Validity and Limitations of the Detection Matrix to Determine Hidden Units and Network Size from Perceptible Dynamics

Abstract

Determining the size of a network dynamical system from the time-series of some accessible units is a critical problem in network science. Recent work by Haehne et al. [1] has presented a modelfree approach to address this problem, by studying the rank of a detection matrix that collates sampled time-series of perceptible nodes from independent experiments. Here, we unveil a profound connection between the rank of the detection matrix and the control-theoretic notion of observability, upon which we conclude when and how it is feasible to exactly infer the size of a network dynamical system.

The detection matrix as a model-agnostic tool to estimate the number of degrees of freedom in mechanical systems and engineering structures

“…H2) The size of each partition is larger than the dimension of the system, so that s ≥ 2N. By virtue of this hypothesis and excluding degenerate cases of undamped systems in which the imaginary eigenvalues are spaced by multiples of 2π √ −1/∆t, E (s) is a non-singular Vandermonde matrix 32 so that E (s) is full row rank, that is, rankE (s) = 2N. By using P2 and recalling that V is nonsingular, H1 ultimately implies rank(E T (s) V −T ) = 2N.…”

“…An equivalent derivation can be carried out by releasing the assumption of time-invariance, following the arguments in the Supplemental Material of Ref. 32. The main argument of the proof would be based on the extension of the notion of observability matrix to analytic systems with constant output matrix, which allows one to build a representation of the transition matrix similar to (9).…”

“…For a network of first-order continuous-time dynamical systems, the detection matrix collates the time-series of some of the network nodes across multiple, independent experiments. As analytically demonstrated by Porfiri 32 under quite general conditions, one can infer the total number of nodes in the network without any knowledge of the network itself from the rank of the detection matrix. Although the mathematical derivation relies on the linearity of the networked dynamical system, complex time-varying dynamics can be considered in a continuous-time settings, unlike the typical settings of subspace identification methods that are based on time-invariant discrete-time models 31 .…”

System Size Identification from Sinusoidal Probing in Diffusive Complex Networks

“…In summary, if N is the number of observable units, this method relies on N kM measured values and requires to be able to set the system in M different initial conditions. A recent Letter [16] elegantly draws the link between the detection matrix of [15] and the observability matrix commonly used in control theory [17]. Ref.…”

“…Ref. [16] shows that the approach proposed in [15] can unambiguously determine the system size if and only if the system is observable in the control-theoretic sense.…”