bullet to paragraph Part1

SelimC06 · Jun 7, 2023 · 33b90bc · 33b90bc
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diff --git a/Images/Image7.png b/Images/Image7.png
diff --git a/Images/dots.png b/Images/dots.png
diff --git a/Images/dots2.png b/Images/dots2.png
diff --git a/Images/image5.png b/Images/image5.png
diff --git a/Images/image6.png b/Images/image6.png
diff --git a/Part1.Rmd b/Part1.Rmd
diff --git a/Part1.html b/Part1.html
diff --git a/Part2.Rmd b/Part2.Rmd
@@ -0,0 +1,58 @@
+---
+title: "**Applications of Topology in Networks**"
+output: html_document
+---
+
+```{r setup, include=FALSE}
+knitr::opts_chunk$set(echo = TRUE)
+```
+## Part 2
+
+<br>
+
+#### **Graph Persistent Homology**
+
+**#Edge Activations**
+
+  * Consider the graph below where edge weights indicate distances.
+
+  * Lets use a Vietoris Rips complex: denoted as Ripsɑ(X), consists of simplices [x0,...,xk] satisfying the condition that the distance between any pair of points, dx (xi, xj), is less than or equal to a for all (i, j).
+<center>
+  ![Let’s define a filtration scale parameter which we will increase from a minimum to a maximum.](Images/dots.png){width=30%}
+</center>
+
+  * Min scale = 0 (we could start from 0.5 or any other value as well)
+
+  * Max scale = 100 (we could go longer, but it would not change anything)
+<center>
+  ![](Images/dots2.png){width=80%}
+</center>
+
+  * Multiple questions:
+    * Question 1: How do we define a distance based on edge weights? In the previous slide, the edge weights represented distances, eliminating the need for explicit distance definition.
+    * Question 2: What happens if the graph is unweighted?
+      1. Graph resistance distance: In a graph, the resistance between two vertices measures the level of difficulty for electric current to flow between them.
+      2. Shortest path distance (works with weighted edges as well):  a measure of the minimum number of edges along the shortest path between two vertices in a graph.
+      3. Ricci curvature (works with weighted edges as well): the concept of Ricci curvature is a generalization of Ricci curvature from Riemannian geometry. It provides a way to quantify the geometric properties of the network based on its connectivity and edge weights.
+<center>
+  ![](Images/image5.png){width=60%}
+</center>
+  * Question 3: What happens if the graph is directed?
+    * Use distance measures introduced in the first question, but define distances based on incoming or outgoing edges only
+
+**#Node Activation**
+
+  * Consider the graph below where node features indicate ages.
+<center>
+  ![](Images/image6.png){width=30%}
+</center>
+  * We will use sub or super level filtration. 
+    * Sublevel: filtration from minimum to maximum scales
+    * Superlevel: filtration from maximum to minimum scales
+
+  * Min scale = 10 (we could start from 15 or any other value as well)
+
+  * Max scale = 80 (we could go longer, but it would not change anything)
+<center>
+  ![](Images/Image7.png){width=80%}
+</center>
diff --git a/Part2.html b/Part2.html
diff --git a/Topological Data Analysis Project.Rmd b/Topological Data Analysis Project.Rmd
@@ -14,7 +14,7 @@ output: html_document
  * The Language of TDA
  * Persistent Homology in TDA
 
-#### [**Part 2**](https://www.dropbox.com/home/Sevilla?preview=Akcora+day+2.pptx)
+#### [**Part 2**](https://selimc06.github.io/TDA-on-Graphs/Part2.html)
 
  * Graph Persistent Homology
  * A few tips and insights for TDA on graphs

diff --git a/Topological-Data-Analysis-Project.html b/Topological-Data-Analysis-Project.html
@@ -367,7 +367,7 @@ <h4><a href="https://selimc06.github.io/TDA-on-Graphs/Part1.html"><strong>Part
 </ul>
 </div>
 <div id="part-2" class="section level4">
-<h4><a href="https://www.dropbox.com/home/Sevilla?preview=Akcora+day+2.pptx"><strong>Part
+<h4><a href="https://selimc06.github.io/TDA-on-Graphs/Part2.html"><strong>Part
 2</strong></a></h4>
 <ul>
 <li>Graph Persistent Homology</li>