To use Dijkstra's you'd have to modified it accept the start as the destination, ensured all positive weights (doesn't work for negative cycles), and progressively updated the weights on the edges that you've already found to be part of a cycle so they wouldn't be detected again; but then some edges will be part of other cycles..and on and on and on - in other words, it would be a very BAD way to find all of your circles of friends.
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