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Fig. 4 | Applied Network Science

Fig. 4

From: Path homologies of motifs and temporal network representations

Fig. 4

(L) \({\tilde{\beta }}_2 > 0\) for these 17 (of 5984 total) DAGs on six vertices. Note that the DAG in the upper left is a ubiquitous subgraph of the others; the path homology of this graph and others like it is analyzed in Chowdhury et al. (2019). (R) \({\tilde{\beta }}_2 > 0\) for these 17 (of 156 total) undirected graphs on six vertices. The graphs in the first two rows all have \({\tilde{\beta }}_\bullet = (0,0,1,0,\dots )\), and the remaining seven graphs (from left to right, top to bottom) respectively have \({\tilde{\beta }}_\bullet = (1,0,1,0,\dots )\); \({\tilde{\beta }}_\bullet = (0,1,1,0,\dots )\); \({\tilde{\beta }}_\bullet = (0,1,1,0,\dots )\); \({\tilde{\beta }}_\bullet = (0,2,1,0,\dots )\); \({\tilde{\beta }}_\bullet = (0,2,1,0,\dots )\); \({\tilde{\beta }}_\bullet = (0,2,1,0,\dots )\), and \({\tilde{\beta }}_\bullet = (1,2,1,0,\dots )\). The common “bow tie” motif here appears to be the cause for emergence of 2-homology in transportation networks as capacities are filtered (this will be elaborated on in future work); meanwhile, polygons with \(\ge 5\) sides have too many sides for paths in opposing directions to “destructively interfere.” That is, although these graphs are undirected, the directed paths of length 4 through them exhibit more coherence than in other graphs of the same size

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