Abstract
Networks in nature have complex interactions among agents. One significant phenomenon induced by interactions is synchronization of coupled agents, and the interactive network topology can be tuned to optimize synchronization. Previous studies showed that the optimized conventional network with pairwise interactions favors a homogeneous degree distribution of nodes for undirected interactions, and is always structurally asymmetric for directed interactions. However, the optimal control on synchronization for prevailing higherorder interactions is less explored. Here, by considering the higherorder interactions in a hypergraph and the Kuramoto model with 2hyperlink interactions, we find that the network topology with optimized synchronizability may have distinct properties. For undirected interactions, optimized networks with 2hyperlink interactions by simulated annealing tend to become homogeneous in the nodes’ generalized degree. We further rigorously demonstrate that for directed interactions, the structural symmetry can be preserved in the optimally synchronizable network with 2hyperlink interactions. The results suggest that controlling the network topology of higherorder interactions leads to synchronization phenomena beyond pairwise interactions.
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Introduction
Complex interactions are ubiquitous in physical^{1,2}, biological^{3,4}, and social systems^{5,6}. The interactions form a complex network of the coupled agents. While systems with coupled agents can be modeled by networks with pairwise interactions^{7,8}, where nodes of the network are connected by links, higherorder interactions are prevailing in various systems, including the network of neurons^{9,10}, the contagion network^{11,12}, and social networks^{13}. An emerging direction in network science started to uncover the significance of higherorder interactions^{14,15,16,17,18,19,20}, which induce diverse phenomena beyond pairwise interactions.
Synchronization is one of the remarkable behaviors for coupled agents^{21,22,23,24}. Previous studies revealed that synchronization depends on the topology of the network with pairwise interactions. For the undirected interaction, the more synchronizable networks of identical coupled agents tend to be homogeneous in the nodes’ degree distribution^{25,26}. For the directed interaction, except for the fullyconnected network, the optimal network in synchronizability is always structurally asymmetric^{27}. Different from the pairwise interaction, higherorder interactions induce intriguing effects on synchronization^{28,29,30,31}. However, except for the specific network structure, e.g., starclique topology^{28}, how the network topology of higherorder interactions affects synchronization has been seldom explored. Then, the question raises: for optimal synchronization, do the conclusions on the network topology with pairwise interaction^{25,26,27} still hold when higherorder interactions are present?
In this paper, we investigate the effect of higherorder network topology on the phase synchronization of coupled oscillators on a type of hypergraph. By treating cycles on conventional networks as hyperlinks, a corresponding hypergraph can be obtained^{14,32}. Therein, higherorder interactions can be formulated as higherorder hyperlinks^{15,29,30,31}, such as 1hyperlink of two nodes (firstorder interaction, pairwise interaction), 2hyperlink of three nodes (secondorder interaction), etc (Fig. 1a). The higherorder interactions from hyperlinks considered here are similar to the simplex interactions in^{28}, and are different from simplicial complexes^{33} or the multilayer network^{34}. To analyze the network topology for optimal synchronization, we consider the Kuramototype coupling function for identical phase oscillators, focus on 2hyperlink interactions, and search for the optimal network in synchronizability. Through analytical treatments and numerical estimations, we find that 2hyperlink interactions can lead to distinct properties of the optimized networks compared with 1hyperlink interactions.
For the undirected interaction, we rewire 2hyperlink interactions and use simulated annealing to optimize synchronizability by minimizing the eigenratios of the generalized Laplacian matrices^{28,30}. Similar to the conclusion for 1hyperlink interactions^{25,26}, the optimized networks with 2hyperlink interactions become homogeneous in the generalized nodes’ degree. For the directed interaction, we provide an example of optimally synchronizable network with directed 2hyperlink interactions that preserves structural symmetry (in the sense that each node has the same number and same type of higherorder interactions). We rigorously demonstrate that the optimally synchronizable network with higherorder interactions can be symmetric, which is different from the result for 1hyperlink interactions^{27}. Still, the optimally synchronizable directed networks found are typically asymmetric by further numerical optimizations. Overall, the present result uncovers that the properties of synchronizable networks with pairwise interactions may or may not hold for higherorder interactions, indicating that novel behaviors can emerge in higherorder networks.
Results
Synchronizability of coupled phase oscillators with higherorder interactions
In this section, we first present the Kuramoto model with higherorder interactions. The generalized Laplacians are introduced by linearizing the Kuramoto model to study synchronization. We then focus on the case with 2hyperlink interactions to demonstrate the effect of higherorder interactions on synchronization.
The model and generalized Laplacians for higherorder interactions
We first present the formulation of the coupled oscillator system to study synchronization with higherorder interactions and the generalized higherorder Laplacians. We consider the Kuramototype model with the following set of ordinary differential equations for N interacting phase oscillators (N is also the network size):
where θ_{i} ∈ [0, 2π) is the onedimensional state variable (the phase) for the ith oscillator, f describes the local dynamics, K_{d} (d = 1, 2, …, D; D ≤ N − 1) are the coupling constants. Synchronization of the oscillators’ phases is under consideration, which can be extended to state’s synchronization by the master stability analysis^{30}.
