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Coupled Dynamics on Hypergraphs: Master Stability of Steady States and Synchronization

Raffaella Mulas, Christian Kuehn, Jürgen Jost

arXiv:2003.13775v2math.DSnlin.AO

TL;DR

Graph-based master stability analysis does not capture all higher-order interactions, motivating its extension to hypergraphs. The paper derives stability conditions for hypergraph dynamics and finds that hypergraph Laplacian spectra can yield stability patterns beyond those familiar from graphs.

Abstract

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In the study of dynamical systems on networks/graphs, a key theme is how the network topology influences stability for steady states or synchronized states. Ideally, one would like to derive conditions for stability or instability that instead of microscopic details of the individual nodes/vertices rather make the influence of the network coupling topology visible. The master stability function is an important such tool to achieve this goal. Here we generalize the master stability approach to hypergraphs. A hypergraph coupling structure is important as it allows us to take into account arbitrary higher-order interactions between nodes. As for instance in the theory of coupled map lattices, we study Laplace type interaction structures in detail. Since the spectral theory of Laplacians on hypergraphs is richer than on graphs, we see the possibility of new dynamical phenomena. More generally, our arguments provide a blueprint for how to generalize dynamical structures and results from graphs to hypergraphs.

INTRODUCTION

The master stability formalism connects stability conditions for steady or synchronized dynamics to network data, under assumptions such as shared node dynamics or sufficient symmetry. This paper extends that approach to hypergraphs, which can represent higher-order interactions that graphs or simplicial complexes may not capture.

  • Master stability conditions express stability of steady or synchronized solutions through network data, given sufficient symmetry and/or common node dynamics.Source: INTRODUCTION · p. 1
  • Hypergraphs represent higher-order interactions that may not fit graphs or the more restrictive structure of simplicial complexes.Source: INTRODUCTION · p. 1
  • The paper extends master stability to hypergraphs, deriving linear stability conditions for synchronized dynamics and studying Laplace-type interactions.Source: INTRODUCTION · p. 1
  • The framework is also presented as a blueprint for generalizing dynamical structures and results from graphs to hypergraphs.Source: INTRODUCTION · p. 2

SETTING: STABILITY FOR SYSTEMS OF ODES

The section frames linear stability through the evolution of perturbations around a solution. It distinguishes time-independent steady states from synchronized states, and focuses its subsequent analysis on synchronization.

INTERACTION ON NETWORKS

For graph networks, stability analysis reduces pairwise-coupled dynamics to network spectral modes under tractable coupling assumptions. In the Laplacian case, synchronization depends jointly on node dynamics and the network spectrum.

INTERACTION ON NETWORKS

  • The analysis focuses on pairwise neighbor interactions and simpler coupling subclasses, since general systems lack enough mathematical structure for a general stability result.These subclasses are also common in modeling, including neuroscience and synchronization problems.Source: INTERACTION ON NETWORKS · p. 2
  • Assuming a diagonalizable coupling matrix decomposes linearized perturbations into eigenmodes; the constant eigenvector defines the synchronization mode, and stability requires decay of all other modes.For f = h, each mode evolves with coefficient (1 + μ_k)Df(x∗).Source: INTERACTION ON NETWORKS · p. 3
  • Even when node dynamics are unstable (ℓ_f > 1), synchronization may be stable if the network eigenvalues satisfy the stability condition.The condition links the dynamics to the coupling matrix through its eigenvalues.Source: INTERACTION ON NETWORKS · p. 3
  • For the normalized Laplace operator, the stability condition requires |1 − σλ_k|ℓ_f < 1 for every mode indexed k = 2, . . . , N, while the eigenfunction for λ_1 = 0 is constant.Source: INTERACTION ON NETWORKS · p. 3
  • The first nonzero and largest Laplacian eigenvalues jointly constrain coupling: disconnection gives λ₂ = 0, while bipartiteness gives λ_N = 2 and can support antiphase oscillations.Disconnected components may synchronize internally without synchronizing with one another.Source: INTERACTION ON NETWORKS · pp. 3–4

INTERACTION ON HYPERGRAPHS

The paper extends network dynamics to hypergraphs by modeling interactions over sets of vertices and deriving stability conditions using a hypergraph Laplacian. Its richer spectrum permits stable dynamics beyond synchronization, while requiring care about zero modes and the limits of linear stability analysis.

INTERACTION ON HYPERGRAPHS

  • Hyperedge interactions generalize pairwise graph interactions, with the number of arguments in each interaction function determined by hyperedge size.The system uses vertices and hyperedges that are sets of vertices.Source: INTERACTION ON HYPERGRAPHS · p. 4
  • Linearizing these dynamics uses an N × M incidence matrix indicating vertex membership in hyperedges.The paper introduces the incidence matrix before presenting the linearized system.Source: INTERACTION ON HYPERGRAPHS · p. 4
  • The tractable interaction class assumes each size-dependent function g_m is normalized and symmetric, with the total incident-hyperedge count constant across vertices.Source: INTERACTION ON HYPERGRAPHS · p. 4
  • The chemical hypergraph Laplacian is orientation-invariant and reduces to the graph Laplacian for an oriented graph edge.Chemical hypergraphs with only inputs or only outputs correspond to unoriented classical hypergraphs.Source: INTERACTION ON HYPERGRAPHS · pp. 4–5
  • The hypergraph stability condition couples the dynamics’ Lyapunov exponent to the hypergraph Laplacian eigenvalues and coupling parameter.The Laplacian replaces the graph Laplacian in the stability condition.Source: INTERACTION ON HYPERGRAPHS · p. 5
  • A connected hypergraph may have multiple zero eigenvalues, allowing stable zero-mode dynamics beyond local synchronization; conversely, an all-positive spectrum can preclude synchronization.In the described example, a zero-mode state assigns activity at one vertex equally across two others.Source: INTERACTION ON HYPERGRAPHS · p. 5
  • The largest hypergraph Laplacian eigenvalue has no general upper bound of 2; it can equal N, and the stated bipartite comparison has equality only for bipartite hypergraphs.The comparison fixes the number of hyperedges and the number of inputs and outputs in each hyperedge.Source: INTERACTION ON HYPERGRAPHS · p. 5
  • A master stability condition applies when the hypergraph Laplacian enters the dynamics directly, but the conditions concern linearization; locally nonlinear bifurcations are more involved.The scheme also applies to time-discrete dynamics.Source: INTERACTION ON HYPERGRAPHS · p. 5

CONCLUSION AND OUTLOOK

The paper extends the master stability framework from graphs to hypergraphs and identifies how hypergraph spectral properties alter stability conclusions. The richer spectrum includes stable dynamics beyond synchronization, while the proposed strategy offers a way to generalize graph results.

CONCLUSION AND OUTLOOK

  • The framework extends master stability analysis to hypergraphs, where Laplacian spectral properties shape stability conditions.Source: CONCLUSION AND OUTLOOK · p. 5
  • The richer hypergraph spectrum can produce stable dynamics beyond synchronization, including cases with multiple zero eigenvalues or a positive smallest eigenvalue.Source: CONCLUSION AND OUTLOOK · p. 6
  • The proposed generalization strategy replaces graph adjacency or Laplacian roles with analogous hypergraph objects.Source: CONCLUSION AND OUTLOOK · p. 6

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