Social networks transmitting covert or sensitive information cannot use all ties for this purpose. Rather, they can only use a subset of ties that are strong enough to be “trusted”. This paper addresses whether it is still possible, under this restriction, for information to be transmitted widely and rapidly in social networks. We use transitivity as evidence of strong ties, requiring one or more shared contacts in order to count an edge as strong. We examine the effect of removing all non-transitive ties in two real social network data sets, imposing varying thresholds in the number of shared contacts. We observe that transitive ties occupy a large portion of the network and that removing all other ties, while causing some individuals to become disconnected, preserves the majority of the giant connected component. Furthermore, the average shortest path, important for the rapid diffusion of information, increases only slightly relative to the original network. We also evaluate the cost of forming transitive ties by modeling a random graph composed entirely of closed triads and comparing its connectivity and average shortest path with the equivalent Erdös–Renyi random graph. Both the empirical study and random model point to a robustness of strong ties with respect to the connectivity and small world property of social networks.
