This paper deals with a 'differentiated consensuses' problem in a distributed stochastic network system with priorities and cost constraints on system topology. The network is considered as a set of heterogeneous agents that process incoming tasks with different importance (priority) levels. The observations about neighbors' states are supposed to be obtained with random noise and delays and the topology could switch over time. Several consensus objectives are to be achieved. To maintain almost balanced load under cost constraints on network topology, i.e. approximate consensus for every priority class across the network, a new family of control protocols that use different step-size parameters for each task class is introduced. An instrument for choosing optimal step-sizes for the proposed control strategy is given. In addition, a numerical example that illustrates the proposed control strategy and the results of simulations are provided.