Set some connections to not be notified about their activities on linkedin

If he was your 3rd degree connection, this does not mean you now have 'friends in common'. That would have made him a 2nd degree (you->common friend -> him). What you are looking for is a person whom he and another one of your friends have now in common.

Or to put it in different words - a 2nd degree connection which at least two of your friends share.

I don't know of a way to get this through to the interface. The nearest is to go to Advanced Search, chose 2nd connection and sort by connections. This should show at the bottom how many connections you have to that person. You are looking for two or more. But that's still thousands of people in the 2nd degree network. And you have to expand the connection list every time to check whether your new friend is one of those two+ connections.

Otherwise, you have to use LinkedIn API but that does not seem to be a very friendly interface.

To my knowledge, there is no all-in-one removal tool that is able to suggest who may be a good candidate to boot from your network (something similar to ManageFlitter for Twitter).

However, I can suggest two tools to assist you with trimming & organizing your network (which I personally use):

• InMaps: Visualizes your network and identifies clusters/sub-networks, and relative "distance" (relevance) of connections to each network. It allows you to easily spot "fringe" connections that may be not that relevant (or possible in reverse to be "bridge" connection to new clusters). And it is a part of LinkedIn Labs.

• MyWebCareer: More of an analytical dashboard-type tool (something of a Klout type). Also gives you a visual representation of your network which allows you to view sub-networks individually and indicating how a certain person is part of that network (current position vs. past position in a company). This can also suggest candidates for removal. However, in relation to your task it only be a complimentary tool to InMaps, I think. But it gives you some other functionality like giving scores about stuff, integrating Facebook and StackExchange-type sites into analysis.

Overall, I do agree that a smart network management tool is in order for LinkedIn. Let's hope it comes along soon!