Apparently, JavaScript CAN make you new friends.

linkedin_auto_connect.js

/**
 * Linkedin auto-connect thingy.
 * 
 * Step 1) Log in to Linkedin and go to https://www.linkedin.com/people/pymk
 *         (This is the "People You May Know page")
 * Step 2) Open your browser console and run this javascript.
 */
(function($) {
	var processedCards = [];	
	var lih = function(){
		// Remove cards that require email
		$('ul.people-cards-list li.card.guest').remove();

		// Get the other buttons that have loaded
		var buttons = $('ul.people-cards-list li.card:not(.guest) button.bt-request-buffed.buffed-blue-bkg-1:not(.processed)');

		if (buttons.length){
			// Scroll down to the last one to cause ajax loading of the next set of buttons.
			$('html body').stop().animate({
			    scrollTop: buttons.last().offset().top
			}, 3000);

			// Set random intervals for clicking the buttons.
			var timer = 0;
			buttons.addClass('processed');
			buttons.each(function(){
				var id = $(this).parent().find('button.bt-incommon').attr('data-memberid'),
					name = $(this).parent().find('.profile-info .name-wrapper .name').clone().find('span').remove().end().text();
				if (typeof(processedCards[id]) != 'undefined'){
					// Already clicked this person
					$(this).parent().parent().parent().remove();

					console.log('Already connected: ' + name);
				} else {
					// New person, goodie...
					processedCards[id] = true;

					var rand = Math.random() * (5000 - 1000) + 1000,
						t = $(this);
					timer += rand;

					setTimeout(function(){
						console.log('Connecting: ' + name);
						t.trigger('click');
					}, timer);
				}
			});
		}
	}
	setInterval(function(){ lih(); }, 6000);
	lih();
})(jQuery);