Visual detection of structural changes in time-varying graphs using persistent homology

PKU blog about this paper

Basic knowledge: 

1. what is time-varying graphs?

time-varying graph VS static graph. 

a time-varying graph - an ordered sequence of graph instances. 

2. how to measure similarity or dissimilarly between graphs?

way1: map the graphs into a feature space and then define distances on this space. 

way2: graph comparison with known node correspondences. 

way3: graph comparison without unknown node correspondences.

3. how to visualize time-varying graphs?

two major categories: animation and timelines. 

4. how to analysis static graph and visualize static graphs?

Static graph visualization: use variations on node-link visualizations to display graphs. 

For dense/clutter graphs, -------> edge bundling.  

Questions: 

2. how to use persistent homology to capture topological features of time-varying graphs.

1. how to embed the graph into a metric space.  then topological techniques can be applied into this metric space. 

Methodology:

basic rules: use persistent homology to identify and compare features in a time-varying graph. 

Their visual design goal: identify high-level structural changes in a time-varying graph. 

time-varying graph: an ordered sequence of static graph instances. 

Methods:  

1. each graph is embedded into a metric space. G = {G_1, G_2,... G_i...} This yields a symmetric distance matrixd d_i,  d_sp(x,y) is the shortest path distance between node x and node y. 
2. extract topological features from each G_i by using persistnet homology to its metric space. 
3. calculate the distance between persistence diagrams and then project them by using classical multi-dimensional scaling(MDS). 

Knowledgebase: 

1. Dijkstra's algotithm: 迪杰斯特拉算法使用了广度优先搜索解决赋权有向图的单源最短路径问题。

2. Quantitative Journey blog about TDA: topological data analysis