  A stochastic incremental subgradient algorithm for the minimization of a sum
of convex functions is introduced. The method sequentially uses partial
subgradient information and the sequence of partial subgradients is determined
by a general Markov chain. This makes it suitable to be used in networks where
the path of information flow is stochastically selected. We prove convergence
of the algorithm to a weighted objective function where the weights are given
by the Ces\`aro limiting probability distribution of the Markov chain. Unlike
previous works in the literature, the Ces\`aro limiting distribution is general
(not necessarily uniform), allowing for general weighted objective functions
and flexibility in the method.
