Model-free inference of direct network interactions from nonlinear collective dynamics

Abstract

The topology of interactions in network dynamical systems fundamentally underlies their function. Accelerating technological progress creates massively available data about collective nonlinear dynamics in physical, biological, and technological systems. Detecting direct interaction patterns from those dynamics still constitutes a major open problem. In particular, current nonlinear dynamics approaches mostly require to know a priori a model of the (often high dimensional) system dynamics. Here we develop a model-independent framework for inferring direct interactions solely from recording the nonlinear collective dynamics generated. Introducing an explicit dependency matrix in combination with a block-orthogonal regression algorithm, the approach works reliably across many dynamical regimes, including transient dynamics toward steady states, periodic and non-periodic dynamics, and chaos. Together with its capabilities to reveal network (two point) as well as hypernetwork (e.g., three point) interactions, this framework may thus open up nonlinear dynamics options of inferring direct interaction patterns across systems where no model is known.

Accelerated reference frames (ARFs) reveal networks from time series data

“…, are diagonal and their entries indicate which other units in the network directly affect the dynamics of unit i [15]. Specifically, if a unit j does not directly affect the dynamics of i, we have f x 0 Our aim is to transform the dynamics (1) to new variables t y( ) such that these exhibit convergence to a fixed point if the original variables t x( ) exhibit convergence to some form of synchronized state (see below for details).…”

“…its node degree in a graph-theoretic perspective. Now the task is to distinguish the case where the link is actually present in the network, 15) intuitively suggests that the distinction between present and absent links is simpler if the network is sparser (and thus the n j are smaller) and, intriguingly, if the network is larger. The above analysis together with the intuitive arguments suggest that the average network state indeed is an appropriate ARF, at least for systems exhibiting synchronizing and locked-like dynamics.…”

“…These results are further supported by simulations of networks of Lorenz (19a) and Goodwin (20a) oscillators. Specifically, we compared the reconstruction quality on such systems using ARF and using the model-free algorithm for Inferring Network Interactions (ARNI) [15], figure 6. The results show that the ARFbased theory proposed here requires smaller data sets than ARNI to reveal the full topology of both systems.…”

“…n N 0 i  -. Whereas exactly synchronous or phase-locked dynamics in principle can generally not reveal the complete network topology, inferring from transient dynamics towards synchrony or locking was so far restricted to driving-response settings with known signals [16] or to general model-free approaches using a large repertoire of functions [11,12,14,15]. While the former strategy allows to create linear mappings from recorded dynamics to network topology, the latter allows to infer links from transient dynamics following an unknown driving or perturbation.…”

“…Such approaches require the entire dynamics to admit a sparse representation in the chosen repertoire, which is difficult to satisfy if no prior information is provided. More recent approaches bypass the need for sparsity and reconstruct the full interaction topology by either imposing functional decompositions in grouped variables [15], or by driving the network dynamics with known constant signals [16]. While the former strategy carries along a high computational complexity that may become intractable in large networks, the latter demands an accurate (and often infeasible) control of network dynamics.…”