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Multiorder Laplacian for synchronization in higher-order networks

Maxime Lucas, Giulia Cencetti, Federico Battiston

arXiv:2003.09734v3nlin.AOphysics.soc-ph

TL;DR

Higher-order oscillator interactions are poorly captured by pairwise networks, and existing analytical results have mostly been restricted to all-to-all settings. The paper introduces a multiorder Laplacian for arbitrary hypergraphs and coupling orders, showing that its spectrum determines synchronized-state stability. Analytical all-to-all results, heterogeneous-topology examples, and macaque-brain applications demonstrate the framework’s scope.

  • Problem

    Pairwise networks cannot faithfully represent many group interactions, while analytical treatments of higher-order oscillator synchronization have largely been limited to all-to-all coupling.

    Source: §I · pp. 1–2

  • Method

    The paper generalizes Kuramoto dynamics to arbitrary hypergraphs and defines a multiorder Laplacian whose spectrum characterizes stability of the synchronized state.

    Source: §I · p. 2, §VI · p. 12

  • Results

    The framework yields analytical stability characterizations for higher-order all-to-all systems and is applied to heterogeneous star-clique and macaque-brain topologies.

    Source: §VI · pp. 11–12

  • Takeaways & Limitations

    The multiorder Laplacian extends pairwise stability analysis to structured group interactions of arbitrary order and finite network size.

    Source: §VI · p. 12

Abstract

from arXiv · show

Traditionally, interaction systems have been described as networks, where links encode information on the pairwise influences among the nodes. Yet, in many systems, interactions take place in larger groups. Recent work has shown that higher-order interactions between oscillators can significantly affect synchronization. However, these early studies have mostly considered interactions up to 4 oscillators at time, and analytical treatments are limited to the all-to-all setting. Here, we propose a general framework that allows us to effectively study populations of oscillators where higher-order interactions of all possible orders are considered, for any complex topology described by arbitrary hypergraphs, and for general coupling functions. To this scope, we introduce a multi-order Laplacian whose spectrum determines the stability of the synchronized solution. Our framework is validated on three structures of interactions of increasing complexity. First, we study a population with all-to-all interactions at all orders, for which we can derive in a full analytical manner the Lyapunov exponents of the system, and for which we investigate the effect of including attractive and repulsive interactions. Second, we apply the multi-order Laplacian framework to synchronization on a synthetic model with heterogeneous higher-order interactions. Finally, we compare the dynamics of coupled oscillators with higher-order and pairwise couplings only, for a real dataset describing the macaque brain connectome, highlighting the importance of faithfully representing the complexity of interactions in real-world systems. Taken together, our multi-order Laplacian allows us to obtain a complete analytical characterization of the stability of synchrony in arbitrary higher-order networks, paving the way towards a general treatment of dynamical processes beyond pairwise interactions.

I. INTRODUCTION

The paper addresses synchronization when oscillators interact in groups larger than pairs, where conventional networks can miss structurally distinct interactions. It introduces a hypergraph-based Kuramoto generalization and targets analytical stability results for arbitrary higher-order topologies.

I. INTRODUCTION

  • Higher-order interactions distinguish group collaborations from separate pairwise links, and can alter dynamics including contagion, diffusion, random walks, and evolutionary games.Source: §I · p. 1
  • Prior oscillator studies linked higher-order coupling to cluster synchronization, explosive transitions, and exotic dynamics, but analytical results largely remained limited to all-to-all settings.Source: §I · pp. 1–2
  • The proposed framework represents arbitrary group interactions with hypergraphs and uses a multiorder Laplacian whose eigenvalues determine fully synchronized-state stability.Source: §I · p. 2

II. GENERALIZED HIGHER-ORDER INTERACTIONS

  • The model generalizes Kuramoto dynamics to identical oscillators with undirected interactions at every order up to D=N−1, encoded by adjacency tensors.Source: §II · p. 2
  • The coupling functions are symmetric in the other oscillator indices, vanish at synchrony, and include sinusoidal forms motivated by phase reduction.Source: §II · p. 2
  • The fully synchronized solution exists as θ_i(t)=ωt, and its stability is analyzed by moving to a rotating frame and linearizing heterogeneous perturbations.Source: §II · p. 3

III. MULTIORDER LAPLACIAN

The multiorder Laplacian extends node-based pairwise stability analysis to higher-order interactions by expressing each order through a generalized Laplacian and combining orders in a weighted matrix.

