Figure 1

Schematic illustration of data-based linearization from a merging process. (a) The original binary-state time series, where the dark blue squares denote the 1 state and the white squares denote the 0 state. The variable $s_{-i}\left(t\right)$ consists of $s_j\left(t\right)$ for all $j\ne i$. Only strings with $s_i\left(t\right)=0$ (highlighted by the green frames) contain useful information for reconstruction. We identify the time steps with $s_i\left(t\right)=0$ and use $s_i(t+1)$. (b) Method of choosing bases. We first construct a network where the vertices denote strings of $s_{-i}\left(t\right)$ with $s_i\left(t\right)=0$ (green squares) and the edges are weighted by the normalized Hamming distance $H$ between the strings. We then eliminate the edges with weight smaller than the threshold $\mathrm{\Delta }$. Setting another threshold $\sigma$, we obtain the top $\sigma$ percentage vertices with large degrees (yellow squares) and remove the other vertices together with their edges. Finally, we pick out the vertices with smaller degree (red squares) according to the number of base strings needed for reconstruction. (c) Selection of subordinate strings from a base. We take $t_1$ as a base ${\widehat{t}}_1$ and calculate $H$ between $s_{-i}\left(t_1\right)$ and other strings $s_{-i}\left(t\right)$ so as to sort out the time steps satisfying $H[s_{-i}\left(t_1\right),s_{-i}(·)]<\mathrm{\Delta }$ in this set. (d) Establishing average node states. We calculate the average value $〈s_{-i}\left(\widehat{t}\right)〉$ to represent the state of the data set subject to the base, and the average value $〈s_i(\widehat{t}+1)〉$ to linearize the switching probability $P_i^{01}\left(t\right)$ [Eqs. (4)]. The average values are shown in blue. Similarly, we obtain a sequence of ${\widehat{t}}_M$ and the associated average values for reconstructing network structure by employing the lasso to solve ${\mathbf{Y}}_i={\mathrm{\Phi }}_i\times {\mathbf{X}}_i$ (see Methods for details).