1. If you want to connect with a stranger, include a message!
I never ever add anyone without sending a message. To send a message, I look at the person’s profile to see what shared angles we both have. I also highly recommend checking out a few of that person's latest posts / comments. I am confident that will spark something authentic and interesting for you to say.
To do that: Step 1: Go to the person's LinkedIn page and click on "See All" under "Activity" Step 2: You can sort the Activity by "Posts" or "Articles" to filter out all of their other activity (which includes comments and likes).
This picture is an example of an intro message I sent, which worked really well; and this person and I have already exchanged several more messages.
2. If a stranger adds you with a message, write them one!
Sometimes people add me without sending a message. I make sure I message them and start establishing a connection. (Otherwise it’s just pings in the ether, who will remember?) I saw today that a UC Berkeley professor had added me. I sent them this message to start our connection.
3. Social Media is Social: comment, like, post!
One great way to increase your LinkedIn profile's visibility, and expand your connections, is to leave (thoughtful) comments under posts of people who have many followers. Try it, you will immediately get some additional connection requests; you will also start getting on people's radars (including the author), which makes it easier for you to add them later.