III. MULTIORDER LAPLACIAN

  • The proposed operator differs from earlier edge-, triangle-, or simplex-based Laplacians because it describes higher-order coupling from the perspective of the participating nodes.Source: §III · p. 4

A. Laplacian

  • Each higher-order interaction term can be rewritten using an order-specific Laplacian, and the complete linearized system uses a multiorder Laplacian.Source: §III · p. 4, A. Laplacian · p. 4
  • The order-d construction generalizes pairwise degree and adjacency: degrees count incident d-simplices, while adjacency counts shared d-simplices between node pairs.Source: A. Laplacian · pp. 4–5
  • The multiorder Laplacian weights each order by its coupling strength normalized by average degree, reduces to the usual Laplacian for D=1, and remains bounded by network size.Source: A. Laplacian · p. 4
  • For general topologies, the multiorder matrix is analytically defined but its eigenvalues generally require numerical computation because eigenvalues do not combine linearly across orders.Source: A. Laplacian · pp. 4–5
  • The generalized Laplacian is symmetric, has zero row sums, and has non-negative eigenvalues, while its construction extends from order 2 to arbitrary order d.Source: A. Laplacian · pp. 4–5
  • The same derivation accommodates other symmetric coupling functions that vanish at synchrony, with their coefficient c0/d scaling implying slower convergence to synchronization.Source: A. Laplacian · p. 5

B. Stability and Lyapunov exponents

Stability is determined by the Laplacian spectrum: synchronized perturbations decay when Lyapunov exponents are negative, and the smallest nonzero exponent quantifies resilience to perturbations.

B. Stability and Lyapunov exponents

IV. STABILITY IN ALL-TO-ALL HIGHER-ORDER NETWORKS

The higher-order all-to-all model provides the paper’s analytically tractable benchmark. Its homogeneous structure yields an explicit full-synchrony spectrum and separates the effects of different interaction orders.

IV. STABILITY IN ALL-TO-ALL HIGHER-ORDER NETWORKS

  • Higher-order all-to-all coupling includes every possible (d+1)-oscillator interaction at every order d, generalizing ordinary global pairwise coupling.Source: §IV · p. 6
  • In the homogeneous setting, each order acts through the same nonlinear mean field with strength R_1^d and a dth harmonic, which can support stable cluster states.Source: §IV · p. 6
  • Unlike earlier studies focused on clusters or transitions, this analysis gives the full-synchrony spectrum and examines variable order counts, attractive and repulsive couplings, and decaying strengths.Source: §IV · p. 6
  • Because complex topology is absent, the full Lyapunov spectrum is a linear combination of pure-order spectra, unlike the nonlinear spectral aggregation expected for complex topologies.Source: §IV · p. 6

A. Higher-order Laplacians are proportional to the traditional pairwise Laplacian

In higher-order all-to-all systems, each order-d Laplacian is proportional to the usual pairwise Laplacian, enabling analytical stability calculations. This proportionality generally fails to yield a linear spectrum for complex coupling schemes.

A. Higher-order Laplacians are proportional to the traditional pairwise Laplacian

  • Higher-order attractive interactions make synchronization more stable: larger d and larger D produce more negative Lyapunov exponents.For pure interactions, 2-simplex coupling synchronizes faster than pairwise coupling, and adding attractive orders further stabilizes the synchronized state.Source: Figure 2 · p. 7, A. Higher-order Laplacians are… · p. 7
  • For every interaction order d, the higher-order all-to-all Laplacian is proportional to the usual pairwise Laplacian.Its entries grow linearly with d and with the number of d-simplex interactions per oscillator.Source: A. Higher-order Laplacians are… · pp. 6–7
  • The all-to-all analytical spectrum is a limiting case because complex coupling schemes do not generally permit a full analytical derivation.The eigenvalue operator is nonlinear, so the multiorder spectrum is not generally a linear combination of order-specific spectra.Source: A. Higher-order Laplacians are… · p. 7
  • The all-to-all multiorder Laplacian depends on the pairwise Laplacian, each coupling strength γd, and the corresponding interaction order d.Because the order-specific Laplacians are proportional, the all-to-all multiorder spectrum can be obtained analytically.Source: A. Higher-order Laplacians are… · pp. 6–7
  • With mixed coupling signs, higher orders can reverse stability: attractive higher-order interactions can overcome repulsive pairwise coupling, while repulsive higher-order interactions can destabilize attractive pairwise coupling.For alternating signs, stability alternates with the parity of the highest included order.Source: A. Higher-order Laplacians are… · p. 8, Figure 3 · p. 8

B. Spectrum and stability

The spectrum of the multiorder Laplacian determines synchronized-state stability, with coupling signs and interaction orders controlling the Lyapunov exponents. Attractive higher orders stabilize synchrony, while mixed signs can produce alternating or reversed stability.

