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Explosive higher-order Kuramoto dynamics on simplicial complexes
Ana P. Millán, Joaquín J. Torres, Ginestra Bianconi
TL;DR
The paper examines synchronization transitions in higher-order Kuramoto dynamics on simplicial complexes, develops an approximate analytical framework, and finds discontinuous explosive transitions under adaptive coupling.
Problem
The synchronization-transition behavior of simple and explosive higher-order Kuramoto models requires an analytical framework distinguishing continuous from discontinuous transitions.
Method
The paper formulates higher-order Kuramoto dynamics and analyzes its synchronization transitions using an approximate analytical framework.
Results
The explosive higher-order Kuramoto model displays a discontinuous transition, with σc = 1.7760... in the illustrated case.
Takeaways & Limitations
The transition behavior remains robust when simplicial complexes are constructed from experimental networks.
Takeaways & Limitations
The analytical framework is approximate, and its critical value depends on the frequency and topology of the simplicial complex.
Abstract
from arXiv · showhide
The higher-order interactions of complex systems, such as the brain are captured by their simplicial complex structure and have a significant effect on dynamics. However, the existing dynamical models defined on simplicial complexes make the strong assumption that the dynamics resides exclusively on the nodes. Here we formulate the higher-order Kuramoto model which describes the interactions between oscillators placed not only on nodes but also on links, triangles, and so on. We show that higher-order Kuramoto dynamics can lead to an explosive synchronization transition by using an adaptive coupling dependent on the solenoidal and the irrotational component of the dynamics.
SUPPLEMENTAL MATERIAL · I. ADDITIONAL BACKGROUND MATERIAL ON TOPOLOGY · Boundary map and incidence matrix
The supplemental background defines boundary maps on simplicial-chain groups and represents them as incidence matrices once ordered simplex bases are chosen. These definitions and basis choices fully determine the matrices B[n].
- SUPPLEMENTAL MATERIAL · I. ADDITIONAL BACKGROUND MATERIAL ON TOPOLOGY · Boundary map and incidence matrix: n-chains are elements of a free abelian group C_n with basis given by the n-dimensional simplices.The chain group is defined in terms of the simplicial complex’s n-dimensional simplices.
- SUPPLEMENTAL MATERIAL · I. ADDITIONAL BACKGROUND MATERIAL ON TOPOLOGY · Boundary map and incidence matrix: The boundary map ∂_n: C_n → C_n−1 is linear and maps each n-simplex to a signed combination of its oriented (n−1)-dimensional boundary faces.The signs are specified by the alternating deletion formula in Eq. (S-1).
- SUPPLEMENTAL MATERIAL · I. ADDITIONAL BACKGROUND MATERIAL ON TOPOLOGY · Boundary map and incidence matrix: The boundary of an n-simplex is constructed by deleting each vertex in turn and weighting the resulting oriented face by (−1)^p.This is the alternating expression given in Eq. (S-1).
- SUPPLEMENTAL MATERIAL · I. ADDITIONAL BACKGROUND MATERIAL ON TOPOLOGY · Boundary map and incidence matrix: Equations (S-1) and (S-2), together with the chosen bases of C_n and C_n−1, fully determine the incidence matrices B[n].The matrix representation therefore depends on both the boundary formula and the basis choices.
- SUPPLEMENTAL MATERIAL · I. ADDITIONAL BACKGROUND MATERIAL ON TOPOLOGY · Boundary map and incidence matrix: Choosing ordered bases of the n- and (n−1)-simplices defines the incidence matrix B[n] representing the boundary map.The matrix has dimensions N[n−1] × N[n].
- SUPPLEMENTAL MATERIAL · I. ADDITIONAL BACKGROUND MATERIAL ON TOPOLOGY · Boundary map and incidence matrix: The basis for C_n consists of the ordered n-simplices, while the basis for C_n−1 consists of the ordered (n−1)-simplices.These ordered simplex sets provide the coordinate systems for representing the boundary map.
