Higher-order interactions can better optimize network synchronization

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 composite Laplacian matrix that encodes multiorder 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 pairwise 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.

Figure 1

Weighted simplicial complexes encode the balancing of multiorder interactions. An illustration of a small network with (a) 1- vs (b) 2-simplex dominated coupling. Shading indicates the relative strength of dyadic and triadic interactions which after rescaling by the respective mean degree conserve the total coupling strength such that the bias parameter $\alpha$ equitably tunes 1- vs 2-simplex interactions

Figure 2

Optimal synchronization in networks with higher-order interactions. (a) The synchronization error $1-r$ vs $K$ for random (open symbols) and optimal (closed symbols) frequencies for two choices of the bias parameter: $\alpha =0$ (red triangles) and 0.8 (blue circles), representing cases where interactions are exclusively defined by 1-simplexes and dominated by 2-simplexes, respectively, for a noisy geometric network (see text). (b) The synchrony alignment function (SAF) $J(\mathit{\omega },L)$ as a function of $\alpha$ for randomly allocated (open squares) and optimal (closed squares) frequency averages over ${10}^3$ networks.

Figure 3

Spectral properties of a composite Laplacian. (a) The eigenvalue spectrum $P\left(\lambda \right)$ of the composite Laplacian $L$ for $\alpha =0$ (solid blue), 0.4 (dashed red), and 0.8 (dot-dashed green) obtained from ${10}^3$ networks of size $N=500$ with mean degree $\langle k\rangle =10$ and $p=0.25$. (b) The variance of the eigenvalue spectrum along with the extremal eigenvalues (c) ${\lambda }_2$ and (d) ${\lambda }_N$ from the same ensemble. (e) 2-simplex degrees $k^{\left(2\right)}$ vs 1-simplex degrees $k^{\left(1\right)}$ for a single network realization and (f) the quantities $\langle k^{\left(1\right)2}\rangle /{\langle k^{\left(1\right)}\rangle }^2$ and $\langle k^{\left(2\right)2}\rangle /{\langle k^{\left(2\right)}\rangle }^2$ (blue circles and red crosses, respectively) obtained from an ensemble of ${10}^3$ networks as a function of the parameter $p$.

Figure 4

Geometric consistency in spectral properties of the composite Laplacian. (a) For networks optimized at ${\alpha }^{*}=0,0.2,\cdots ,1$ (blue to yellow), the SAF $J({\mathit{\omega }}^{*},L)$ of this solution as a function of $\alpha$ given the optimal frequency vector ${\mathit{\omega }}^{*}$. Points along the curves at $\alpha ={\alpha }^{*}$ and the local minimum are denoted in open and closed circles, respectively. (b) The logarithm (base 10) of the squared projections ${\langle {\mathit{v}}^j\left(0.2\right),{\mathit{v}}^i\left(0.8\right)\rangle }^2$. Results are obtained from an ensemble of ${10}^3$ networks of size $N=100$ with mean degree $\langle k^{\left(1\right)}\rangle =10$ and $p=0.25$.

Figure 5

Constrained optimization. The SAF $J(\mathit{\omega },L)$ as a function of $\alpha$ obtained after optimal perturbations of sizes $\parallel \delta \mathit{\omega }\parallel /\parallel \mathit{\omega }\parallel =0$ (blue circles), 0.4 (red triangles), 0.8 (green crosses), and 1.2 (black squares) are applied to a randomly drawn vector of frequencies (a) without and (b) with preprocessing the frequency vector using a (near) optimal permutation. Results are obtained from an ensemble of ${10}^2$ networks of size $N=100$ with mean degree $\langle k^{\left(1\right)}\rangle =10$ and $p=0.25$.