Investigation of social networks formed by individuals in various contexts provides numerous interesting and important challenges for researchers and practitioners in multiple disciplines. Within the field of variationist sociolinguistics, social networks are analyzed in order to reveal the patterns of language variation and change while taking the social, cultural, and geographical aspects into account. In this field, traditional approaches usually focusing on small, manually collected data sets can be complemented with computational methods and large digital data sets extracted from online social network and social media sources. However, increasing data size does not immediately lead to the qualitative improvement in the understanding of such data. In this position paper, we propose to address this issue by a joint effort combining variationist sociolinguistics and computational network analyses with information visualization and visual analytics. In order to lay the foundation for this interdisciplinary collaboration, we analyse the previous relevant work and discuss the challenges related to operationalization, processing, and exploration of such social networks and associated data. As the result, we propose a roadmap towards realization of visual sociolinguistic network analysis.