1. Graph Analysis by Persistent Homology
Dingkang Wang

2. Backgroud
Objectives
Preprocessing
Distance Metrics
Simplices Extraction & Comparison
Results

3. Background What is graph analysis? Commonly used methods:
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
Data Analysis or Network Analysis is a big area of large research interest, previously,
most of the network analysis is based on traditional metrics, e.g., the degree distribution,
graph diameter and so on. However, these metrics are vulnerable to noise data points,
and may not be able to reveal the hidden dierence. Topological analysis by persistence
homology is a new excellent tool for network analysis, although most of the time, it's used
for analyzing data embedded in Euclidean space, however, for node-link network, we can
still make use of it after some distance calculation. This method is stable, which means
a little perturbation won't make a signicant eect on the output diagram, in addition,
it can nd out the topological features of a graph, which cannot be shown through other
methods.

4. Objectives Part I Part II
Characterize the structure of graphs in different categories.
Part II
Find out the “interesting” year in senate
voting graphs.
stanford network analysis project
Slashdot Social network
Arxiv High Energy Physics paper citation network
Gnutella peer to peer network from August

5. 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
In my project, I defined the Jaccard Index of an edge (u; v), or equivalently, for two nodes u; v with a connection as follows: where N(u) represents the neighbor nodes of node u.
Now we can set a threshold, when the Jaccard Index of an edge is less than the threshold, we can consider it as a noise edge and remove that edge. Intuitively, that means the
similarity between the two endpoints are too low, so there should not be an edge between these two nodes.

6. Distance Metrics Shortest Path Distance Diffusion Distance
Easier to calculate
More sensitive
Lack of variety
Diffusion Distance
Step 1
Step 2
Step 3
we define a pairwise similarity matrix between points, for example using a Gaussian kernel with width σ2
a diagonal normalization matrix, make it a transition probability matrix
Calculate the distance with the specific choice w(y)=1/φ0(y) for the weight function, which takes into account the (empirical) local density of the points, and puts more weight on low density points.
This distance is robust to noise, since the distance between two points depends on all possible paths of length  between the points.
Parameters are hard to choose, so we try different parameters and find a suitable one.

7. Simplices Extraction & Diagram Comparison
Using Rips Complex
Using Phat to generate persistent diagrams
Using two different metrics, bottleneck distance and Wasserstein distance
In this step, we will only extract simplicial complexes up to 3-dimension, i.e, we only focus
on nodes, edges, triangles and tetrahedrons, which is due to the limit of time complexity.
Also, we need to include the birth date of these complexes, more specically, we use Rips
complex.
For Rips complex, we add a k-simplex when its boundary simplices are all in, e.g, there
are 3 nodes, a, b, c, and their corresponding edges ab, bc, ac have weight 4; 5; 6, we add
these edges at time 4; 5; and 6. Once you all all the edges, the triangle abc will be added,
so triangle abc will be in at time 6.
First the bottleneck distance used in my project can be dened as follows:
In other words, it will nd out a bijection between two node set, mapping x in X to (x) in
Y, and try to minimize the maximum of the pairwise distances. When number of two node
For wasserstein distance, which is more stable than the bottleneck distance, can be dened
as follows:
We can see that the wasserstein distance uses the sum instead of the maximum of all
pairwise distances, so in some sense, it will be less sensitive to the outliners.

8. Results

9. Results

10. Results

11. Thank you! Any questions?