I am not very familiar with the field, but a quick Google search brought up this paper ("The Happiness Paradox: Your Friends are Happier Than You" from Bollen, Goncalves, Leemput, and Ruan) which had some interesting results and conclusions. This paper considers a happiness paradox within the concept of friendships, and a friendship paradox within the concept of happiness. It also includes correlation coefficients between happiness and popularity, which may address your question. I'll note that the sample of this study isn't exactly what you are looking for: it is done via Twitter data (and, as noted in the supplementary materials of the paper, the sample only includes users with greater than 15 friends).
Results seem to show that there is a mild correlation for "happy people", but there is not much of a correlation for "unhappy people" (very slight, though statistically significant, negative correlation). For more details, I encourage you to read the article (which includes some nice graphics depicting their theories and the distributions of outcomes).
Perhaps a start in the right direction?