  A regularized optimization problem over a large unstructured graph is
studied, where the regularization term is tied to the graph geometry. Typical
regularization examples include the total variation and the Laplacian
regularizations over the graph. When applying the proximal gradient algorithm
to solve this problem, there exist quite affordable methods to implement the
proximity operator (backward step) in the special case where the graph is a
simple path without loops. In this paper, an algorithm, referred to as "Snake",
is proposed to solve such regularized problems over general graphs, by taking
benefit of these fast methods. The algorithm consists in properly selecting
random simple paths in the graph and performing the proximal gradient algorithm
over these simple paths. This algorithm is an instance of a new general
stochastic proximal gradient algorithm, whose convergence is proven.
Applications to trend filtering and graph inpainting are provided among others.
Numerical experiments are conducted over large graphs.
