Predicting network dynamics without requiring the knowledge of the interaction graph

Abstract

A network consists of two interdependent parts: the network topology or graph, consisting of the links between nodes and the network dynamics, specified by some governing equations. A crucial challenge is the prediction of dynamics on networks, such as forecasting the spread of an infectious disease on a human contact network. Unfortunately, an accurate prediction of the dynamics seems hardly feasible, because the network is often complicated and unknown. In this work, given past observations of the dynamics on a fixed graph, we show the contrary: Even without knowing the network topology, we can predict the dynamics. Specifically, for a general class of deterministic governing equations, we propose a two-step prediction algorithm. First, we obtain a surrogate network by fitting past observations of every nodal state to the dynamical model. Second, we iterate the governing equations on the surrogate network to predict the dynamics. Surprisingly, even though there is no similarity between the surrogate topology and the true topology, the predictions are accurate, for a considerable prediction time horizon, for a broad range of observation times, and in the presence of a reasonable noise level. The true topology is not needed for predicting dynamics on networks, since the dynamics evolve in a subspace of astonishingly low dimension compared to the size and heterogeneity of the graph. Our results constitute a fresh perspective on the broad field of nonlinear dynamics on complex networks.

Duality between predictability and reconstructability in complex systems

“…This experiment was performed in ref. 51 for deterministic dynamics on graphs, to show that high prediction accuracy of time series can sometimes be achieved without the knowledge of the true graph. In Fig.…”

“…2 a, we use the mean absolute error—the same measure as in ref. 51 —to perform the comparison. In turn, we associate the high predictive capabilities of the true conditional model where the error with the graph-independent model is high.…”

“…In ref. 51 , it was shown that time series generated by a deterministic dynamics evolving on a specific graph can be accurately predicted by a broad range of other graphs. These findings highlight how poor our intuition can be with regard to the relationship between predictability and reconstructability.…”

“…Our objective is to combine these ideas into a single framework, motivated by recent works involving spin dynamics on lattices 79 , 80 and deterministic dynamics 51 . Our contributions are fourfold.…”

Data-driven inference of complex system dynamics: A mini-review

“…By updating the sampling and NN's parameters to meet the desired outputs, the network topology can be revealed [67,68]. The method developed by Prasse et al does not require prior knowledge of network topology but needs surrogate networks, relying on proper orthogonal decomposition (POD) [69]. Resilience and criticality are important properties of complex network dynamics [70,71], and Eroglu et al constructed effective networks to recover the structure, dynamics and predicted critical transitions [70].…”