1. Other Random Graph Models
Figure from: The political blogosphere and the 2004 U.S. election: divided they blog (Adamic and Glance, 2005)

2. Community structure of political blogs over the 2 month period preceding the 2004 US Presidential Election. The colors reflect political orientation, red for conservative, and blue for liberal. Orange links go from liberal to conservative, and purple ones from conservative to liberal. The size of each blog reflects the number of other blogs that link to it.

3. Questions:
What is the degree of interaction between liberal and conservative blogs?
Are there structural differences between the behavior of liberal and conservative blogs?

4. Stochastic Block Model
𝑝 1,1 =0.6
Divide the vertices into two disjoint communities 𝐶 1 and 𝐶 2 .
Connect two vertices in 𝐶 1 with probability 𝑝 1,1
Connect two vertices in 𝐶 2 with probability 𝑝 2,2
Connect a vertex in 𝐶 1 to a vertex in 𝐶 2 with probability 𝑝 1,2 = 𝑝 2,1
All edges are independent
𝑝 1,2 =0.1
𝑝 2,2 =0.8

5. Stochastic Block Model (general)
𝐶 3
Divide the vertices into disjoint communities 𝐶 1 , 𝐶 2 ,…, 𝐶 𝑘 .
Connect two vertices in 𝐶 𝑖 with probability 𝑝 𝑖,𝑖
Connect a vertex in 𝐶 𝑖 to a vertex in 𝐶 𝑗 with probability 𝑝 𝑖,𝑗 = 𝑝 𝑗,𝑖
All edges are independent
𝐶 1
𝐶 2
𝐶 4

6. Machine Learning Application
Community detection and recovery
Detection: determine whether a graph has community structure
Recovery: if it does, determine the communities
Image from: Community Detection and Stochastic Block Models (Abbe 2017)

7. Degrees of Separation
The number of degrees of separation between two individuals is the smallest number of social network connections necessary to connect them
People you personally know have one degree of separation from you
People who know someone you know (but whom you don’t know) have two degrees of separation from you
Etc…

8. Bernhard Caesar Einstein
Degrees of Separation
Question: How many degrees separate you from Albert Einstein?
Answer: (no more than) 5!
You
Lila
T. Einstein
Mr. Einstein
Bernhard Caesar Einstein
Albert Einstein
“Six degrees of separation”

9. Small-world Networks
Start with all vertices connected to their 𝑘 nearest neighbors
Randomly re-wire edges with probability 𝑝
Image from: Collective dynamics of ‘small-world’ networks (Watts and Strogatz, 1998)

10. Small-world Networks 𝐿(𝑝)= typical path length between vertices
𝐿(0) is very large, 𝐿(1) is very small
𝐿(0.1) is almost as small as 𝐿(1)!
𝐿(𝑝) 𝑝
Image from: Collective dynamics of ‘small-world’ networks (Watts and Strogatz, 1998)

11. Small-world Networks 𝐶(𝑝)= “cliquishness” (measure of clustering)
𝐶(0) is very large, 𝐶(1) is very small
𝐶 drops off much more slowly
𝐶(𝑝) 𝑝
Image from: Collective dynamics of ‘small-world’ networks (Watts and Strogatz, 1998)

12. Small-world Networks
Small-world phenomenon: small path length between vertices, but high levels of clustering
𝐶(𝑝)
𝐿(𝑝) 𝑝
Small-world
Image from: Collective dynamics of ‘small-world’ networks (Watts and Strogatz, 1998)

13. Some NetworkX random graph generators:
Watts-Strogatz
Barabási-Albert
Random Lobster
d-Regular
Gaussian Random Partition