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

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.

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

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

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

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)

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…

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”

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)

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)

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)

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)

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