Before revision:  In this work, we utilize the PageRank function on the lower-star filtration of the graph as input to persistent homology to study the problem of graph similarity. By representing each graph as a persistence diagram,we can then compare outputs using the bottleneck distance .
After revision:  In this work, we utilize the PageRank function on the lower-star filtration of the graph as input to persistent homology to obtain a scalable graph descriptor and utilize it to compare the similarities between graphs. For a given graph G(V,E), our descriptor can be computed in O(|E|\alpha(|V|)), where \alpha is the inverse Ackermann function which makes it scalable and computable on massive graphs .