Proof of the Eq. (5) of the main text
The proof rewrites Eq. (5) to expose that the boundary of a boundary is null. It illustrates this with the simplex and establishes the result in full generality through the boundary-map definition.
- Eq. (5) is rewritten to reveal the principle that the boundary of the boundary is null.
- For the simplex, its boundary is ∂2[1, 2, 3] = [2, 3] −[1, 3] + [1, 2].
- Applying the boundary map to these links gives ∂1∂2[1, 2, 3] = 0.
- Using the boundary-map definition, the corresponding cancellation identity is derived in full generality.
II. KURAMOTO DYNAMICS EXPRESSED IN TERMS OF THE INCIDENCE MATRIX
The node-based Kuramoto dynamics can be reformulated using the network’s incidence matrix. Because the incidence matrix is nonzero only for incident node–link pairs, this formulation is equivalent to the original dynamics.
- The Kuramoto equations describe node phases θ_i using internal frequencies ω_i and coupling constant σ among linked oscillators.
- The dynamics can be expressed equivalently in terms of the incidence matrix B[1], with simplices of dimension n collected in S_n and counted by N[n].The supplementary formulation corresponds to the matrix notation of Eq. (8) in the main text.
- For a link ℓ = [i, j], the incidence-matrix expression gives the same contribution under either link orientation.The orientations ℓ = [i, j] and ℓ = [j, i] lead to the same final expression in the dynamics.
- The incidence-matrix formulation is equivalent to the original Kuramoto equations because nonzero entries occur only between mutually incident nodes and links.
III. HIGHER-ORDER KURAMOTO DYNAMICS ON SMALL SIMPLICIAL COMPLEXES
This section examines simple higher-order Kuramoto dynamics on three small two-dimensional simplicial complexes. Although these systems cannot exhibit a true thermodynamic synchronization transition, their trajectories display non-trivial periodic states.
- Small simplicial complexes: The study considers three dimension-2 simplicial complexes: a full triangle, an empty triangle, and a third complex shown in Fig. S-1.These examples are used to elucidate fundamental properties of the dynamics rather than a thermodynamic phase transition.
- Model specification: The simple higher-order Kuramoto dynamics assigns a phase θα to each n-dimensional simplicial complex α.The section also specializes the dynamics to link phases for simplicial complex A.
- Observed dynamics: Exemplary link-phase trajectories show that the simple higher-order Kuramoto dynamics on the three complexes produces non-trivial dynamics and periodic states.Figure S-2 illustrates these behaviors in the phase space defined by the link phases.
IV. THE SYNCHRONIZATION TRANSITION OF SIMPLE AND EXPLOSIVE HIGH-ORDER KURAMOTO MODEL
An approximate analytical framework captures the main phenomenology of simple and explosive higher-order Kuramoto models. It explains their continuous versus discontinuous phase transitions and estimates the critical coupling value σ_c.
- Analytical framework: An approximate analytical framework describes the main observed phenomenology of both simple and explosive higher-order Kuramoto models.The framework is analytical but approximate.
- Transition nature: The simple higher-order Kuramoto model undergoes a continuous phase transition.The framework identifies the transition as continuous for the simple model.
- Transition nature: The explosive higher-order Kuramoto model undergoes a discontinuous phase transition.The framework identifies the transition as discontinuous for the explosive model.
- Critical value: The framework provides an estimate for the critical value σ = σ_c.This estimate is derived within the approximate analytical approach.
A. Simple higher-order Kuramoto model
The simple higher-order Kuramoto model separates harmonic phase components, which remain unsynchronized, from orthogonal components that can synchronize. Its projected dynamics exhibit continuous synchronization transitions at σc = 0 for both components.
- Harmonic and orthogonal components: Harmonic phase components oscillate at proper frequencies, keeping the standard order parameter near zero, R ≃0, while orthogonal components can synchronize.The variables θ[+] and θ[−] filter out the harmonic component of θ.
- Projected dynamics: The projected phase equations for θ[+] and θ[−] are independent and have a similar structure.The analysis focuses on θ[+], whose order parameter is denoted R[+].
