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A dynamical network, a graph whose nodes are dynamical systems, is usuallycharacterized by a large dimensional space which is not always accesible due tothe impossibility of measuring all the variables spanning the state space.Therefore, it is of the utmost importance to determine a reduced set ofvariables providing all the required information for non-ambiguouslydistinguish its different states. Inherited from control theory, one possibleapproach is based on the use of the observability matrix defined as theJacobian matrix of the change of coordinates between the original state spaceand the space reconstructed from the measured variables. The observability of agiven system can be accurately assessed by symbolically computing thecomplexity of the determinant of the observability matrix and quantified bysymbolic observability coefficients. In this work, we extend the symbolicobservability, previously developed for dynamical systems, to networks made ofcoupled $d$-dimensional node dynamics ($d>1$). From the observability of thenode dynamics, the coupling function between the nodes, and the adjacencymatrix, it is indeed possible to construct the observability of a large networkwith an arbitrary topology.

Observability of dynamical networks from graphic and symbolic approaches