Homophily - UCF CAP-6318

Computational Analysis of Social Complexity
Fall 2026, Spencer Lyon

Prerequisites

Introduction to Graphs
Strong and Weak Ties

Outcomes

Understand the concept of homophily
Practice working through “by hand” examples of diagnosing homophily
Be prepared to computationally diagnose homophily in a large network

References

Easley and Kleinberg chapter 4 (especially section 4.1)

Datasets

Florentine family relationships: https://www.cs171.org/2018/assets/instructions/lab8/Lab8.html

Introduction¶

Main Idea¶

Consider your friends. Do they tend to
- Enjoy the same movies, music, hobbies as you?
- Hold similar religious or political beliefs?
- Come from similar schools, workplaces, or socio-economic settings?
What about a random sample of people in the world?
If you are like me, your answers likely indicate that you have more in common with your friends than you would expect to have with a random sample of people
This concept -- that we are similar to our friends -- is called homophily

Homophily in Graphs¶

In the context of graphs or networks, homophily means that nodes that are connected are more similar than nodes at a further distance in the graph
But what do we mean by more similar?
- Idea: We might have common friends.
  - This is an intrinsic force that led to node formation (e.g. triadic closure)
- Alternative: We may share characteristics or properties that are not represented in the graph -- external forces.
  - Examples: same race, gender, school, employer, sports team, etc.
These external forces are what homophily captures

Context¶

To identify if homophily is active in a network, we must have access to context on top of list of nodes and edges
One way to represent this context would be with a DataFrame in addition to a graph:
- One row per node
- One column indicating the node identifier (or just use row number)
- One column for additional characteristic

Suppose these six people are friends. We record whether each person plays soccer or chess.
Question: We haven’t changed any friendships. Could the characteristic we choose change what we find?
In Figure 1, match each node to its entry in the table. Both characteristics occur in three people, but their cross-type friendships differ.

Two copies of the same six-person friendship cycle, colored by soccer and chess participation. Soccer has two cross-type edges; chess has six. — Figure 1:The friendships stay fixed. Changing the characteristic changes which edges connect different types.

using DataFrames, Graphs, GraphPlot



df1 = DataFrame(
    family=[
        "Acciaiuoli", "Albizzi", "Barbadori", "Bischeri", "Castellani",
        "Ginori", "Guadagni", "Lamberteschi", "Medici", "Pazzi",
        "Peruzzi", "Ridolfi", "Salviati", "Strozzi", "Tornabuoni"
    ],
    wealth=[10, 36, 55, 44, 20, 32, 8, 42, 103, 48, 49,  27, 10, 146, 48],
    priorates=[53, 65, missing, 12, 22, missing, 21, 0, 53, missing, 42, 38, 35, 74, missing],
)

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It will be easier to do our homohpily calculations with binary data,
we’ll create new columns, high_wealth and high_power if the wealth and priorates columns, respectively, are above the column medians

using Statistics
df1[!, :high_wealth] = df1.wealth .> median(df1.wealth)
df1[!, :high_power] = df1.priorates .> median(df1.priorates[.!(ismissing.(df1.priorates))])
df1[ismissing.(df1.priorates), :high_power] .= false
df1

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marriages = [
    0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
    0 0 0 0 0 1 1 0 1 0 0 0 0 0 0
    0 0 0 0 1 0 0 0 1 0 0 0 0 0 0
    0 0 0 0 0 0 1 0 0 0 1 0 0 1 0
    0 0 1 0 0 0 0 0 0 0 1 0 0 1 0
    0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 1 0 1 0 0 0 1 0 0 0 0 0 0 1
    0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
    1 1 1 0 0 0 0 0 0 0 0 1 1 0 1
    0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
    0 0 0 1 1 0 0 0 0 0 0 0 0 1 0
    0 0 0 0 0 0 0 0 1 0 0 0 0 1 1
    0 0 0 0 0 0 0 0 1 1 0 0 0 0 0
    0 0 0 1 1 0 0 0 0 0 1 1 0 0 0
    0 0 0 0 0 0 1 0 1 0 0 1 0 0 0
]
g1 = Graph(marriages)

{15, 20} undirected simple Int64 graph

gplot(g1, nodelabel=df1.family)

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Measuring Homophily¶

Our discussion of homophily so far has been conceptual... let’s make it precise
We’ll compare observed edge frequencies with a random-formation benchmark
This resembles a null-hypothesis comparison from statistics, but we won’t carry out a formal significance test here

