, differentiates Dirac synchronization from the standard Kuramoto model, where coherent phases may be stationary despite drifting oscillators. These findings suggest a new topological mechanism for generating macroscopic rhythms with oscillatory order parameters that could be relevant for understanding brain rhythms and cortical oscillations, where local, time-dependent phase lags and edge-level signals are biologically plausible. The analytical predictions (σ_c, σ_b, phase classification, and finite-size effects) are corroborated by extensive simulations, highlighting the predictive power of the theory and the role of topology in shaping collective dynamics.

The paper introduces Dirac synchronization, a locally coupled, topology-driven framework that couples node and link phases via the topological Dirac operator and adaptive, time-dependent phase lags. On fully connected networks, the model exhibits an explosive forward transition at σ_c = 2.14623…, a continuous backward transition at σ_b ≈ 1.66229…, and a stable hysteresis loop. The coherent phase is predominantly non-stationary (rhythmic), with oscillatory order parameters persisting over a broad parameter range and becoming the only coherent phase as N→∞. Theoretical analysis (continuity equation with generalized OA ansatz, stability of incoherence, and approximate treatment of the rhythmic phase) aligns with numerical simulations, including finite-size scaling. These results showcase a genuinely topological route to explosive synchronization distinct from previously proposed mechanisms and suggest potential relevance for modeling brain rhythms. Future research directions include: (i) extending analysis beyond fully connected graphs to sparse, heterogeneous, and real-world networks (e.g., brain connectomes); (ii) generalizing to simplicial complexes with higher-order Dirac operators to couple signals on higher-dimensional simplices; and (iii) exploring data-driven applications and parameter inference for empirical systems with edge-level dynamics.

• Network topology: The analytical treatment and most simulations focus on fully connected networks; behavior on sparse or structured networks (heterogeneous degrees, community structure) remains to be systematically characterized.
• Frequency distributions: Stationary coherent states require bounded frequency support; with Gaussian (unbounded) frequencies, stationary coherence disappears as N→∞, limiting stationary analyses to finite-size systems.
• Rhythmic phase theory: The non-stationary phase is addressed with approximate arguments and numerical observations; a full closed-form theory for oscillation frequencies and amplitudes under general conditions is not provided.
• Parameter choices: Numerical demonstrations use specific integration schemes, step sizes, and adiabatic protocols; while robust across tests, sensitivity to protocol variations in other topologies is not fully explored.
• Empirical validation: The study is theoretical/numerical; experimental validation in systems with measurable edge signals (e.g., synaptic dynamics, power-grid flows) is an open direction.