Fig. 1
From: Inferring collective dynamical states from widely unobserved systems
Spatial subsampling. a In complex networks, such as the brain, often only a small subset of all units can be sampled (spatial subsampling); figure created using TREES57. b In a branching network (BN), an active unit (e.g., a spiking neuron, infected individual, or defaulting bank) activates some of its neighbors in the next time step. Thereby activity can spread over the system. Units can also be activated by external drive. As the subsampled activity a t may significantly differ from the actual activity At, spatial subsampling can impair inferences about the dynamical properties of the full system. c In recurrent networks (BN, Bak-Tang-Wiesenfeld model (BTW)), the conventional estimator (empty symbols) substantially underestimates the branching ratio m when less units n are sampled, as theoretically predicted (dashed lines). The novel multistep regression (MR) estimator (full symbols) always returns the correct estimate, even when sampling only 10 or 1 out of all N = 104 units. d For a BN with m = 0.99, the conventional estimator infers \(\hat m\) = 0.37, \(\hat m\) = 0.1, or \(\hat m\) = 0.02 when sampling 100, 10, or 1 units, respectively. Kalman filtering based estimation returns approximately correct values under slight subsampling (n = 100), but is biased under strong subsampling. In contrast, MR estimation returns the correct \(\hat m_{}^{}\) for any subsampling. e MR estimation is exemplified for a subcritical branching process (m = 0.9, h = 10), where active units are observed with probability α. Under subsampling (gray), the regression slopes r1 are smaller than under full sampling (blue). f While conventional estimation of m relies on the linear regression r1 and is biased under subsampling, MR estimation infers \(\hat m\) from the exponential relation r k ∝ mk, which remains invariant under subsampling