For each order d, \({a}_{i{l}_{1}\ldots {l}_{d}}^{(d)}\) are adjacency tensors. For example, the firstorder interaction (1hyperlink) has the conventional adjacency matrix: \({a}_{i{l}_{1}}=1\) if the oscillators (i, l_{1}) have a pairwise interaction and 0 otherwise; the secondorder interaction (2hyperlink) has \({a}_{i{l}_{1}{l}_{2}}=1\) if the oscillators (i, l_{1}, l_{2}) have a 2hyperlink interaction and 0 otherwise; etc. The interactions are undirected if the adjacency tensors are invariant under all permutation of indices^{28}, and correspondingly they are directed if such invariance does not hold, i.e., the adjacency tensors are variant under some permutation of indices.
The functions g^{(d)} are coupling functions for synchronization, which is assumed to be noninvasive (g^{(d)}(θ, θ, …, θ) = 0 for ∀ d)^{31}. The Kuramoto type of coupling functions^{21,22,35} have \({g}^{(1)}({\theta }_{i},{\theta }_{{l}_{1}})=\sin ({\theta }_{{l}_{1}}{\theta }_{i})\), \({g}^{(2)}({\theta }_{i},{\theta }_{{l}_{1}},{\theta }_{{l}_{2}})=\sin ({\theta }_{{l}_{1}}+{\theta }_{{l}_{2}}2{\theta }_{i})\), …, \({g}^{(D)}({\theta }_{i},{\theta }_{{l}_{1}},\ldots ,{\theta }_{{l}_{D}})=\sin (\mathop{\sum }\nolimits_{d = 1}^{D}{\theta }_{{l}_{d}}D{\theta }_{i})\). Then, the master stability equation^{30,36} only depends on the adjacency tensors, i.e., generalized Laplacians, as the Jacobian terms in the master stability equation are constant. Besides, the present coupling constant K_{d} can be denoted by the coefficients in^{28} as K_{d} = γ_{d}/(d〈K^{(d)}〉).
Based on the linearized equation in the master stability analysis (Methods), the generalized Laplacian matrix of the dorder interaction can be defined as^{28}:
where \({k}_{ij}^{(d)}\) is the generalized dorder degree between the nodes i, j, i.e., the number of dhyperlink between i, j, and \({k}_{i}^{(d)}\) is the generalized dorder degree of node i. Note that for higherorder cases (d > 2), the Laplacian here is different from another definition of the Laplacian Eq. (38). A detailed comparison on the various definitions of the generalized Laplacians^{28,30,31} is in Methods. Further, we use the adjacency tensors to represent higherorder interactions, and employ higherorder Laplacians defined from adjacency tensors. Alternatively, higherorder interactions can be introduced by the boundary matrix acting on simplicial complexes^{33,37}, which leads to a different way to define higherorder Laplacians.
To study the dependence of synchronizability on the network structure with higherorder interactions, we focus on the Kuramoto type of coupling function for the coupled oscillator. Then, the master stability equation, Eq. (34), belongs to the case of Eq. (15) in^{30}. As noted below Eq. (15) there, the situation is conceptually equivalent to synchronization in networks with only pairwise interactions. The summation of higherorder Laplacians now plays the same role of the conventional Laplacian from pairwise interactions. Thus, synchronizability depends on generalized Laplacian matrices^{38,39}, and can be characterized by the eigenvalues of Laplacian matrices^{25,27,40}.
The case with 2hyperlink interactions
We demonstrate the higherorder effect by studying 2hyperlink interactions, i.e, the interaction among three nodes (d = 2). We further consider the identical oscillators, i.e., each oscillator has an identical frequency^{27}, where the function f(θ) = ω in Eq. (5), with ω denoting the natural frequency of the oscillators. Then,the dynamical equation becomes (Methods):
Its linearized synchronization dynamics is:
Synchronizability is determined by the secondorder Laplacian matrix:
With 2hyperlink interactions only, all the definitions of the generalized Laplacians^{28,30,31} are the same (Methods). Then, Eq. (6) can be rewritten as:
In the next sections, we will separately study the optimized undirected and directed interactions for synchronizability.
Optimized synchronizable networks with undirected 2hyperlink interactions
Optimizing synchronizability by the eigenratio of the generalized Laplacian
In this subsection, we present the framework to optimize synchronization of the network with undirected interactions. For the synchronized state of the system Eq. (5), we consider its bounded and connected stability region, where synchronizability of coupled oscillators can be quantified in terms of the eigenratio of Laplacian matrices Eq. (2), which was used mainly for networks with firstorder interaction^{25,27,40}. We optimize the networks with 2hyperlink interactions for the linearized system Eq. (6) to determine synchronizability.
Specifically, we calculate the eigenvalues of higherorder Laplacian matrices Eq. (2), which can be arranged as 0 = λ_{1} < λ_{2} ≤ ⋯ ≤ λ_{N}. Note that the eigenvalues are all real, as generalized Laplacians are symmetric. The smallest nonzero eigenvalue λ_{2} is known as the spectral gap. The eigenratio
quantifies synchronizability^{30}. By diagonalizing the Laplacian matrix Eq. (2), we can get its eigenvalues and the eigenratio.
We make remarks about the search for the optimally synchronizable networks. First, we focus on using 2hyperlink interactions to exemplify the effect of higherorder interactions in the optimized synchronizable network topology. The implementation can be extended to cases with higherorder interactions by a similar procedure, including the case with multiorder interactions by using the sum of higherorder Laplacians^{28}. Second, since we investigate the optimal network topology for synchronizability, we have considered an identical frequency for the oscillators. Under this case, the system displays global synchronization instead of cluster synchronization^{41,42}, and synchronizability is determined by the eigenratio.
