Fig. 4: Pipeline description.

Estimation of transition probability matrices between balanced and unbalanced triads over a sequence of time periods. This figure illustrates the preprocessing steps, the optimization problem, and the results. Step 1: for all time periods t from 1 to T − 1 (where T is the maximum number of periods in that dataset), each edge is labeled using aggregated majority in that period as positive (+) or negative (−). Step 2: we compute the proportion of triads in every period and estimate the transition matrix at time t as the unknown matrix that multiplied by the vector of proportion of triads at period t gives the corresponding vector at period t + 1. Step 3: we estimate all transition matrices together using a time-varying Markov model. This step ensures that the results provide a holistic view of the available longitudinal data. Step 4: we visualize the probability of transition from balanced and unbalanced triads in time-varying estimated transition matrices. By allowing null ties, there are 138 types of triads. P_t represents the unknown transition matrix at time t and \(\hat{{P}_{t}}\) represents the empirically estimated transition matrix at time t, and \({\widetilde{P}}_{t}\) represents the time-varying estimated transition matrix at time t. The four quadrants of \(\hat{{P}_{t}}\) show the average estimated transition probabilities (see “Methods” for details). The result of this experiment is shown in Fig. 5.