# Observability of dynamical networks from graphic and symbolic approaches

## TL;DR

This paper extends symbolic observability methods to complex dynamical networks, enabling assessment of how well the network's states can be reconstructed from limited measurements using symbolic and graph-based approaches.

## Contribution

It introduces a method to evaluate network observability from node dynamics, coupling functions, and topology, advancing analysis techniques for large, complex networks.

## Key findings

- Extended symbolic observability to networks with coupled node dynamics.
- Provided a framework to assess network observability from topology and node interactions.
- Demonstrated the approach's applicability to large networks with arbitrary topology.

## Abstract

A dynamical network, a graph whose nodes are dynamical systems, is usually characterized by a large dimensional space which is not always accesible due to the impossibility of measuring all the variables spanning the state space. Therefore, it is of the utmost importance to determine a reduced set of variables providing all the required information for non-ambiguously distinguish its different states. Inherited from control theory, one possible approach is based on the use of the observability matrix defined as the Jacobian matrix of the change of coordinates between the original state space and the space reconstructed from the measured variables. The observability of a given system can be accurately assessed by symbolically computing the complexity of the determinant of the observability matrix and quantified by symbolic observability coefficients. In this work, we extend the symbolic observability, previously developed for dynamical systems, to networks made of coupled $d$-dimensional node dynamics ($d>1$). From the observability of the node dynamics, the coupling function between the nodes, and the adjacency matrix, it is indeed possible to construct the observability of a large network with an arbitrary topology.