  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.
