How can you connect with anyone on Linkedin?
As I'm sure you realize, you have many different kinds of connections on Linkedin.
You can request a connection with anyone who is a member of any of your groups, and you can connect with anyone who is a 1st or 2nd level connection.
LinkedIn used to not let you connect to your third level contacts. Your 3rd level connections are the people connected to your 2nd level connections. This is usually your largest pool of contacts on LinkedIn. It it the exponential effect at work.
Here is a method to connect to your 3rd level connections. Do an advanced search in LinkedIn for their name. Even if there is no last name copy that information into Google. (Do not search yet). Then copy their position, their company, and location from their profile. Chances are good that you will find a link that has their email address, (which you will need to connect to them).
Copy and paste all this information step by step into Google, then search. You should see their LinkedIn profile with their last name come up on Google. Copy their first and last name from their profile on Google, and go back to LinkedIn, and search for them that way. You should see their profile with an option to connect to them in the usual way.
The reason this works is that LinkedIn assumed that if you have their first and last name, you must have some kind of relationship with that person, and they allow you to connect. This is a great way to expand your connections since your 3rd level connection represent the largest group outside your network.
This should allow you to greatly expand your network beyond your current number of connections.
Good luck in your endeavors!