Time-series–based prediction of complex oscillator networks via compressive sensing

Abstract

Complex dynamical networks consisting of a large number of interacting units are ubiquitous in nature and society. There are situations where the interactions in a network of interest are unknown and one wishes to reconstruct the full topology of the network through measured time series. We present a general method based on compressive sensing. In particular, by using power series expansions to arbitrary order, we demonstrate that the network-reconstruction problem can be casted into the form X = G • a, where the vector X and matrix G are determined by the time series and a is a sparse vector to be estimated that contains all nonzero power series coefficients in the mathematical functions of all existing couplings among the nodes. Since a is sparse, it can be solved by the standard L1-norm technique in compressive sensing. The main advantages of our approach include sparse data requirement and broad applicability to a variety of complex networked dynamical systems, and these are illustrated by concrete examples of model and real-world complex networks.

Network Dynamics as an Inverse Problem

“…Previous studies on inferring structural connections from time series have focused on the reconstruction of networks with known local and coupling functions [22,[68][69][70]73,74]. Such prior knowledge reduces the problem of inferring connections to a standard linear algebra problem, where one has to solve a linear system of equations to reveal the network connections, cf.…”

“…These were previously used in [70,71,74] for inferring physical links utilizing the compressed sensing framework, cf. Ref.…”

“…Here, in contrast to [70,71,74], we take a more general perspective. Specifically, rather than assuming that expansions may contain the actual coupling functions, we instead assume that coupling functions are only formed by linear combinations of basis functions.…”

“…Still, researchers often select particular models for h i , involving known functions and specific types of interactions to explain the network dynamics [22,69,70,73,74]. Subsequently, based on these models, one may reconstruct the network connectivity from recorded measurements.…”

“…In particular, knowing the actual parameters and functions generating the network dynamics reduces the problem of inferring network structural connections to just fitting a collection of unknown parameters (in this case the connections) to measured data [34,35,40,[68][69][70][71][72][73]. Such approaches are especially reliable when estimating connections from time series [68][69][70][71][72][73][74], even in the presence of challenging dynamics (e.g.…”