The aim of this thesis is to study communication algorithms in networks where nodes and/or communication links fail in a random dependent way. In order to capture fault dependencies, we introduce the neighborhood fault, , and swamping communication models. In the neighborhood fault model for arbitrary networks, damaging events, called spots, occur randomly and independently with probability p at nodes of a network, and cause faults in the given node and all of its neighbors. Under this model, faults at nodes within distance 2 are correlated. In the ranged fault model for geometric networks, spots occur with Poisson arrival rate λ on a plane and cause faults in all nodes at distance at most s. Under this model, faults at nodes within distance 2s are spatially correlated. In the swamping communication model for geometric radio networks, nodes can communicate only with nodes that are at a distance greater than the swamping distance s and at most at the communication range r. Under this model, nodes experience reception disturbances when nodes within distance s transmit, hence transient message reception failures are spatially correlated. Under each of these models, we present results concerning the connectivity and the diameter of networks as well as the time of communication.