Detecting Hidden Units and Network Size from Perceptible Dynamics

Abstract

The number of units of a network dynamical system, its size, arguably constitutes its most fundamental property. Many units of a network, however, are typically experimentally inaccessible such that the network size is often unknown. Here we introduce a detection matrix that suitably arranges multiple transient time series from the subset of accessible units to detect network size via matching rank constraints. The proposed method is model-free, applicable across system types and interaction topologies and applies to non-stationary dynamics near fixed points, as well as periodic and chaotic collective motion. Even if only a small minority of units is perceptible and for systems simultaneously exhibiting nonlinearities, heterogeneities and noise, exact size detection is feasible. We illustrate applicability for a paradigmatic class of biochemical reaction networks.

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

“…We propose that the relationship between the detection matrix [1] and the concept of observability uncovered in 355 this work could beget similar methodological and theoretical advances. This paper shows that the success of the detection matrix in exactly estimating the size of a network is, in fact, conditional to the complete observability of the system from its perceptible dynamics.…”

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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.

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“…Detecting the number of hidden nodes from a few perceptible nodes was the open question addressed in Ref. 1. Therein, the authors put forward a promising modelfree approach to estimate the true network size from the rank of a detection matrix that comprises the sampled time-series of perceptible nodes from independent experiments.…”

Network Reconstruction Problem for an Epidemic Reaction-Diffusion

“…We have studied the case when the observations we have consist of all the trajectories in all the nodes. Other studies have considered partial observations, as in the presence of missing nodes (Tyrcha and Hertz 2014; Haehne et al 2019), or more generally partial observations (Nitzan, Casadiego, and TImme 2017; Ioannidis, Romero, and Giannakis 2018; Ioannidis, Shen, and Giannakis 2019). This would be an interesting extension to further our work.…”

“…Structural and qualitative analysis of the model is therefore of much importance to better understand the dynamics at hand, and guide the reconstruction work. Moreover, such analysis may provide insights for other models incorporating diffusion as well (Haehne et al 2019). Our contributions are therefore the following.…”