Topological Learning and Its Application to Multimodal Brain Network Integration

Topological Data Analysis of Human Brain Networks Through Order Statistics

“…Let G 1 ( p , u ) and G 2 ( p , v ) be two networks and the corresponding barcodes (or persistent diagrams) be P 1 and P 2 . Then, the 2- Wasserstein distance on barcodes is given by over every possible bijection τ between P 1 and P 2 [23, 60, 61]. For graph filtrations, barcodes are 1D scatter points.…”

“…For graph filtrations, barcodes are 1D scatter points. Therefore, the bijection τ can be simplified to the norm between the sorted birth values of connected components or the sorted death values of cycles [23].…”

“…Since the barcodes completely characterize the topology of a network [14], the topological similarity between two such networks can be quantified using a distance metric between the corresponding 0D or 1D barcodes [56]. Often used metric is the Wasserstein distance [23,[57][58][59]. Let G 1 (p, u) and G 2 (p, v) be two networks and the corresponding barcodes (or persistent diagrams) be P 1 and P 2 .…”

“…over every possible bijection τ between P 1 and P 2 [23,60,61]. For graph filtrations, barcodes are 1D scatter points.…”

“…The PH based topological distances are found to consistently outperform traditional graph based metrics [21]. The main idea of using PH to brain networks is to generate a sequence of nested networks over every possible threshold through a graph filtration, which builds the hierarchical structure of the brain networks at multiple scales [10,[22][23][24].…”