Figure 1: Schematic illustration of building up Y and Φ from binary time series.
From: Reconstructing propagation networks with natural diversity and identifying hidden sources
(a) Fourteen snapshots of data at time instants t1−t14 of eight nodes in a sample network, where Si is the time series of node 2 and S−i denotes the strings of other nodes at different times. The neighbourhood of node 2 is to be reconstructed. Only the pairs 00 and 01 in the time series of Si (i=2) and the corresponding S−i contain useful information about the network, as marked by different colours. (b) As Si(t+1) is determined by the neighbours of i and S−i(t), we sort out Si(t+1) and S−i(t) in the coloured sections of the time series in a. According to the threshold parameters Δ=3/7 and Θ=3/7, we calculate the normalized Hamming distance between each pair of strings S−i(t), finding two base strings at 
and 
with 
. We separate the coloured strings into two groups that are led by the two base strings, respectively. In each group, the normalized Hamming distance 
between the base string and other strings is calculated and the difference from 
in each string is marked by red. Using parameter Δ, in the group led by 
, S−i(t5) and Si(t6) are preserved, because of 
. In contrast, S−i(t7) and Si(t8) are disregarded because 
. In the group led by 
, due to 
, the string is preserved. The two sets of remaining strings marked by purple and green can be used to yield the quantities required by the reconstruction formula. (Note that different base strings are allowed to share some strings, but for simplicity, this situation is not illustrated here. See Supplementary Note 3 for a detailed discussion.) (c) The average values 
and 
used to extract the vector Y and the matrix Φ in the reconstruction formula, where 
, 
, 
and 
based on the remaining strings marked in different colours (see Methods for more details). CST can be used to reconstruct the neighbouring vector X of node 2 from Y and Φ from Y=Φ·X.