- Synchronization transition: σc = 0 marks a continuous synchronization transition for R[+] in the simple higher-order Kuramoto model.The analytical result agrees with simulation results and is illustrated for Ω= 2 and A = B = 1.
- Synchronization transition: σc = 0 also characterizes the transition of R[−] in the simple higher-order Kuramoto model.The same analytical approach is stated to apply to the negative projected component.
B. Explosive higher-order Kuramoto model
The explosive higher-order Kuramoto model yields coupled order-parameter dynamics for R[+] and R[−] that undergo a discontinuous transition. Its critical coupling can be determined analytically and depends on frequency and simplicial-complex topology.
- Model equations: The model begins with equations for the projected phases θ[+] and θ[−], leading to coupled equations for the order parameters R[+] and R[−].The derivation follows substitutions relating the explosive model’s equations to previously obtained phase equations.
- Discontinuous transition: The coupled system displays a discontinuous phase transition for Ω= 2, A[+] = A[−] = 1 and B[+] = B[−] = 2.The transition is demonstrated by numerically integrating Eq. (S-47) and Eq. (S-48).
- Critical point: The transition point is obtained analytically by imposing that the Jacobian matrix has determinant equal to zero.The critical point follows from solving the resulting condition together with the coupled order-parameter equations.
- Critical point: For the Fig. S-4 case, σc = 1.7760 . . . and R[+]c = 0.7981 . . ..In general, σc depends on Ω and simplicial-complex topology through A[±] and B[±].
V. HIGHER-ORDER KURAMOTO DYNAMICS ON LARGE SIMPLICIAL COMPLEXES
The section tests higher-order Kuramoto dynamics on large three-dimensional simplicial complexes beyond the main configuration-model setting. It reports that the phase-transition nature remains unchanged under more uniform degree distributions and nontrivial network geometry.
- V. HIGHER-ORDER KURAMOTO DYNAMICS ON LARGE SIMPLICIAL COMPLEXES: Simulations examine simple and explosive higher-order Kuramoto dynamics of order n = 1 on three-dimensional simplicial complexes generated by a configuration model.The node generalized degree distribution follows a power law.
- V. HIGHER-ORDER KURAMOTO DYNAMICS ON LARGE SIMPLICIAL COMPLEXES: The phase-transition nature remains unchanged when the generalized degree distribution is more uniform or the simplicial complex has nontrivial network geometry.
- V. HIGHER-ORDER KURAMOTO DYNAMICS ON LARGE SIMPLICIAL COMPLEXES: Similar phase diagrams arise for simple and explosive order n = 1 Kuramoto dynamics on a Network Geometry with Flavor having d = 3, s = −1, and inverse temperature ˆβ = 0.The simulations are referenced as Figures S −7 and S −8.
VI. HIGHER ORDER KURAMOTO DYNAMICS ON CONNECTOMES OF HOMO SAPIENS AND C. ELEGANS
The study tests simple and explosive higher-order Kuramoto dynamics on simplicial complexes derived from Homo Sapiens and C. Elegans connectomes. The transition behavior remains robust on these experimentally constructed networks.
- Connectome construction: Connectome simplicial complexes represent nodes, links, and triangles as 0-, 1-, and 2-simplices, respectively.They are generated by identifying each (n + 1)-clique with an n-dimensional simplex.
- Transition robustness: Both simple and explosive higher-order Kuramoto dynamics preserve their respective transition natures on simplicial complexes constructed from experimental networks.The results indicate that the nature of the phase transitions is robust on these networks.
- Connectome datasets: The Homo Sapiens connectome contains N[0] = 66 nodes, N[1] = 254 links, and N[2] = 291 triangles.The C. Elegans connectome contains N[0] = 277 nodes, N[1] = 1918 links, and N[2] = 2699 triangles.
- Measured dynamics: The connectome analysis compares projections of n = 1 dynamics onto (n −1)- and (n + 1)-dimensional faces using R[+] and R[−].It also plots R, R[1], and R[2] against the coupling constant σ for both dynamics.