  Almost all statistical and machine learning methods in analyzing brain
networks rely on distances and loss functions, which are mostly Euclidean or
matrix norms. The Euclidean or matrix distances may fail to capture underlying
subtle topological differences in brain networks. Further, Euclidean distances
are sensitive to outliers. A few extreme edge weights may severely affect the
distance. Thus it is necessary to use distances and loss functions that
recognize topology of data. In this review paper, we survey various topological
distance and loss functions from topological data analysis (TDA) and persistent
homology that can be used in brain network analysis more effectively. Although
there are many recent brain imaging studies that are based on TDA methods,
possibly due to the lack of method awareness, TDA has not taken as the
mainstream tool in brain imaging field yet. The main purpose of this paper is
provide the relevant technical survey of these powerful tools that are
immediately applicable to brain network data.
