We present a link-centric approach to study variation in the mobile phone communication patterns of individuals. Unlike most previous research on call detail records that focused on the variation of phone usage across individual users, we examine how the calling and texting patterns obtained from call detail records vary among pairs of users and how these patterns are affected by the nature of relationships between users. To demonstrate this link-centric perspective, we extract factors that contribute to the variation in the mobile phone communication patterns and predict demographics-related quantities for pairs of users. The time of day and the channel of communication (calls or texts) are found to explain most of the variance among pairs that frequently call each other. Furthermore, we find that this variation can be used to predict the relationship between the pairs of users, as inferred from their age and gender, as well as the age of the younger user in a pair. From the classifier performance across different age and gender groups as well as the inherent class overlap suggested by the estimate of the bounds of the Bayes error, we gain insights into the similarity and differences of communication patterns across different relationships.