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Fig. 3 | Applied Network Science

Fig. 3

From: Non-backtracking cycles: length spectrum theory and graph mining applications

Fig. 3

Aggregating the values of the length spectrum. a A graph G with two nodes highlighted in red and blue. These two nodes are used as basepoints to construct two versions of the fundamental group: π1. b The set of all cycles based at the red node (left) and blue node (right). For either set of cycles, we encircle together those that are homotopy equivalent, thus forming a homotopy class. We annotate each class with its length, and we highlight the representative with minimal length. Note that the lengths of corresponding cycles (those that share all nodes except for the basepoints) can change when the basepoints change. c We have kept only the highlighted representative in each class in b) and encircled together those that are conjugate. In each conjugacy class, we highlight the (part of) each cycle that corresponds to the free homotopy loop. d By taking one representative of each conjugacy class, and ignoring basepoints, we arrive at the free homotopy classes, or equivalently, at the set of non-backtracking cycles. We annotate each cycle with its length. We arrive at the same set regardless of the basepoint. The ellipses inside the closed curves mean that there are infinitely many more elements in each set. The ellipses outside the curves mean that there are infinitely many more classes or cycles

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