  Classification of large and dense networks based on topology is very
difficult due to the computational challenges of extracting meaningful
topological features from real-world networks. In this paper we present a
computationally tractable approach to topological classification of networks by
using principled theory from persistent homology and optimal transport to
define a novel vector representation for topological features. The proposed
vector space is based on the Wasserstein distance between persistence barcodes.
The 1-skeleton of the network graph is employed to obtain 1-dimensional
persistence barcodes that represent connected components and cycles. These
barcodes and the corresponding Wasserstein distance can be computed very
efficiently. The effectiveness of the proposed vector space is demonstrated
using support vector machines to classify simulated networks and measured
functional brain networks.