Next, we provide the numerical protocol of optimizing networks with undirected 2hyperlink interactions. We start with various randomly initialized networks, rewire secondorder interactions with keeping a fixed number of 2hyperlinks, and numerically search for optimal networks by simulated annealing^{27} to minimize the eigenratio, Eq. (11) of the Laplacian matrix Eq. (7).
For the initialization, we randomly generate different networks with certain numbers of 2hyperlinks. For Nnode networks, there are \({C}_{N}^{3}=N(N1)(N2)/6\) combinations of three nodes, i.e., the number of 2hyperlinks. To demonstrate the optimization procedure, we initialize the network by first adding the 2hyperlink interactions for the nodes i, i + 1, i + 2(i = 1, …, N). It ensures that each node has at least one 2hyperlink interaction, such that the network does not have isolated nodes. Then, we randomly add N 2hyperlink interactions to the network, such that each realization of optimization starts with these N 2hyperlinks generated differently. Note that the 5node case by such an initialization is a fullyconnected network and is already optimal in synchronization. Therefore, when rewiring undirected interactions, we have chosen the minimal network size to be 6.
When rewiring the network, we delete randomly a fixed proportion of 2hyperlink interactions, such as 20% of the existing 2hyperlinks, and add the same number of 2hyperlink interactions to three randomly chosen nodes which did not have an interaction before. For the 2hyperlink interaction of the nodes l_{1}, l_{2}, l_{3} to be deleted, \({a}_{{l}_{i},{l}_{j},{l}_{k}}^{(2)}=0\) with i, j, k being all the permutations of 1, 2, 3. After each rewiring step, we calculate the eigenratio of the rewired network, and a Metropolis acceptreject step is used for the rewired network with smaller eigenratio^{27}. The chosen proportion of rewiring at each step would not dramatically affect the final optimization result, when the ratio of the rewiring was not too larger (e.g., ≥40%) and a sufficient number of optimization steps were conducted (Supplementary Fig. 1), because the eigenratios converge to a stable range during the optimization. However, the global minimum of eigenratios is not always guaranteed, and the numerical minimization may gradually lead to the global minimum with an expense of longer computation.
The optimization procedure is repeated until the eigenratio is smaller than a chosen target value, which is initially set as the smallest possible eigenratio 1. The target eigenratio cannot be too small, because otherwise it may not be reached due to the sparsity of 2hyperlink interactions (2N) when the number of nodes increases. We thus increase the target eigenratio by 10% if the number of rewiring steps runs over 100 times without reaching the current target eigenratio. This procedure automatically increases the target eigenratio, to reduce the computational time of searching for too small eigenratio that may not be achieved for the sparse large networks. It still ensures to optimize synchronizability by minimizing the eigenratio.
The above completes one realization of optimization, and 1000 realizations are conducted with various configurations of initialized networks. In total, we have two hyperparameters about the iteration. The first is the maximum number of iterations to reduce the eigenratio to the target value (100 times) before increasing the target value. The second is the number of realizations (1000), i.e., the number of different initialized networks. The computational time increases with these two hyperparameters, and also increases with the number of nodes. When operating on a personal desktop computer, the computation time can be hours when the network size exceeds 100. Larger values of these two hyperparameters can be used to enable the search for smaller eigenratios, with a cost of more computational resources.
The network is rewired instead of simply deleting the 2hyperlink interactions, because after deleting the 2hyperlinks the eigenratio may be similar while all eigenvalues continue to become smaller. It eventually leads to an optimal network with much fewer 2hyperlink interactions than the initial network. The optimization with only adding 2hyperlink also gives less constrained network structures. Thus, we choose to rewire the network by adding the same number of 2hyperlink interactions after the deletion, which preserves the total number of 2hyperlink interactions. Different types of constraints can be employed in the optimization procedure, to search for the optimized network with desired properties.
Optimizing synchronizability of a 6node network
In this subsection, we give an example with its secondorder adjacency tensor and generalized Laplacian to exemplify the network with 2hyperlink interactions. We further demonstrate the optimization procedure by this example.
Specifically, we consider the network with 6 nodes and 8 2hyperlinks, and randomly initialize the 2hyperlink interactions. It has in total 8 2hyperlinks connecting the nodes:
Its generalized Laplacian matrix L^{(2)} by Eq. (7) is:
The eigenvalues of the generalized Laplacian matrix in ascending order are 0, 6.171, 8.167, 10.000, 10.549, 13.111, and the eigenratio is 2.125.
We next optimize synchronizability of this example. In each step of the optimization, one or two 2hyperlinks may be rewired to generate a network with a smaller eigenratio. An illustration is given in Fig. 1(b). After conducting the numerical optimization with preserving 6 nodes and 8 2hyperlinks, the resultant optimized network has the following 8 2hyperlinks connecting the nodes:
The generalized Laplacian matrix L^{(2)} by Eq. (7) is:
The eigenvalues in ascending order are 0, 8, 9, 9, 11, 11, and the eigenratio is 1.375.
We note that the optimized network may not be the best in synchronization until sufficient numerical search is conducted. However, with the present numerical optimization, this network is at least near to the optimal network in synchronizability as its eigenratio is close to 1.
