Are people associating based on gender similarity?

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

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

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

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

Naïve Approach – calculate the fraction of same gender ties 1 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

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

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

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

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

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

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

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

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

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

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

Measuring the Presence of Homophily – Calculating modularity If there is no homophily effect, we should expect to see 10.36 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

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

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

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

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

Making Sociology Relevant: What do we want to say?

Scalar Characteristics Assortative Mixing by Scalar Characteristics