The social element of blogs and the connectedness of the blogosphere is a defining characteristic of blogging as a communication platform. It is how blogging began and how it has evolved into what it is today. Blogs feed off each other and are dependent on each other for their existence. Readers are tuned into this sense of connectedness and have come to expect it from the blogs they follow. The relationships between blogs are an intrinsic part of the reader experience (this is how they link to new content and become habitual consumers of the blogging medium).
You need to be part of this in order to truly call yourself a blogger. A blog isn't a blog unless it's a fully connected part of the blogosphere. No blog is an island.