Fig. 9: Map of the Los Angeles multi-lane network and the resulting 24-hour Koopman.

a Map of the multi-lane network obtained from Map © data 2019 Google. The highways studied are highlighted and data for three lanes of both north–south and east–west directions are collected for the entire day of December 20th, 2018. b The 24-hour Koopman mode revealing the order of congestion within the network. Specifically, by observing the staggering of the amplitude one can see that during the morning rush the westbound I-105 and I-10 along with the northbound I-405 and I-110 jam first. In the afternoon traffic switches directions and it is the eastbound and southbound directions of the previously mentioned highways which are jammed. This corresponds with the well-known fact the morning commuters generally travel from San Bernardino (east to west) and Orange County (southeast to northwest) into Los Angeles. c A plot of the average phase of the mode sorted by highway confirming the previously mentioned synchrony of congestion. Specifically, the north and west directions seem to be in phase with each other and likewise for the south and east directions. d The magnitude of the 24-hour mode along with magenta-colored lines used to visually divide the differing highways. The magnitude of the mode reveals that the eastbound I-10 and I-105 highways are more occupied than the other highways.