Reconstructing higher-order interactions in coupled dynamical systems

Abstract

Higher-order interactions play a key role for the operation and function of a complex system. However, how to identify them is still an open problem. Here, we propose a method to fully reconstruct the structural connectivity of a system of coupled dynamical units, identifying both pairwise and higher-order interactions from the system time evolution. Our method works for any dynamics, and allows the reconstruction of both hypergraphs and simplicial complexes, either undirected or directed, unweighted or weighted. With two concrete applications, we show how the method can help understanding the complexity of bacterial systems, or the microscopic mechanisms of interaction underlying coupled chaotic oscillators.

Fig. 1. Reconstructing higher-order interactions in a microbial ecosystem.

a The underlying weighted hypergraph of a Lotka–Volterra system with N = 7 species and two- and three-body interactions, which we want to reconstruct from (b) the time evolution of the seven species abundance x_i(t). c Quality of the reconstruction is measured by reporting the error $\mathcal{E}$ as a function of the ratio between the length M of the trajectories and the number H of interactions to reconstruct. d Error $\mathcal{E}$ for the various approximations of the derivatives.

Fig. 2. Testing the reconstruction method on a system of N = 34 coupled Rössler oscillators.

a The underlying simplicial complex consists of 78 links and 45 2-simplices. b Reconstruction error $\mathcal{E}$ defined in Eq. (7) as a function of M/H when derivatives are known. c The computed values of the components of the arrays ${\widehat{\mathcal{A}}}_i,i=1,\dots ,N$ for M/H = 34 for the OLS method (blue dots) and for the NNLS method (red dots). d Reconstruction error $\mathcal{E}$ as a function of M/H when derivatives are not known and a fourth-order approximation of the derivatives is used.