Graph Analysis by Persistent Homology Dingkang Wang l
Graph Analysis by Persistent Homology Dingkang Wang
l Backgroud l Objectives l Preprocessing l Distance Metrics l Simplices Extraction & Comparison l Results 2
Background What is graph analysis? • • Characterize structures of a large graph in terms of nodes and edges. Make it possible to compare two large graphs. Commonly used methods: • • • Degree distribution Diameter, shortest path distance distribution Community structures In my project: • Persistent Homology 3
Objectives Part I Characterize the structure of graphs in different categories. Part II Find out the “interesting” year in senate voting graphs. 4
Preprocessing Denoising by Jaccard Index: Landmark Sampling when input graph is too large: • • Pick the first landmark randomly Pick the next landmark, which is farthest from the chosen landmarks until you get enough landmarks Other nodes will be assigned to one of the landmarks based on the distance Two landmarks will have connection while their communities have connections 5
Distance Metrics Shortest Path Distance • • • Easier to calculate More sensitive Lack of variety Diffusion Distance • • • Step 1 Step 2 Step 3 6
Simplices Extraction & Diagram Comparison • Using Rips Complex • Using Phat to generate persistent diagrams • Using two different metrics, bottleneck distance and Wasserstein distance 7
Results 8
Results 9
Results 10
Thank you! Any questions? 11
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