The optimized networks with various sizes
In this subsection, we provide the optimization result for undirected networks with various sizes. Examples of the initial and rewired network are given in Fig. 2a, b, with the number of nodes N = 7, 8, 9, 10. It demonstrates that the optimization procedure rewires 2hyperlink interactions and reduces eigenratios (Fig. 2c). For illustration, we have shown networks with a small number of nodes. In Fig. 3a, the eigenratios before and after the optimization are provided, where the numbers of nodes include 6 and those from 10 to 100 with a step size 5. The optimized networks have smaller eigenratios after the optimization, showing better synchronizability. Indeed, directly simulating the Kuramoto dynamics in Eq. (5) validate that the optimized network has better synchronizability than the initial network (Supplementary Fig. 2).
With a fixed number of 2hyperlinks, the specific configuration of the optimized network does not dramatically affect the final eigenratio when conducting multiple numerical replicates, such as 10 times, on the network structures, as shown by the errorbars in Fig. 3b. The number of 2hyperlinks is a crucial factor in determining synchronizability. The numbers of initialized 2hyperlinks are 2N, which gives a sparse network and only allows a relatively large eigenratio after the optimization. The eigenratios are smaller when the number of 2hyperlinks increases to improve synchronizability, i.e., larger network densities lead to relatively smaller eigenratios (Supplementary Fig. 3). Specifically, besides the density of 2hyperlinks 2N in Fig. 3, we used a different number of triangles, e.g., 3N, 4N, 5N in Supplementary Fig. 3. The eigenratios of initialized networks become smaller with the increasing density, because more links improve the synchronizability of networks. At the same time, there is less room to rewire the network for optimizing synchronizability if the number of 2hyperlinks becomes abundant. Under the chosen densities, our numerical algorithm can still find better synchronizability and reduce the eigenratio by rewiring the networks. In addition, the optimized networks under various densities also tend to have more homogeneous nodes’ degrees.
Different ways of initialization can be implemented to investigate the dependence on the initialized networks. For example, one may generate random hypergraphs with 2N 2hyperlinks and only use those connected hypergraphs to do the optimization. The results in Supplementary Figs. 4, 5 show a same qualitative conclusion as Figs. 2, 3. This indicates that the specific way of initialization does not affect the optimization result, once sufficient rewiring steps are conducted during the optimization.
We have further calculated the generalized degree \({k}_{i}^{(2)}\) in Eq. (8) of each node, which quantifies the number of 2hyperlinks participated by each node. The distribution of the generalized degree for the optimized networks with various sizes is in Fig. 4a–f. When considering a identical frequency distribution of oscillators, the optimal network tends to be more homogeneous in the nodes’ degree for the network with firstorder interactions^{25,26}. Similarly, the optimized network with 2hyperlink interactions also becomes more homogeneous, as the nodes’ degree distribution concentrates to fewer values of degree in Fig. 4. We further calculate the homogeneity parameter^{43,44,45,46}:
where κ is the mean degree of the network defined as \(\kappa \equiv {N}^{1}\mathop{\sum }\nolimits_{i}^{N}{k}_{i}^{(2)}\) for the 2hyperlink interactions and σ^{2} is the variance of the nodes’ degrees. Note that g becomes a delta distribution for fully homogeneous network, and tends to 0 for networks with more heterogeneous degree distributions. The homogeneity parameter for rewired network is small compared with the initial network (Fig. 4), showing that the rewired networks are more homogeneous. Other measures such as generalizing the clustering coefficient^{47}, degreedegree correlations^{43}, characteristic path length and heterogeneity measures^{48} to highorder networks can help quantify and control synchronizability of the network.
Optimized synchronizable networks with directed 2hyperlink interactions
In this section, we study directed 2hyperlink interactions to demonstrate the effect of directed higherorder interactions. We focus on the relation between synchronizability and structural symmetry of the network^{27}. A network is regarded as structurally symmetric when each node has the same number and same type of (higherorder) interactions.
The optimized directed network with only firstorder interactions has been studied^{27}: it has been proved that the optimally synchronizable directed network is always structurally asymmetric (except for the fullyconnected network). The proof was done by establishing a contradiction showing that the structurally symmetric network cannot be optimal in synchronizability. However, whether this conclusion holds for networks with higherorder interactions remains unknown.
For a higherorder directed network, we find that the symmetry may hold for the network with directed higherorder interactions, i.e., there are structurally symmetric networks which are optimal in synchronizability. Intuitively, higherorder interactions provide more capacity to reach optimal network design, enabling optimally synchronizably directed networks to preserve structural symmetry.
Before presenting the result, we recapitulate the definition of the directed hyperlinks. The directed 2hyperlinks are defined such that each permutation of three nodes leads to a distinct 2hyperlink’s direction: a 2hyperlink interaction is “directed” once the tensor \({a}_{i,j,k}^{(2)}\), with i, j, k being all the permutations of 1, 2, 3, has one of its six elements to be nonidentical. Then, each \({a}_{i,j,k}^{(2)}\) with i, j, k having a specific order can be regarded as a directed hyperedge, i.e., an ordered pair of disjoint subsets of vertices^{49}. The ordering i, j, k specifies the direction, as illustrated in Fig. 5a. For example, with \({a}_{i,j,k}^{(2)}=1\), i is a source node while j, k are its target nodes, and j is a source node while k is its target node. By using the adjacency tensors for directed interactions, the linearized synchronization dynamics is also given by Eq. (6), and synchronizability depends on eigenvalues of the generalized Laplacian matrix in Eq. (7).
