Fig 1.
Schematic representation of physics (parallel) and logistics (sequential) synchronization arising in a minimal model of excitable dynamics on graphs.
Fig 2.
Evolution of pair densities resulting from pair-approximation (second-order) mean-field SER dynamics.
Each circle represents a possible state of a pair of nodes and arrows represent state transitions. The cases ES and EE are emphasized with their evolution equations. The quantity Θnn(t) is given in Eq 10.
Fig 3.
Scatter plots of physics synchronization and logistics synchronization.
(a) Synchronization values observed for recovery probability p = 0.15 and rate of spontaneous activation f = 0.001. (b) Synchronization values for p = 0.15 and f = 0.1. Each point in the graph represents the value for one network realization of a 100 node BA graph with m = 3, the same network is used for all the 40 runs involved in the computation of synchronization measures.
Fig 4.
Space-time plot showing the state of the nodes (vertical axis) across time (horizontal axis) in a 1000 steps simulation of SER dynamics on a 100-node BA graph.
(a) shows a realization for high physics and low logistics synchronization, whereas (b) shows a realization for a low physics and high logistics synchronization. The two panels illustrate two extreme situations originating from two different runs at fixed rate of spontaneous activation f = 0.001 and recovery probability p = 0.15. The first 300 time steps have been discarded as transient.
Fig 5.
Correlation between logistics and physics synchronization as a function of recovery probability p.
(a) shows the correlation for a low rate of spontaneous machine activation f = 0.001 (b) shows the correlation for a high rate of spontaneous machine activation f = 0.1. Pearson and Spearman correlations have been calculated from 40 runs of SER dynamics on the same BA graph with 100 nodes and m=3. The mean and standard deviation (hatched zones) has then been calculated by repeating the same experiment 30 times.
Fig 6.
A heatmap showing the correlation between logistics and physics synchronization across a range of values of p and four values of f.
The transition of negative correlation to uncorrelated/weak positive correlation can be clearly seen for the different values of f. The network used is a BA graph with 100 nodes and m = 3. Pearson correlation has been computed from 40 individual runs on the same network realization, followed by averaging over 30 such numerical experiments.
Fig 7.
Scatterplot of logistics synchronization and wave pattern strength.
Time series of 5000 steps have been simulated on 20 different BA graphs (number of nodes N = 100, connection parameter m = 5) for f = 0.005 and p ranging from 0.05 to 1 (varied in steps of 0.05). Wave pattern strength has been computed according to [61] with a hub set size of N/2. The correlation coefficient between logistics synchronization and wave pattern strength is 0.66.
Fig 8.
Correlation between logistics and physics synchronization as a function of recovery probability p in real production networks.
(a) company A network (b) company B network for low spontaneous machine activation rate f = 0.001 (left) and high rate of spontaneous machine activation rate f = 0.1. (right). The mean and standard deviation (hatched zones) has then been calculated by repeating the same experiment 30 times. Note that the range can end slightly below −1 in some cases due to the definition of the hatched zone (mean ± standard deviation).
Fig 9.
(a) cE,E (b) cE,S obtained in the simulation, and the steady-state solution of mean-field Eq 11 (given in red) for varying values of the recovery probability p. Error bars on simulation points have been obtained from the last 500 time steps of a 1000 time-step simulation starting from random initial conditions.
Fig 10.
Correlations between average excitation change upon edge removal and synchronization measures.
(a) Correlation rEL between average excitation change and logistics synchronization for rate of machine activation f=0.001 (left) and f = 0.1. (right) (b) Correlation rEP between average excitation change and physics synchronization for rate of spontaneous machine activation f=0.001 (left) and f = 0.1 (right). Simulations have been performed on an ER graph with 100 nodes and 500 edges for varying values of the recovery probability p. Error bars on simulation points have been obtained from 50 runs for each type of correlation (Pearson or Spearman).