Hello Cristinaalp,
One thing I must emphasize is that the distinctive characterisitcs between communities and social networks should not be be viewed a strict definition. Being in academician yourself, I think you might have heard people at conference saying that there are 19 definitions of community and the only common thing among them is people (I forgot was it 19 or some other numbers). There will always come up with exceptions. So what I've outlined are not really strict definitions, they are merely distinctive characteristics of these two social structure.
I think your question is more or less in the same vein as the comment by Nigel Sarbutts and comment by mark_kaufman. You can take a look at my reply to Nigel and reply to Mark respectively to see if they help addressing your question.
In short, social network can form within communities. In fact, if you go to the 3rd post in this mini-series, you will find that community is where relationships are developed and become part of an individual's personal social netowrk. So social network naturally form within communities. On the other hand, there are communities within the social network as well. The topological structure of these two social structure are are actually quite complex and interwoven. This seem to be a concept that is particularly confusing. Maybe I will write another post to explain it further.
Thanks for your comment, LinkedIn, and Retweets. Hope to see you next time and have a Happy July 4th.