Directed network systems

If one adds directionality to the connections in a network it becomes particularly easy to set up rules for clusters of nodes that cannot overlap. For no two clusters whose dangling connections all point inwards can ever overlap, at least so long as neither of these clusters themselves contain subclusters whose dangling connections similarly all point inwards. The pictures below show a few examples of such clusters. Note that in a random network of n nodes, about n/8 such clusters typically occur.