  We study the evolution of networks through `triplets' -- three-node
graphlets. We develop a method to compute a transition matrix of these triplets
to describe their evolution in temporal networks. To identify the network
dynamics' non-pairwise interactions, we compare both artificial and real-world
data to a pairwise interaction model. The significant differences between the
computed matrix and the calculated matrix from the fitted parameters
demonstrate that non-pairwise interactions exist for various real-world data
sets. Furthermore, different entries of the matrix differences reveal the
real-world systems have different higher-order interaction patterns which are
seldomly reported in the previous researches.
  We then use these transition matrices as the basis of a link prediction
algorithm. We investigate our algorithm's performance on four temporal networks
comparing our approach against ten other link prediction methods. Our results
show that higher-order interactions play a crucial role in the evolution of
networks as we find our method along with two other methods based on non-local
interactions, give the best overall performance. The results also confirm the
concept that the higher-order interaction patterns, i.e., triplet dynamics, can
help us understand and predict different real-world systems' evolution.
