I have been a member of the professional social network Linkedin for a couple of years now. My profile there. Many of my colleagues, retired or back at the firm are also members. Have made a number of interesting and profitable connections based on my membership. The psych of it has interested me from the very beginning. Just read a good piece on how you should use it to generate massive success. Linkedin recommends that you only accept invites from those you know. The article basically suggests that the more in your network the merrier.
I have never invited people I did not know at all, but I have invited people I knew through others, or wanted to connect with better. I have been ignored a few times, but only gotten the reply 'I do not know you' once ... strangely I did know the person. What is the psych of casual rejection?
So what is the best size of a professional network? There is not much effort in sending an invitation. I don't think I would invite someone who was in a completely different industry, though my expertise, business improvement, could play everywhere.
One commenter in the article says they are suspicious of people who have hundreds of people in their network. I have a couple of hundred. I don't think that is too many, I have valuably connected with many hundreds in my company, vendors, suppliers, academia and organizations over the years. A few hundred is still pretty selective.
Speaking from the view of someone with many years of experience and now consulting, I come down on the side of the more, reasonably considered, connections you have, the more value you create. Especially if you are seeking consulting and information connections. Even more so in the case of group memberships, I have joined a number of them and ended up with people contacting me because I joined and that has led to unusual and very useful connections.