An example of optimally synchronizable network with structural symmetry
First, we demonstrate that a network with N = 7 nodes and a symmetric structure can be optimal in synchronizability. As illustrated in Fig. 5(b), the network has the 3 × 2 × 7 directed 2hyperlinks given by the following nonzero elements in the adjacency tensor:
Other elements in the tensor a^{(2)} are zero.
From this adjacency tensor, its generalized Laplacian matrix by Eq. (7) is:
The Laplacian has all nonzero eigenvalues identical as 7 and eigenratio 1.
This network with 2hyperlink interactions has the structural symmetry, as each node has the same number and type of 2hyperlink interactions: each line of the 2hyperlinks in Eq. (17) belongs to the same type of 2hyperlink for the first node (source node). Therefore, it is a counterexample of only the structurally asymmetric network being optimal in synchronizability. Higherorder interactions make it possible to have an optimally synchronizable network with symmetry. We expect that larger networks can have more symmetric optimal structure.
The directed 2hyperlink interactions enable structural symmetry
The unweighted higherorder network is most synchronizable when the real eigenratio (Eq. (21)) is the smallest. That is, the nonzero eigenvalues of the Laplacian matrix satisfy λ_{2} = λ_{3} = ⋯ = λ_{N} when the network is most synchronizable^{40}. For pairwise interactions, this condition implies that the eigenvalues of the optimally synchronizable network are all real, such that they can be ordered, even though the directed networks generally have complex eigenvalues. This property can be extend to the higherorder cases with using generalized Laplacian^{30}. By a similar procedure in the Supplementary of^{40}, we found that the identical eigenvalues are integers as follows. Specifically, we first define \(\bar{\lambda }\equiv \mathop{\sum }\nolimits_{i = 2}^{N}{\lambda }_{i}/(N1)\), and \(\bar{\lambda }={\lambda }_{2}={\lambda }_{3}=\cdots ={\lambda }_{N} \, > \, 0\) when the network is most synchronizable. The characteristic polynomial of the generalized Laplacian is: \(\det ({L}^{(2)}\lambda I)=\lambda {(\lambda \bar{\lambda })}^{N1}={(1)}^{N}{\lambda }^{N}+\cdots {\bar{\lambda }}^{N1}\lambda\), where I is the identity matrix. As L^{(2)} has all integer entries, the coefficients of the characteristic polynomial should be all integers, and thus \(C\equiv {\bar{\lambda }}^{N1}\) is an integer. According to the definition of the generalized Laplacian L^{(2)} in Eq. (7), \(tr({L}^{(2)})=l=(N1)\bar{\lambda }\), where l denotes the number of elements 1 in the adjacency tensor a^{(2)}. Then, C = [l/(N−1)]^{N−1} ≡ (s/t)^{N−1}, where integers s and t do not have common factors. For Ct^{N−1} = s^{N−1}, any prime factor p of C must be a factor of s^{N−1} and consequently a factor of s. Since there are no common factors in s and t, p^{N−1} needs to be a factor of k. Therefore, any factor of C has multiplicity N − 1, giving C = q^{N−1} with an integer q. This leads to \({\bar{\lambda }}^{N1}={q}^{N1}\) and that \(\bar{\lambda }\) is an integer.
By using the above properties of the eigenvalues to establish a contradiction, the authors^{27} found that an optimally synchronizable network with firstorder interaction (except for the fullyconnected network) must be structurally asymmetric. Below, we provide an attempt to establish such a contraction for the network with directed 2hyperlink interactions, and find that the contraction can no longer be established.
On the one hand, in a symmetric network, the nodes are structurally identical, i.e., each node has the same number and type of higherorder interactions. It implies that the indegrees and outdegrees from the 2hyperlink interactions of all nodes must be equal. Thus, l needs to be divisible by N if the network is symmetric. As an example, for the network with 4 nodes and 2hyperlink interactions, the condition of structural identity requires each node to have the same degree and the same number of 2hyperlinks. Then, this 4node case needs to have a fullyconnected network, with all the 4 2hyperlinks and l = 12 divisible by 4.
On the other hand, when the network is optimal in synchronizability, the eigenvalues are integers and equal: \({\lambda }_{2}={\lambda }_{3}=\cdots ={\lambda }_{N}=\bar{\lambda }\), and \(tr({L}^{(2)})=l=(N1)\bar{\lambda }\). These properties imply that \(\bar{\lambda }=l/(N1)\) and that l must be divisible by N − 1 if the network is most synchronizable. Therefore, l must be divisible by (N − 1)N and then
For the network with 2hyperlink interactions,
The two conditions can be satisfied simultaneously when (N − 1)N ≤ l ≤ (N − 2)(N − 1)N. It no longer constrains the network to be fullyconnected as the case of the network with only firstorder interactions^{27}. Thus, the contradiction that the structurally symmetric network is not optimal in synchronizability cannot be reached for the network with 2hyperlink interactions.
We remark that our proof can be extended to higherorder interactions. For example, the 3hyperlink would have l ≤ (N − 3)(N − 2)(N − 1)N in Eq. (20). Then, following the same procedure of proof, the 3hyperlink also allows the structurally symmetric network to be optimal in synchronizability, different from the firstorder interaction.
