  The truss, a relaxation of the clique based on triangles, serves to identify
clusters of actors in a way that is easy to interpret and is computationally
attractive. This paper introduces the 4-cycle-based relative to the truss,
called the trapeze, presents a weighted extension of both the truss and
trapeze, and offers the refinements of strong trusses and trapezes and summit
trusses and trapezes. Use of trapezes permits the application to bipartite
graphs, while the weighted versions permit variation of support due to natural
edge weights without increasing computational complexity. Finally, strong and
summit versions make for easy determination of communities across graphs of
varying density. Each of these structures offers guaranteed computation in
polynomial time, is motivated by a natural observation of social cohesion, and
is related nicely to other standard structures.
