node2vec random walks are tuneable random walks that come from the popular algorithm node2vec which is used for network embedding. The transition probabilities of the random walks depend on the previous visited node and on the triangles that contain the current and the previous node. In the node2vec algorithm, node2vec random walks are used to sample neighborhoods for each node of the network and by comparing these an embedding of the network into a Euclidean space can be computed. Since the parameters of the random walks can be tuned to create different types of neighborhoods, this approach is very flexible and advantageous over just using simple random walks.

Even though the algorithm is widely used in practice, mathematical properties of node2vec random walks almost have not been investigated and even basic questions such as how the stationary distribution depends on the walk parameters and if the random walk is recurrent are nearly unexplored. In this talk, we study the behavior of node2vec random walks on regular graphs. By going to a higher-order state space, the space of directed wedges, we can prove a simple expression of the stationary distribution on this space which is determined by the transition type of the wedge. We also formalize a pullback mechanism to retrieve the stationary distribution on the original state space. Further, we show that on infinite regular graphs, node2vec random walks are recurrent if and only if the simple random walk is recurrent.