The social phenomenon of familiar strangers was identified by Stanley Milgram in 1972 with a small-scale experiment. However, there has been limited research focusing on uncovering the phenomenon at a societal scale and simultaneously investigating the social relationships between familiar strangers. With the help of the large-scale mobile phone records, we empirically show the existence of the relationship in the country of Andorra. Built upon the temporal and spatial distributions, we investigate the mechanisms, especially collective temporal regularity and spatial structure that trigger this phenomenon. Moreover, we explore the relationship between social distances on the communication network and the number of encounters and show that larger number of encounters indicates shorter social distances in a social network. The understanding of the physical encounter network could have important implications to understand the phenomena such as epidemics spreading and information diffusion.