  This work addressed the problem of learning a network with communication
between vertices. The communication between vertices is presented in the form
of perturbation on the measure. We studied the scenario where samples are drawn
from a uniform ergodic Random Graph Process (RGPs for short), which provides a
natural mathematical context for the problem of interest. For the binary
classification problem, the result we obtained gives uniform learn-ability as
the worst-case theoretical limits. We introduced the structural Rademacher
complexity, which naturally fused into the VC theory to upperbound the first
moment. With the martingale method and Marton's coupling, we establish the tail
bound for uniform convergence and give consistency guarantee for empirical risk
minimizer. The technique used in this work to obtain high probability bounds is
of independent interest to other mixing processes with and without network
structure.
