# Structural Invertibility and Optimal Sensor Node Placement for Error and Input Reconstruction in Dynamic Systems

## TL;DR

This paper investigates how the structure of complex dynamic networks affects the ability to reconstruct unknown inputs and model errors, proposing a sensor placement algorithm for optimal observability.

## Contribution

It introduces a framework linking invertibility to network structure and develops an algorithm for optimal sensor placement to enhance input and error reconstruction.

## Key findings

- Sparse scale-free networks are hardest to invert.
- Invertibility depends on influence graph properties.
- The sensor placement algorithm optimizes measurement locations.

## Abstract

Despite recent progress in our understanding of complex dynamic networks, it remains challenging to devisesufficiently accurate models to observe, control or predict the state of real systems in biology, economics or other fields. A largely overlooked fact is that these systems are typically open and receive unknown inputs from their environment. A further fundamental obstacle are structural model errors caused by insufficient or inaccurate knowledge about the quantitative interactions in the real system. Here, we show that unknown inputs to open systems and model errors can be treated under the common framework of invertibility, which is a requirement for reconstructing these disturbances from output measurements. By exploiting the fact that invertibility can be decided from the influence graph of the system, we analyse the relationship between structural network properties and invertibility under different realistic scenarios. We show that sparsely connected scale free networks are the most difficult to invert. We introduce a new sensor node placement algorithm to select a minimum set of measurement positions in the network required for invertibility. This algorithm facilitates optimal experimental design for the reconstruction of inputs or model errors from output measurements. Our results have both fundamental and practical implications for nonlinear systems analysis, modelling and design.