Abstract
Collective behavior plays a key role in the function of a wide range of physical, biological, and neurological systems where empirical evidence has recently uncovered the prevalence of higher-order interactions, i.e., structures that represent interactions between more than just two individual units, in complex network structures. Here, we study the optimization of collective behavior in networks with higher-order interactions encoded in clique complexes. Our approach involves adapting the Synchrony Alignment Function framework to a new composite Laplacian matrix that encodes multi-order interactions including, e.g., both dyadic and triadic couplings. We show that as higher-order coupling interactions are equitably strengthened, so that overall coupling is conserved, the optimal collective behavior improves. We find that this phenomenon stems from the broadening of a composite Laplacian's eigenvalue spectrum, which improves the optimal collective behavior and widens the range of possible behaviors. Moreover, we find in constrained optimization scenarios that a nontrivial, ideal balance between the relative strengths of pair-wise and higher-order interactions leads to the strongest collective behavior supported by a network. This work provides insight into how systems balance interactions of different types to optimize or broaden their dynamical range of behavior, especially for self-regulating systems like the brain.
Highlights
Complex networks provide the structural architecture for dynamical processes from a wide array of disciplines, and their study constitutes an important fundamental area of research in physics, mathematics, biology, and engineering [1,2,3]
To quantify the optimal collective behavior supported by a given network structure with higher-order interactions, we introduce a composite Laplacian matrix, which encodes the collective dynamics and network structure at multiple orders in a weighted simplicial complex and generalizes the synchrony alignment function (SAF) framework [36] to this case
To explain and further illustrate the improvement that occurs in collective network dynamics as a result of increased higher-order interactions, we investigate the spectral properties of the composite Laplacian L = (1 − α)L(1) + αL(2)
Summary
Complex networks provide the structural architecture for dynamical processes from a wide array of disciplines, and their study constitutes an important fundamental area of research in physics, mathematics, biology, and engineering [1,2,3]. In addition to typical pairwise or dyadic interactions in network-coupled systems, recent work points to the presence of higher-order, e.g., triadic, interactions in both brain networks [17,18,19,20,21] and generic limit-cycle oscillator systems [22,23]. To quantify the optimal collective behavior supported by a given network structure with higher-order interactions, we introduce a composite Laplacian matrix, which encodes the collective dynamics and network structure at multiple orders in a weighted simplicial complex and generalizes the synchrony alignment function (SAF) framework [36] to this case. We close by exploring a realistic constrained optimization problem where local dynamics are not freely tunable, but must be allocated from a predefined set, and it is revealed that a network’s ideal configuration is realized by a nontrivial, critical balance between the strength of dyadic and triadic interactions
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