  Topological data analysis can reveal higher-order structure beyond pairwise
connections between vertices in complex networks. We present a new method based
on discrete Morse theory to study topological properties of unweighted and
undirected networks using persistent homology. Leveraging on the features of
discrete Morse theory, our method not only captures the topology of the clique
complex of such graphs via the concept of critical simplices, but also achieves
close to the theoretical minimum number of critical simplices in several
analyzed model and real networks. This leads to a reduced filtration scheme
based on the subsequence of the corresponding critical weights, thereby leading
to a significant increase in computational efficiency. We have employed our
filtration scheme to explore the persistent homology of several model and
real-world networks. In particular, we show that our method can detect
differences in the higher-order structure of networks, and the corresponding
persistence diagrams can be used to distinguish between different model
networks. In summary, our method based on discrete Morse theory further
increases the applicability of persistent homology to investigate the global
topology of complex networks.
