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Fig. 3 | SpringerPlus

Fig. 3

From: The topology of the directed clique complex as a network invariant

Fig. 3

The directed clique complex. a The directed clique complex of the represented graph consists of a 0-simplex for each vertex and a 1-simplex for each edge. There is only one 2-simplex (123). Note that ‘2453’ does not form a 3-simplex because it is not fully connected. ‘356’ does not form a simplex either, because the edges are not oriented correctly. b The addition of the edge (52) to the graph in a does not contribute to creating any new 2-simplex, because of its orientation. The edges connecting the vertices 2, 3 and 5 (respectively 2, 4 and 5) are oriented cyclically, and therefore they do not follow the conditions of the definition of directed clique complex stated in Eq. (1). c By reversing the orientation of the new edge (25), we obtain two new 2-simplices: (235) and (245). Note that we do not have any 3-simplex. d We added a new edge (43), thus the sub-graph (2435) becomes fully connected and is oriented correctly to be a 3-simplex in the directed clique complex. In addition, this construction gives two other 2-simplices: (243) and (435)

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