1. Are people associating based on gender similarity?

2. First Make a Block Model
Attribute N1 N2 N3 N4 N5 N6 N7 N8 N9
Male 1
Female

3. First Make a Block Model
Attribute N1 N2 N3 N4 N5 N6 N7 N8 N9
Female 1
Male

4. First Make a Block Model
Attribute N2 N4 N9 N1 N3 N5 N6 N7 N8
Female 1
Male

5. First Make a Block Model
Block Densities
Attribute N2 N4 N9 N1 N3 N5 N6 N7 N8
Female 1
Male

6. Naïve Approach – calculate the fraction of same gender ties1 2 3 4 5 6 7 8 9 10 N1 N6 N3 N2 N7 N4 N8 N5 N9
72% (13/18) of the edges are between vertices of the same gender

7. Finding the number of same-class ties
(“Turn off the mixed-class ties with a Kronecker Delta”)
Kronecker Delta
𝛿 𝑐 𝑖 , 𝑐 𝑗 = 0, 𝑖𝑓 𝑐 𝑖 ≠ 𝑐 𝑗 &1, 𝑖𝑓 𝑐 𝑖 = 𝑐 𝑗

8. Finding the number of same-class ties
(“Turn off the mixed-class ties with a Kronecker Delta”)
Kronecker Delta
Actual number of same-class ties
𝛿 𝑐 𝑖 , 𝑐 𝑗 = 0, 𝑖𝑓 𝑐 𝑖 ≠ 𝑐 𝑗 &1, 𝑖𝑓 𝑐 𝑖 = 𝑐 𝑗
𝑒𝑑𝑔𝑒𝑠 (𝑖,𝑗) 𝛿 𝑐 𝑖 , 𝑐 𝑗 = 1 2 𝑖𝑗 𝐴 𝑖𝑗 𝛿 𝑐 𝑖 , 𝑐 𝑗 =13

9. Kleinberg’s method of estimating the number of expected edges…

10. Proportion of Males and Females
P(male)
p = 6/9 N3 N2
P(Female)
q = 3/9 N7 N4 N8 N5 N9

11. Probability of Selecting a Male or Female
P(male)
p = 6/9
p = 2/3 N3 N2
P(Female)
q = 3/9
q = 1/3 N7 N4 N8 N5 N9

12. Probability of a Male selecting a
Male-Male, Female-Female, Male-Female N1 N6
P(male)
p = 6/9
p = 2/3 N3
P(m-m)
p2 =4/9 N2
P(Female)
q = 3/9
q = 1/3 N7 N4
P(f-f)
q2 =1/9 N8 N5 N9
P(male-female)
P(female-male)
2pq = 4/9

13. Male-Male, Female-Female, Male-Female Ties
Expected number of
Male-Male, Female-Female, Male-Female Ties N1 N6
P(male)
p = 6/9
p = 2/3 N3
P(m-m)
p2 =4/9
p2 =8/18 N2
P(Female)
q = 3/9
q = 1/3 N7 N4
P(f-f)
q2 =1/9
q2 =2/18 N8 N5 N9
P(male-female)
P(female-male)
2pq = 4/9
2pq = 8/18

14. Expected number of Male-Male, Female-Female, Male-Female Ties
P(male)
p = 6/9
p = 2/3 N3
P(m-m)
p2 =4/9
p2 =8/18
8 M-M N2
P(Female)
q = 3/9
q = 1/3 N7 N4
P(f-f)
q2 =1/9
q2 =2/18
2 F-F N8 N5 N9
P(male-female)
P(female-male)
2pq = 4/9
2pq = 8/18
8 M-F
Total expected # of same gender ties: 10

15. Newman’s approach
“make connections at random while preserving the vertex degrees. Ignoring vertex degrees and making connections truly at random has been show to give much poorer results”
1 2 𝑖𝑗 𝑘 𝑖 𝑘 𝑗 2𝑚 𝛿 𝑐 𝑖 , 𝑐 𝑗

16. Expected number of same-class ties
1 2 𝑖𝑗 𝑘 𝑖 𝑘 𝑗 2𝑚 𝛿 𝑐 𝑖 , 𝑐 𝑗 =10.36

17. Measuring the Presence of Homophily – Calculating modularity
If there is no homophily effect, we should expect to see same gender ties.
Since we see 13 same gender ties instead of 10.36, there is some evidence of homophily
We see about 3 more same gender ties than we would expect if gender had no effect on tie formation.
1 2 𝑖𝑗 𝐴 𝑖𝑗 𝛿 𝑐 𝑖 , 𝑐 𝑗 − 1 2 𝑖𝑗 𝑘 𝑖 𝑘 𝑗 2𝑚 𝛿 𝑐 𝑖 , 𝑐 𝑗 = 1 2 𝑖𝑗 𝐴 𝑖𝑗 − 𝑘 𝑖 𝑘 𝑗 2𝑚 𝛿 𝑐 𝑖 , 𝑐 𝑗

18. Measuring the Presence of Homophily - Calculating modularity
If there is no homophily effect, we should expect to see 57% same gender ties.
Since we see 72% same gender ties instead of 57%, there is some evidence of homophily
We see 14.6% more same gender ties than what we would expect if gender had no effect on tie formation.
The modularity score is 0.146
𝑄= 1 2𝑚 𝑖𝑗 𝐴 𝑖𝑗 − 𝑘 𝑖 𝑘 𝑗 2𝑚 𝛿 𝑐 𝑖 , 𝑐 𝑗 =0.146

19. A much easier way to calculate modularity using a “Mixing Matrix”

20. Making Sociology Relevant: What do we want to say?
A few empirical facts: Some racially heterogeneous schools are socially segregated

21. Making Sociology Relevant: What do we want to say?
A few empirical facts: … while other heterogeneous schools are socially integrated. Why?

22. Making Sociology Relevant:
What do we want to say?

23. Scalar Characteristics
Assortative Mixing by
Scalar Characteristics