  We study multidimensional Cram\'er-Lundberg risk processes where agents,
located on a large sparse network, receive losses form their neighbors. To
reduce the dimensionality of the problem, we introduce classification of agents
according to an arbitrary countable set of types. The ruin of any agent
triggers losses for all of its neighbours. We consider the case when the loss
arrival process induced by the ensemble of ruined agents follows a Poisson
process with general intensity function that scales with the network size. When
the size of the network goes to infinity, we provide explicit ruin
probabilities at the end of the loss propagation process for agents of any
type. These limiting probabilities depend, in addition to the agents' types and
the network structure, on the loss distribution and the loss arrival process.
For a more complex risk processes on open networks, when in addition to the
internal networked risk processes the agents receive losses from external
users, we provide bounds on ruin probabilities.
