Fig. 2

a The flowchart of constructing graph filtration, and persistent features in graph filtration can be visualized using barcode. For β₀ barcode, there are 6 connected components in filtration distance 0, which are corresponding to 6 bars. After integrating two connected components into one unit, one of the bars that represents the original components disappears; the other as the new connected component extends to next step. Therefore, the bars in step 1 become 3 that are corresponding to 3 connected components in G_1. For an example of β₁ barcode, we observe that the cycle with yellow is born in filtration step 2, and continues to step 4 before it is completely covered in filtration step 5. Subsequently, frequency network can be obtained by integrating all cycle structures into a network, where P is the persistent interval, the values in frequency network represent probability-of-appearance of an edge. b We use functional data analysis (FDA) to fit β₀ barcode to β₀ curve, and further calculate the corresponding derivative curve. c Persistence landscapes is applied to convert β₁ barcode to landscape layers; furthermore, we can compute the unique average for a collection of persistent landscapes