Fig. 1

a Five colour-coded paths between a set of six nodes (A–F) in a networked system. We split the paths into a training set ( , , ) and a validation set ( , ). The topology of the networked system—depicted in grey—limits which transitions can exist. All paths start in A or B. They are of variable length as they end either after a single transition in C, or after three transitions in E or F. Consider aiming to predict if a path arriving in C continues to D or ends. Only paths starting in A can end in C, whereas all paths starting in B continue to D. Hence, to make an informed prediction, we need to account for the temporal ordering of transitions, i.e., we require memory recording the sequence of visited nodes before arriving at C. b To fit MOGen, we estimate a set of models encoding the observed transitions with different maximum memory lengths. c We apply MOGen’s model selection algorithm to determine the optimal maximum order. d We find the optimal model for the given data. e We use this optimal model to predict the paths in the validation data