  Information networks are becoming increasingly popular to capture complex
relationships across various disciplines, such as social networks, citation
networks, and biological networks. The primary challenge in this domain is
measuring similarity or distance between networks based on topology. However,
classical graph-theoretic measures are usually local and mainly based on
differences between either node or edge measurements or correlations without
considering the topology of networks such as the connected components or holes.
In recent years, mathematical tools and deep learning based methods have become
popular to extract the topological features of networks. Persistent homology
(PH) is a mathematical tool in computational topology that measures the
topological features of data that persist across multiple scales with
applications ranging from biological networks to social networks. In this
paper, we provide a conceptual review of key advancements in this area of using
PH on complex network science. We give a brief mathematical background on PH,
review different methods (i.e. filtrations) to define PH on networks and
highlight different algorithms and applications where PH is used in solving
network mining problems. In doing so, we develop a unified framework to
describe these recent approaches and emphasize major conceptual distinctions.
We conclude with directions for future work. We focus our review on recent
approaches that get significant attention in the mathematics and data mining
communities working on network data. We believe our summary of the analysis of
PH on networks will provide important insights to researchers in applied
network science.
