This paper investigates characteristics of implicit brand networks extracted from a large dataset of user historical activities on a social media platform. To our knowledge, this is one of the first studies to comprehensively examine brands by incorporating user-generated social content and information about user interactions. This paper makes several important contributions. We build and normalize a weighted, undirected network representing interactions among users and brands. We then explore the structure of this network using modified network measures to understand its characteristics and implications. As a part of this exploration, we address three important research questions: (1) What is the structure of a brand-brand network? (2) Does an influential brand have a large number of fans? (3) Does an influential brand receive more positive or more negative comments from social users? Experiments conducted with Facebook data show that the influence of a brand has (a) high positive correlation with the size of a brand, meaning that an influential brand can attract more fans, and, (b) low negative correlation with the sentiment of comments made by users on that brand, which means that negative comments have a more powerful ability to generate awareness of a brand than positive comments. To process the large-scale datasets and networks, we implement MapReduce-based algorithms.