  Clustering based on the random walk operator has been proven effective for
undirected graphs, but its generalization to directed graphs (digraphs) is much
more challenging. Although the random walk operator is well-defined for
digraphs, in most cases such graphs are not strongly connected, and hence the
associated random walks are not irreducible, which is a crucial property for
clustering that exists naturally in the undirected setting. To remedy this, the
usual workaround is to either naively symmetrize the adjacency matrix or to
replace the natural random walk operator by the teleporting random walk
operator, but this can lead to the loss of valuable information carried by edge
directionality. In this paper, we introduce a new clustering framework, the
Parametrized Random Walk Diffusion Kernel Clustering (P-RWDKC), which is
suitable for handling both directed and undirected graphs. Our framework is
based on the diffusion geometry and the generalized spectral clustering
framework. Accordingly, we propose an algorithm that automatically reveals the
cluster structure at a given scale, by considering the random walk dynamics
associated with a parametrized kernel operator, and by estimating its critical
diffusion time. Experiments on $K$-NN graphs constructed from real-world
datasets and real-world graphs show that our clustering approach performs well
in all tested cases, and outperforms existing approaches in most of them.
