# NIPS: Network Inference with Partial State measurements using forced-delay embedding

## TL;DR

This paper introduces a new method called NIPS to infer network connections from limited data, even when not all network components are observable.

## Contribution

The novel framework NIPS enables network reconstruction from partial-state observations using forced-delay embedding theory.

## Key findings

- NIPS accurately reconstructs networks using only partial-state measurements.
- The method is robust to noise and hidden nodes, as demonstrated on simulated and experimental data.
- The framework is extended to handle networks with unobservable coupling states.

## Abstract

Decoding the connectivity patterns of complex networks from time series measurements is crucial for understanding and controlling their dynamics. Although network inference algorithms have advanced significantly in identifying both pairwise and higher-order interactions, they often rely on the availability of full-state measurements, an assumption that is difficult to satisfy in practice. In this article, we address this limitation by introducing Network Inference from Partial States (NIPS), a framework for network reconstruction from partial-state observations of network units. Focusing initially on networks coupled through observable states, we model coupling inputs as external forcing and utilize forced-delay embedding theory to establish a map that describes the evolution of the node observables as a function of observable state components. Specifically, the dynamics of the observable state of a node depends only on delayed observations of that node itself, not on delayed observations of other nodes. This enables accurate network reconstruction with limited data, which is demonstrated using both simulated and experimental data obtained from a wide range of networks. We evaluate the robustness of NIPS to noisy data and hidden network nodes and subsequently extend the framework to networks coupled through unobservable states.