B. Spectrum and stability

  • The framework defines attractive and repulsive interactions by the sign of γd, with positive coupling stabilizing and negative coupling destabilizing the synchronized state.The stability calculation uses the analytical eigenvalues of the higher-order all-to-all Laplacian.Source: B. Spectrum and stability · pp. 8–9
  • Attractive higher-order interactions stabilize synchrony: higher d yields more negative second Lyapunov exponents, and trajectories converge faster under pure 2-simplex than pairwise coupling.This effect appears despite equal normalization weights across interaction orders.Source: B. Spectrum and stability · p. 8
  • The synchronized state can remain stable with repulsive pairwise coupling when attractive 3-oscillator interactions are sufficiently strong.Conversely, repulsive 3-oscillator interactions can destabilize synchrony despite attractive pairwise coupling.Source: B. Spectrum and stability · p. 9
  • With alternating attractive even-order and repulsive odd-order couplings, finite-order stability alternates with the parity of the highest included order.The alternating series diverges as D approaches infinity, requiring N to approach infinity.Source: B. Spectrum and stability · p. 9
  • When coupling strengths decay geometrically with order, the Lyapunov exponent remains negative and converges to a finite limit, ensuring stability.The limiting expression is −γ1/(γ1 −1)^2 as D and N increase.Source: B. Spectrum and stability · p. 9

V. STABILITY IN HIGHER-ORDER NETWORKS WITH ARBITRARY TOPOLOGY

The framework is applied to heterogeneous higher-order structures through a star-clique toy model and a macaque brain connectome dataset.

V. STABILITY IN HIGHER-ORDER NETWORKS WITH ARBITRARY TOPOLOGY

  • The arbitrary-topology analysis considers a synthetic star-clique model and a real macaque brain dataset with heterogeneous higher-order interactions.These examples extend the analysis beyond higher-order all-to-all networks.Source: §V · p. 10

A. The star-clique model

The star-clique model combines a star and clique whose higher-order structure separates their spectral contributions. Pairwise interactions retain weak coupling between subnetworks, whereas higher-order interactions see only the clique and introduce additional zero modes.

Model construction

  • The star-clique consists of a star and clique joined by one link, making the subnetworks nearly independent while allowing the framework to test complex topology.At every pure order d ≥2, the subnetworks are actually disconnected.Source: A. The star-clique model · p. 10

Pairwise interactions

  • The pairwise Laplacian spectrum is the slightly perturbed union of the star and clique spectra because their subnetworks are almost independent.The corresponding Lyapunov exponents therefore reflect two nearly separate blocks.Source: A. The star-clique model · p. 10

Higher-order interactions

  • At orders d ≥2, the star contributes no higher-order interactions, so the higher-order Laplacians see only the clique and produce multiple zero eigenvalues.For d = 2, the spectrum contains N_s zeros plus the clique spectrum; for d > 2, it contains N_s + 1 zeros and N_c − 1 repeated nonzero values.Source: A. The star-clique model · p. 10

Multiorder Laplacian

  • The multiorder spectrum is close to the sum of pure-order spectra but still reflects the star-clique topology because the pairwise subnetwork is only almost disconnected.Pure higher orders match the analytically characterized all-to-all clique spectrum, while pairwise and combined orders retain the two-subnetwork structure.Source: A. The star-clique model · p. 10

Dynamical implication

  • Different global spectra imply different dynamical timescales under repulsive coupling, even when selected spectral components coincide.The passage specifically contrasts the repulsive case where all γ_d < 0.Source: A. The star-clique model · p. 10

B. Real network: Macaque brain

The macaque connectome provides an empirical test of higher-order synchronization on a heterogeneous simplicial complex. Its interactions are centralized and increasingly sparse across nodes at higher orders, and the resulting Lyapunov spectra differ between multiorder and pairwise projections.

Dataset and construction

  • The 91-node, 628-edge macaque dataset is converted into a simplicial complex by treating each fully connected (d + 1)-node clique as a d-simplex.The dataset is used as empirical ground for testing the theoretical model, not as a realistic functional-brain model.Source: B. Real network: Macaque brain · p. 10

Interaction distribution

  • The complex contains interactions through order D = 10, with fewer than 1000 1-simplices and 10-simplices but over 6000 5-simplices.The distribution of d-simplices follows a bell curve across orders.Source: B. Real network: Macaque brain · p. 10

Interaction heterogeneity

  • At each order, higher-order interactions are highly centralized: a few nodes have many d-simplex interactions, while most have few or none.As d increases, more nodes have zero d-simplex interactions; the structure at orders 4 and 9 is shown in Fig. 5.Source: B. Real network: Macaque brain · p. 11, Figure 5 · p. 11

Pure-order dynamics

  • Pure higher-order spectra contain more zero eigenvalues at larger d because more nodes become disconnected from the main component of the pure d-simplex network.The spectra also show a stage in which higher orders correspond to lower Lyapunov exponents, followed by stronger descending behavior for the last eigenvalues.Source: B. Real network: Macaque brain · p. 11

Multiorder versus projection

  • The full higher-order system is compared with a projected graph that replaces any higher-order connection by a pairwise edge, allowing their stability spectra to be contrasted.The comparison is shown in Fig. 5(f).Source: B. Real network: Macaque brain · p. 11

VI. SUMMARY AND CONCLUSIONS

The paper develops a multiorder Laplacian for analyzing synchronization with arbitrary higher-order interactions on hypergraphs. Analytical results cover all-to-all systems, mixed attractive and repulsive couplings, heterogeneous topologies, and a macaque brain network, while coupling-function effects remain partly open.