Optimizing synchronizability by the real eigenratio of the generalized Laplacian
Generalized Laplacians for higherorder interactions are typically not structurally symmetric, and lead to complex eigenvalues. The eigenvalues of the secondorder Laplacian L^{(2)} can be listed in ascending order of their real parts: λ_{1}, λ_{2}, λ_{3}, …, λ_{N}. In the strong coupling regime, since the stability region of the fullysynchronized state is bounded and connected, synchronizability can be quantified by an eigenratio of the real part^{27}:
where ℜ denotes the real part of the complex eigenvalues.
The network will be more synchronizable if this eigenratio is smaller. Note that here we can extend this property to the higherorder case, because generalized Laplacians play the same role on quantifying synchronizability when Eq. (6) has identical oscillators and specific coupling functions (see the discussion under Eq. (15) of^{30}). We numerically optimize synchronizability of the directed network by minimizing Eq. (21).
We next provide an example of optimized networks by the numerical optimization. The initial network has 6 nodes and the following 2 × 8 undirected 2hyperlinks:
The other elements in the adjacency tensor are zero.
By deleting directed 2hyperlinks, various configurations of the optimal network with R = 1 can be reached by the numerical optimization. For example, one optimized directed network from the numerical optimization has the nonzero elements in the adjacency tensor:
with other elements of a^{(2)} zero. Its generalized Laplacian matrix by Eq. (7) is:
The eigenvalues in ascending order of real parts are 0, 2, 2, 2, 2, 2, and the eigenratio is R = 1.
Optimized directed networks are structurally asymmetric in general
Though the counterexample shows that the optimal network can be structurally symmetric when higherorder interaction presents, it does not mean that optimal directed networks with higherorder interactions generally tend to be symmetric over asymmetric. We use simulated annealing to search for the optimized synchronizable network numerically. For the directed interaction, we separately delete and rewire the directed 2hyperlink interaction randomly to achieve better synchronizability. After removing or rewiring directed higherorder interactions to optimize synchronizability, the network typically becomes structurally asymmetric.
We first present the optimization by removing directed 2hyperlink interactions, such that the network can become directed and asymmetric. By removing directed 2hyperlinks, such as setting a_{1,2,3}, a_{2,3,1} or a_{3,1,2}, etc to zero, the network can have directed 2hyperlink interactions instead of undirected interactions. We employ simulated annealing for the optimization, where the input is a threedimensional tensors for a network with 2hyperlink interactions. We then calculate the generalized Laplacian matrix \({L}_{ij}^{(2)}\) for a given tensor \({a}_{{l}_{1},{l}_{2},{l}_{3}}\) by Eq. (7), modify the network and calculate Eq. (21) after each step of the modification. We have used the same procedure to set the target value in the previous section, where the target value is increased by 10% if the network modification runs over 100 steps. Then, 1000 realizations are conducted with different configurations of initialized networks.
We considered network sizes 6, 10, 20, 50, 80, 100. After optimizing synchronizability by modifying the network, we use two quantities to measures the asymmetry of the optimized network. First, we count the number of directed 2hyperlink interactions of each node, which measures the structural asymmetry from each 2hyperlink interaction for each node:
where A_{1} denotes the first asymmetry measure. Second, we measure the asymmetry on the directed inandout interaction for three nodes of each 2hyperlink interaction:
where A_{2} denotes the second asymmetry measure.
We estimate these asymmetry measures for each node, and then average them over all the nodes. The two asymmetric measures A_{1}, A_{2} of 30 numerical replicates are plotted as violin distributions in Fig. 6a, b. They show that the optimized networks are generally asymmetric, because most numerical replicates generate the optimized network with nonzero asymmetric measures. Note that these measures may not quantify structural asymmetry of having 2hyperlinks symmetrically in the network. We further evaluate the network density for the directed network in Fig. 6c, by the average value of the generalized nodes’ degree Eq. (8), showing the network densities for the optimized directed networks.
The phenomenon that asymmetry enhances synchronization is for the directed networks, which is not contradictory to optimal synchronization of fullyhomogeneous undirected networks^{14,25}. Besides, after deleting 2hyperlink interactions, the eigenvalues overall become smaller. For the optimized network obtained above, we have chosen the minimum number of deletions to reach the target eigenratio.
We have further rewired the directed 2hyperlink interactions without deletion (Supplementary Fig. 6). When the 2hyperlink interactions are rewired without being deleted, the asymmetric measures are larger as more interactions are kept. Regardless of the exact values of the asymmetric measures, the optimized directed networks in general are also structurally asymmetric.
Conclusion
In summary, our result demonstrates that higherorder interactions can lead to distinct properties of optimized synchronizability compared with the conventional network with pairwise interactions. For the undirected interaction, the more synchronizable networks tend to be homogeneous, consistent with pairwise interactions^{25,26}. For the directed interaction, the optimized synchronizable network is structurally asymmetric in general but can be symmetric, beyond the firstorder case^{27}. The optimization on synchronization of higherorder interactions may find uses in real networks, such as controlling the asynchronous state^{50} of brain networks with highorder structures^{9}.
Recent studies^{13,15} revealed significant roles of higherorder interactions in network science. In light of these studies, we investigate how higherorder interactions affect synchronization by optimizing network topology. Specifically, we have demonstrated the effect of higherorder interactions by focusing on the network topology of 2hyperlinks, considering the phase synchronization of coupled oscillators, and employing a Kuramoto type of coupling function. The cases with general coupling function can be studied by the master stability analysis^{30,31,36}, which can determine the stability of the synchronized state. Under that case, an interplay between the coupling function and the network topology needs to be analyzed. However, for a large class of coupled oscillator systems^{27}, where the stability region is bounded and connected, synchronizability can be quantified by the eigenvalues of the generalized Laplacian.
