We conduct a statistical analysis on the social networks of contributors in Open Source Software (OSS) communities using datasets collected from two most fast-growing OSS social interaction sites, Github.com and Ohloh.net. Our goal is to analyze the connectivity structure of the social networks of contributors and to investigate the effect of the different social tie structures on developers' overall productivity to OSS projects. We first analyze the general structure of the social networks, e.g., graph distances and the degree distribution of the social networks. Our analysis confirms that the social networks of OSS communities follow power-law degree distributions and exhibit small-world characteristics. However, the degree mixing pattern shows that high degree nodes tend to connect more with low degree nodes, suggesting collaborations between experts and newbie developers. Second, we study the correlation between graph degrees and the productivity of the contributors in terms of the amount of contribution and commitment to OSS projects. The analysis demonstrates evident influence of the social ties on the developers' overall productivity.