Understanding factors associated with tie strength in social networks is essential in a wide variety of settings. With the internet and cellular phones providing additional avenues of communication, measuring and inferring tie strength has become much more complex. We introduce the social bow tie framework, which consists of a focal tie and all actors connected to either or both of the two focal nodes on either side of the focal tie. We also define several intuitive and interpretable metrics that quantify properties of the bow tie which enable us to investigate associations between the strength of the "central" tie and properties of the bow tie. We combine the bow tie framework with machine learning to investigate what aspects of the bow tie are most predictive of tie strength in two very different types of social networks, a collection of medium-sized social networks from 75 rural villages in India and a nationwide call network of European mobile phone users. Our results show that tie strength depends not only on the properties of shared friends, but also on non-shared friends, those observable to only one person in the tie, hence introducing a fundamental asymmetry to social interaction.