Oh, those are aphid lion eggs? I'll have to move some of them from the milkweeds where they've been laid to the incarnatas, some of which always seem to get an aphid infestation near the blossoms.
p.s. About the thread, I had a strong suspicion one year that a small group of monarch eggs I had in a container got eaten by an aphid lion I found in there. Guess I was careless and it hitchhiked in there on one of the leaves or something, but after a couple of mornings, the eggs were nowhere to be seen and I had no cats either ... just one aphid lion.
So here's my layperson's guess on the thread, it's either to keep the voracious little devils from eating other aphid lion eggs, or to keep other egg-sucking predator species from doing it. I imagine the thread would not hold the weight of some other larvae crawling up it to get at the egg (if it got noticed at all).