  • The multiorder Laplacian determines synchronized-state stability through Lyapunov exponents proportional to its spectrum and applies to arbitrary hypergraphs and finite system sizes.It generalizes the pairwise Laplacian framework to structured group interactions of any order.Source: §VI · pp. 11–12
  • In the analytically tractable all-to-all setting, each order-d Laplacian is proportional to the pairwise Laplacian, so multiorder stability decomposes into weighted contributions from pure d-simplex systems.Numerical simulations confirm this analytical decomposition.Source: §VI · p. 12
  • Repulsive pairwise couplings can be countered by attractive higher-order couplings, while alternating attractive odd and repulsive even orders can stabilize or destabilize the system depending on the highest included order.For attractive couplings decaying with order, the Lyapunov exponents converge as higher orders are added.Source: §VI · p. 12
  • In a star-clique topology, higher-order interactions in the clique alter much of the spectrum, whereas the higher-order-free star does not affect the first nonzero Lyapunov exponent.The example illustrates how topology shapes multiorder spectral properties.Source: §VI · p. 12
  • For the macaque brain topology, including higher-order interactions changes all Lyapunov exponents and generally stabilizes synchronization, especially on the more negative part of the spectrum.The stronger changes are attributed to the dataset's hublike structure.Source: §VI · p. 12
  • The precise effects of different coupling functions on other dynamical regimes in higher-order complex topologies remain an open question.Source: §VI · p. 12
  • The framework extends beyond synchronization analysis and has already been used to generalize master-stability-function ideas to simplicial complexes.Source: §VI · p. 12

Appendix A: Derivation of the Laplacian formulation

The appendix derives order-specific and general d-simplex Laplacians by reducing many-phase interactions to pairwise phase differences. The resulting matrices retain the defining Laplacian properties and support alternative coupling functions.

1. Order 3

  • For 3-simplex interactions, permutation invariance reduces four-phase terms to equivalent two-phase terms before expressing them through an order-3 Laplacian.The combinatorial factor 2! = 3!/3 arises from the intermediate sum of identical terms.Source: §1 · pp. 12–13
  • At order 3, the degree counts incident 3-simplices and the adjacency counts shared 3-simplices between node pairs.These definitions provide the connectivity quantities needed for the Laplacian rewrite.Source: §1 · p. 13
  • The order-3 operator is symmetric, positive semidefinite, and zero-row-sum, so four-oscillator dynamics can be represented by a valid Laplacian.Source: §1 · p. 13

2. Order d

  • For arbitrary order d, the derivation first converts d+1 phases into d equivalent two-phase terms and then rewrites their differences using the order-d Laplacian.Source: §2 · p. 13
  • At every order d, the dynamics of d+1-oscillator interactions is governed by a Laplacian whose eigenvalues are non-negative and include at least one zero.Source: §2 · p. 13

Appendix B: Alternative higher-order coupling functions

  • The framework also accommodates alternative higher-order coupling functions, not only the sinusoidal functions used in the main derivation.These alternatives are treated in Appendix B without changing the framework's generality.Source: Appendix B · p. 13

1. Order 2

The appendix analyzes an asymmetric order-2 coupling and generalizes the reduction to arbitrary order. It also gives combinatorial constructions for higher-order adjacency and degree matrices in the all-to-all case.

1. Order 2

  • For the asymmetric order-2 function sin(2θ_j − θ_k − θ_i), one term vanishes by symmetry, leaving half the symmetric coupling contribution.Source: §1 · pp. 13–14
  • For pure triplets, this asymmetric coupling produces convergence two times slower than the symmetric coupling.Source: §1 · p. 14

2. Order d

  • At arbitrary order d, terms containing oscillator i contribute 1/d of the order-d Laplacian, while terms excluding i vanish by the analogous symmetry argument.Source: §2 · p. 14

Appendix C: Higher-order degree and adjacency matrices for all-to-all coupling

  • In all-to-all coupling, A^(d)_ij counts distinct d+1-oscillator interactions containing nodes i and j, obtained by choosing d−1 additional nodes.Source: Appendix C · p. 14
  • The all-to-all order-d degree K^(d)_i counts the d+1-oscillator interactions containing node i, obtained by choosing d other nodes among the remaining N−1.The degree therefore scales with system size according to the corresponding combinatorial count.Source: Appendix C · p. 14

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