Emergent synchrony in oscillator networks with adaptive arbitrary-order interactions

TL;DR

The paper addresses synchronization in networks with higher-order interactions by formulating an adaptive Kuramoto model that includes arbitrary-order hyperedge coupling. It develops an exact Ott-Antonsen reduction to obtain low-dimensional order-parameter dynamics in the thermodynamic limit and identifies how adaptation functions and phase lags shape synchronization and phase transitions, including explosive transitions and bistability. Numerical simulations on a pairwise-plus-triad system validate the analytical predictions and reveal a rich repertoire of states, including steady, unsteady, and fluctuation-driven transitions, with finite-size effects enabling behavior not seen in the infinite limit. The findings offer insights for applications in epilepsy and information diffusion on social networks and point to future work on empirical hypergraph connectivities and data-driven parameter estimation.

Abstract

Dynamics of complex systems are often driven by interactions that extend beyond pairwise links, underscoring the need to establish a correspondence between interpretable system parameters and emergent phenomena in hypergraph-based networks. The current work formulates an adaptive Kuramoto model that incorporates hyperedges of arbitrary order and explores their effects on synchronization. By deriving the exact order parameter dynamics in the thermodynamic limit, analytical expressions governing the collective dynamics are obtained. Subsequent numerics confirm the analytical predictions, in addition to capturing qualitatively different dynamical regimes and phase transitions. Further investigations based on order parameter distributions demonstrate how fluctuations, arising due to finite system size, can influence the long-term system dynamics. These results provide important insights and can have diverse applications, such as designing optimal surgical procedures for drug-resistant epilepsy and identifying the sources of rumours in a social network.

Figures (6)

• Figure 1: (a): Schematic representation of hyperedges in the brain: In (i), pink circles indicate nodes, or Broadman areas of functional significance, whereas the lines (edges) and coloured regions (hyperedges) denote pairwise and higher-order interactions, respectively. These can appear at multiple spatial scales, such as macroscopic functional connectivity patterns from fMRI-BOLD time series and microscopic neuronal scale, shown in (ii) and (iii), respectively; (b): Number of interactions in a brain network, as a function of its order, relative to pairwise interactions; see Refs. karmelic2022emergentcabral2022metastable for more details; (c): Variation of intracranial (iEEG) signals from different channels bougou2025mesoscaledataset, with the histogram (top-inset) and time series (bottom-inset) of phase synchrony, highlighting its increase during epilepsy, with its onset marked with a vertical line.
• Figure 2: Plot of $f_d(r)$ as a function of $r\in[0,1]$ for (a) $d=1$ and (b) $d=2$, where the colours denote different values of $\gamma$.
• Figure 3: (a)-(d): Left: Graphical solution of Eq. \ref{['final-r-eqn']}. The black and magenta curves denotes the LHS and RHS, respectively, whereas the green and red broken vertical lines (in inset) denote the stable and unstable steady-state solutions respectively, as determined from the eigenvalues of the Jacobian of Eqs. \ref{['ssh1']} and \ref{['ssh2']}; Right: Temporal variation of order parameter $r(t)$. The blue and red curves denote the asynchronous and synchronous initial conditions, respectively, whereas the horizontal line denotes $r_s$. For all figures, $\omega_0=1.5$, $\gamma=0.2$, and $\epsilon_2=5.0$.
• Figure 4: (a)-(j): Variation of the order parameter $r$ as a function of $E=\epsilon_1/\epsilon_2\in[0,2]$ for different values of $\gamma$ (top-row; $\epsilon_2=5.0$), and $\epsilon_2$ (bottom-row; $\gamma=0.2$). The vertical line corresponds to the forward transition point (i.e., $E^c=\epsilon_1^c/\epsilon_2$), whereas the blue and red curves denote the forward and backward variation of $E$, respectively, whereas $\omega_0=1.5$ across all figures.
• Figure 5: (a)-(d): Numerically constructed distribution of the order parameter, i.e., $p(r)$, collapsed across time, for $N=50$, and different values of $\epsilon_1$. The blue and red curves denote completely asynchronous and synchronous initial conditions, respectively, whereas the vertical line denotes $r_s$. For all figures, $(\omega_0,\epsilon_2,\gamma)=(1.5,5.0,0.2)$.