This paper models the cyber-social system as a cyber-network of agents monitoring states of individuals in a social network. The state of each individual is represented by a social node, and the interactions among individuals are represented by a social link. In the cyber-network, each node represents an agent, and the links represent information sharing among agents. The agents make an observation of social states and perform distributed inference. In this direction, the contribution of this paper is threefold: First, a novel distributed inference protocol is proposed that makes no assumption on the rank of the underlying social system. This is significant as most protocols in the literature only work on full-rank systems. Second, a novel agent classification is developed, where it is shown that the connectivity requirement of the cyber-network differs for each type. This is particularly important in finding the minimal number of observations and minimal connectivity of the cyber-network as the next contribution. Third, the cost-optimal design of the cyber-network constraint with distributed observability is addressed. This problem is subdivided into sensing cost optimization and networking cost optimization, where both are claimed to be NP-hard. We solve both the problems for certain types of social networks and find polynomial-order solutions.