Multiorder Laplacian for synchronization in higher-order networks

Abstract

The emergence of synchronization in systems of coupled agents is a pivotal phenomenon in physics, biology, computer science, and neuroscience. Traditionally, interaction systems have been described as networks, where links encode information only on the pairwise influences among the nodes. Yet, in many systems, interactions among the units take place in larger groups. Recent work has shown that the presence of higher-order interactions between oscillators can significantly affect the emerging dynamics. However, these early studies have mostly considered interactions up to four 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 end, we introduce a multiorder 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 multiorder 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 multiorder Laplacian allows us to obtain a complete analytical characterization of the stability of synchrony in arbitrary higher-order networks, paving the way toward a general treatment of dynamical processes beyond pairwise interactions.

Phase transitions and stability of dynamical processes on hypergraphs

“…While f i (x i ) represents the internal dynamics of the node, g j ðx fe j g Þ is the external interaction, expressed as the hyperedge contribution. Linear stability analysis for higher-order systems has been explored both in simplicial complexes 24,25 , where the mutual inclusion constrain imposes a limit on the dynamics, and for particular dynamics with a specific Laplacian for hypergraphs 23 . Despite these efforts, a general theory of dynamical processes on higher-order structures is still largely missing.…”

“…Admittedly, the attention to higher-order systems has proliferated lately 5,6,8,[10][11][12][13][14][15][16][17][18][19][20] , with an increasing focus on hypergraphs in fields such as mathematics 6,12,13,15,16,21,22 , physics 8,10,11,[17][18][19][23][24][25] , and computer science [26][27][28][29][30][31][32][33][34] . In many fields, the dynamical systems approach appears naturally.…”

“…8,17 , where firstand second-order transitions and hysteresis were found. Along the similar lines, the problems of evolutionary game theory 18 , synchronization [23][24][25] , and random walks 19,33,34 were also explored. Each field's focus is different, but it is often useful to use an abstract model that has a direct physical interpretation.…”

Temporal properties of higher-order interactions in social networks

“…For instance scientific authors naturally team up in larger groups to complement the expertise of different members 22 , neurons send and receive stimuli from multiple adjacent partners at the same time 21 , 23 , and the stability of large ecosystems relies on mutual and cooperative partnerships often involving three or more species 24 , 25 . Besides, higher-order interactions were shown to significantly modify the collective behavior of many dynamical processes, from diffusion 26 , 27 and synchronization 28 – 30 to spreading 31 , 32 , social dynamics 33 , 34 and games 35 . For a thorough introduction on the structure and dynamics of these higher-order systems, we refer the interested reader to the comprehensive overview provided in Ref 19 .…”

Stability of synchronization in simplicial complexes

“…A different approach has been proposed by Millán et al, who have formulated a higher-order Kuramoto model in which the oscillators are placed not on the nodes but on higher-order simplices, such as links, triangles, and so on, of a simplicial complex 55 . Finally, Lucas et al have considered an extension of the Kuramoto model to high-order interactions of any order, which is still analytically tractable because all the oscillators have identical frequencies 56 .…”