When calculating eigenvalues of the generalized Laplacian matrices, the analytical solvable cases are restricted to special networks^{28}, and numerical estimations are generally required. In the numerical implementation, the initialization on the network topology in Results ensures a sufficient number of 2hyperlinks to be optimized. On the other hand, abundant 2hyperlinks may cause an ineffectiveness of the optimization, because the network is already near to optimal synchronizability when the number of 2hyperlinks is abundant. Different types of initialized networks can be used to further improve the search of the optimal network, with a cost of longer computational time. Similarly, though the chosen network sizes are sufficient to show the distinct properties of higherorder networks compared with the pairwise interactions, applying our numerical package to larger networks is useful when more computational resources are available.
When oscillatory frequencies are heterogeneous, synchronizability depends on both network structure and oscillators’ frequencies to be optimized. For pairwise interactions^{26}, synchronization can be enhanced by a match between the heterogeneity of frequencies and network structure. For higherorder networks, one also needs to optimize the frequencies and the alignment function simultaneously. Here, we have focused on the network topology and considered identical oscillators. Besides, to study the network topology, we have treated the network as a single cluster, rather than a network with a subcluster coupled to other nodes^{27}.
We have used adjacency tensors to encode higherorder interactions, which can be formulated as simplicial complexes or hypergraphs^{51}. We consider the case with pure 2hyperlink interactions^{28}. Instead, simplicial complexes require collections of simplices^{15}. Extending the present result to simplicial complexes needs to include the various orders of simplices. To include multiorder interactions simultaneously depends on the coupling function. For general cases, there is an issue of diagonalizing the multiorder Laplacian matrices simultaneously^{31}, because the multiorder Laplacian matrices cannot be directly added up due to the coupling function. For the Kuramoto type of coupling function, the Laplacian matrices can be added up^{28}, and then the multiorder eigenvalues determine the stability and quantify synchronizability. The numerical implementation is extendable to higherorder interactions by using generalized Laplacian matrices in Eq. (2). Future work includes to investigate the optimal network design for the case with multiorder interactions^{52,53,54} and with attractive and repulsive terms^{28}.
To define the direction in higherorder networks, we have extended the definition of the directed network with pairwise interactions^{27} to the 2hyperlink case: the interactions are directed if the adjacency tensors are variant under the permutation of the indices. The direction of 2hyperlink is assigned in a same way as that of the triangle in directed simplicial complexes^{55}. Moreover, the direction of hyperedge in general hypergraphs needs to be carefully defined^{56}, and one needs to control hyperedge including their directions in the hypergraph. To extend the present result of 2hyperlink interactions to general directed hypergraphs is another interesting topic, which starts to be uncovered^{57}.
Methods
The master stability equation for the system of coupled oscillators
In this subsection, we present the method of master stability function^{36}, which can be used to obtain the synchronization phase diagram for general coupledoscillator systems Eq. (1). We consider the system with general coupling functions for the first and secondorder interactions, as higherorder interactions can be treated similarly^{30}.
In order to study the stability of the synchronization solution, we consider small perturbations around the synchronous state, i.e., \(\delta {\theta }_{i}={\theta }_{i}{\theta }_{i}^{{{{{{{{\bf{s}}}}}}}}}\), where θ^{s} denotes the steady state. We then perform a linear stability analysis on the vector δθ = (δθ_{1}, δθ_{2}, …, δθ_{N}). The master stability equation^{36} can be obtained as Eq. (11) of^{30}:
where ⨂ is the matrix Kronecker product, I_{N} is the ndimensional identity matrix and the Jacobian terms at the synchronized states are:
After diagonalizing the firstorder Laplacian matrix L^{(1)} by a linear coordinate transformation δθ → δη, the master stability equation becomes Eq. (13) of^{30}:
where i = 2, …, N denotes the different modes transverse to the synchronization manifold, λ_{i} are the eigenvalues of the Laplacian L^{(1)}, and \({\tilde{L}}^{(2)}\) is the transformed secondorder Laplacian L^{(2)}.
With the master stability equation Eq. (31), the master stability function Λ_{max}, i.e., the maximum transverse Lyapunov exponents (the modes except for the first stable mode), can be obtained by numerically tracking the norm of: \(\sqrt{\mathop{\sum }\nolimits_{i = 2}^{N}\mathop{\sum }\nolimits_{j}^{m}{[{\eta }_{i}^{(j)}]}^{2}}\), with \({\eta }_{i}\equiv ({\eta }_{i}^{(1)},{\eta }_{i}^{(2)},\ldots ,{\eta }_{i}^{(m)})\) solved from Eq. (31) under given parameters. It is beyond solely using the eigenvalue of the Laplacian matrices for the Kuramototype coupling function.
When considering the Kuramototype coupling function and the linearized coupling terms^{25}, the interaction term becomes variableindependent and eigenvalues of Laplacian matrix can quantify synchronization. Under this case, one could analyze eigenvalues of the generalized Laplacian matrices Eq. (2) to search for the optimally synchronizable network. Without the linearization, the coupling function is variabledependent. Then, we need to solve Eq. (31) to quantify synchronizability, which prohibits to efficiently search for optimal network structure. We thus consider the Kuramototype coupling function^{28} in the main text. As for the linearization, in the strong coupling regime, the system Eq. (5) will converge to a synchronized state with θ_{i} ≈ θ_{j} for any i, j. The linearized synchronization dynamics for the synchronized state is:
The linearized equation reduces to the same form of Eq. (2) in^{28}. We further use the rotating reference frame by θ_{i} = θ_{i} − ωt, where the synchronized solution is θ_{i}(t) = 0, (i = 1, …, N). Then, the master stability equation for Eq. (5) is:
which determines the linear stability of the synchronized state. This master stability equation is a case of Eq. (15) in^{30}. It leads to Eq. (5) in the main text, where only the 2hyperlink interaction is kept.
