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Fig. 1

From: Unveiling synchronization transitions in networks of coupled oscillators through persistent homology of local structures

Fig. 1The alternative text for this image may have been generated using AI.

Graphical representation of the networks used in this work: (a) Erdös-Rényi (ER, 30 nodes and 48 links), (b) scale-free (SF, 30 nodes and 56 links), (c) random network (RN, 30 nodes and 40 links), and (d) experimental network (EN, 28 nodes and 42 links, colored nodes and edges illustrate a triangle in the network). (e) Illustration of the procedure used to calculate the normalized persistent entropy of zero homology groups, \(NPE(H_0)\), for a point cloud defined by three time series associated to three nodes (oscillators), \(x_1\), \(x_2\), and \(x_3\). The filtration procedure is applied by increasing the maximum edge length, \(\epsilon _i\), to obtain the barcodes (the difference death-born times of each homology class) of the resulting simplicial complex. For each \(\epsilon _i\), the \(NPE_{\epsilon _i}(H_0)\) is calculated with Eq. (2) (see Methods).

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