  Topological data analysis (TDA) delivers invaluable and complementary
information on the intrinsic properties of data inaccessible to conventional
methods. However, high computational costs remain the primary roadblock
hindering the successful application of TDA in real-world studies, particularly
with machine learning on large complex networks.
  Indeed, most modern networks such as citation, blockchain, and online social
networks often have hundreds of thousands of vertices, making the application
of existing TDA methods infeasible. We develop two new, remarkably simple but
effective algorithms to compute the exact persistence diagrams of large graphs
to address this major TDA limitation. First, we prove that $(k+1)$-core of a
graph $\mathcal{G}$ suffices to compute its $k^{th}$ persistence diagram,
$PD_k(\mathcal{G})$. Second, we introduce a pruning algorithm for graphs to
compute their persistence diagrams by removing the dominated vertices. Our
experiments on large networks show that our novel approach can achieve
computational gains up to 95%.
  The developed framework provides the first bridge between the graph theory
and TDA, with applications in machine learning of large complex networks. Our
implementation is available at
https://github.com/cakcora/PersistentHomologyWithCoralPrunit