Generalized Laplacians for higherorder interaction
In this section, we compare the different definitions on generalized Laplacians for higherorder interactions. In short, though their definitions for higherorder interactions in^{28,30,31} may differ, the firstorder and secondorder Laplacians are the same. As this paper focuses on the topology of 2hyperlink interactions, the obtained results are valid for using any one of the generalized Laplacians defined in^{28,30,31}. For convenience, we have used Eq. (35) as in^{28} when presenting the generalized higherorder Laplacians in the main text.

The Laplacian in^{28} is given by their Eqs. (3, 4, 5):
$${L}_{ij}^{(d)}=d{k}_{i}^{(d)}{\delta }_{ij}{k}_{ij}^{(d)},$$(35)$${k}_{i}^{(d)}=\frac{1}{d!}\mathop{\sum }\limits_{{l}_{1}=1}^{N}\ldots \mathop{\sum }\limits_{{l}_{d}=1}^{N}{a}_{i,{l}_{1},\ldots ,{l}_{d}}^{(d)},$$(36)$${k}_{ij}^{(d)}=\frac{1}{(d1)!}\mathop{\sum }\limits_{{l}_{2}=1}^{N}\ldots \mathop{\sum }\limits_{{l}_{d}=1}^{N}{a}_{i,j,{l}_{2},\ldots ,{l}_{d}}^{(d)},$$(37)which is used in Eq. (2) of the main text. We list it here for the comparison with other definitions of Laplacians.

The generalized Laplacians for higherorder interaction terms defined in^{30} is by their Eq. (28):
$${L}_{ij}^{(d)}=\left\{\begin{array}{ccc}&0\hfill&i\ne j,\quad {a}_{ij}^{(1)}=0,\\ &(d1)!{k}_{ij}^{(d)}&i\ne j,\quad {a}_{ij}^{(1)}=1,\\ &d!{k}_{i}^{(d)},\hfill& i=j,\hfill\end{array}\right.$$(38)where \({k}_{ij}^{(d)}\) also denotes the generalized dorder degree of the nodes i, j and \({k}_{i}^{(d)}\) denotes the generalized dorder degree of node i. Therein, their Eqs. (24, 26) give:
$${k}_{i}^{(d)}=\frac{1}{d!}\mathop{\sum }\limits_{{l}_{1}=1}^{N}\ldots \mathop{\sum }\limits_{{l}_{d}=1}^{N}{a}_{i,{l}_{1},\ldots ,{l}_{d}}^{(d)},$$(39)$${k}_{ij}^{(d)}=\frac{1}{(d1)!}\mathop{\sum }\limits_{{l}_{1}=1}^{N}\ldots \mathop{\sum }\limits_{{l}_{d1}=1}^{N}{a}_{i,j,{l}_{1},\ldots ,{l}_{d1}}^{(d)}.$$(40)Here, d specifies the order, which gives a 3dimensional tensor when d = 2. Note that \({k}_{i}^{(d)}\), \({k}_{ij}^{(d)}\) here are the same as those in Eq. (35). However, the definitions of Laplacians are different: Eq. (35) multiplied by (d − 1)! becomes the same as Eq. (38). Still, for the first and second orders (d = 1, 2), the two definitions of Laplacians reduce to the same.

The generalized Laplacian for higherorder interactions used in^{31} are:
$${L}_{ij}^{(1)}={\delta }_{ij}\mathop{\sum }\limits_{k=1}^{N}{a}_{ik}^{(1)}{a}_{ij}^{(1)},$$(41)$${L}_{ij}^{(2)}=\mathop{\sum }\limits_{k=1}^{N}{a}_{ijk}^{(2)}\quad (i\,\ne\, j),\quad {L}_{ii}^{(2)}=\mathop{\sum}\limits_{j\ne i}{L}_{ij}^{(2)},$$(42)to retain the zero rowsum property. It is consistent with that in^{30} for the secondorder.
With using generalized Laplacians for the firstorder and secondorder cases, we obtain the optimized networks with 2hyperlink interactions.
Data availability
The authors declare that the data supporting the findings of this study are available within the paper.
Code availability
The MATLAB code package is available at GitHub (https://github.com/jamestang23/Optimizinghigherordernetworktopology.git). All the simulations were done with MATLAB version R2019a.
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Acknowledgements
The work was supported by the National Natural Science Foundation of China (11622538, 61673150, 12105014) and the Tencent Foundation or XPLORER PRIZE. L.L. also acknowledges the Special Project for the Central Guidance on Local Science and Technology Development of Sichuan Province (2021ZYD0029).
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Tang, Y., Shi, D. & Lü, L. Optimizing higherorder network topology for synchronization of coupled phase oscillators. Commun Phys 5, 96 (2022). https://doi.org/10.1038/s4200502200870x
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DOI: https://doi.org/10.1038/s4200502200870x
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