Random Homophily¶

Start with a thought experiment: edge formation does not depend on characteristic $X$
We have $N$ nodes, and $N_x$ of them have characteristic $X$
- The share with this characteristic is $p_x = N_x/N$
For a simple benchmark, imagine drawing each endpoint’s type independently from these population shares
- Both have $X$ : $p_x^2$
- Neither has $X$ : $(1-p_x)^2$
- One has $X$ and the other does not:

\begin{aligned} \Pr(\text{cross-characteristic pair}) &= p_x(1-p_x) + (1-p_x)p_x \\ &= 2p_x(1-p_x) \end{aligned}

(1)

This is our independent-endpoint benchmark. For a finite graph whose edges join distinct nodes, it is an approximation.
Question: Why do we add two terms for a cross-characteristic pair?
Figure 2 shows the four outcomes. What happens to the cross-type probability when one group gets smaller?

A two-by-two table of independent endpoint types. Two cross-type outcomes each contribute p_x times one minus p_x. — Figure 2:The two highlighted outcomes describe opposite orders of drawing the endpoint types. Together they give $2p_x(1-p_x)$ ; they do not mean we count an undirected edge twice.

Counting Frequencies¶

Now an empirical value...
Let there be $e$ edges
Let...

variable	meaning
$e_{xx}$	# edges between 2 $X$
$e_{yy}$	# edges between 2 not $X$
$e_{xy}$	# edges between 1 $X$ and 1 not $X$

Then $e = e_{xx} + e_{yy} + e_{xy}$
We’ll use these 4 numbers to count frequencies of edges between $X$ types and non- $X$ types

Testing for Homophily¶

We are now ready to compare our network with the benchmark
Recall the two quantities:
- $2p_x(1-p_x)$ : the cross-type probability under the independent-endpoint benchmark
- $e_{xy}/e$ : the observed proportion of cross-characteristic edges
The direction of the difference helps us describe the pattern:

Comparison	Pattern
$e_{xy}/e$ well above $2p_x(1-p_x)$	consistent with inverse homophily
$e_{xy}/e$ near $2p_x(1-p_x)$	near the random-formation benchmark
$e_{xy}/e$ well below $2p_x(1-p_x)$	consistent with homophily

Intuition: fewer cross-type edges than the benchmark means more same-type edges
A difference alone does not tell us whether it is statistically significant or whether the characteristic caused the edges to form

Example: high school relationships¶

Recall the graph of romantic relationships between high school students in Figure 3.
Each node is a student; an edge records a romantic relationship during the study’s 18-month observation period.
Question: does this graph exhibit homophily in gender? Why?

A high-school romantic-relationship network, with blue nodes labeled male and pink nodes labeled female in the original legend. One large connected component appears alongside smaller components. — Figure 3:Romantic relationships in a high school. Original image reproduced from Easley and Kleinberg, *Networks, Crowds, and Markets* (2010), Figure 2.7. Study: Bearman, Moody, and Stovel (2004), *Chains of Affection: The Structure of Adolescent Romantic and Sexual Networks*. Image credited to Mark Newman on the book’s website.

Example: Florentines¶

Let’s work through an example of measuring homophily using the Florentine data
I’ll repeat the data below

df1

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node_color = map(x -> x ? "blue" : "red", df1.high_wealth)
gplot(g1, nodelabel=df1.family, nodefillc=node_color)

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Step 1: Counting frequencies¶

First we need to count frequencies for all our characteristics
We’ll do that here

using DataStructures

function count_frequencies(vals)
    counts = DataStructures.counter(vals)
    total = length(vals)
    Dict(c => v / total for (c, v) in pairs(counts))
end

count_frequencies (generic function with 1 method)

count_frequencies(df1.high_wealth)

Dict(
    n => count_frequencies(df1[!, n])
    for n in names(df1)[4:end]
)

Dict{String, Dict{Bool, Float64}} with 2 entries:
  "high_wealth" => Dict(0=>0.533333, 1=>0.466667)
  "high_power"  => Dict(0=>0.666667, 1=>0.333333)

Step 2: Counting Edges¶

Next we need to count the number of edges of each type
This step is a bit trickier: we’ll need both the graph and the DataFrame
To not spoil the fun, we’ll leave this code as an exercise on the homework
For now we’ll count by hand
Question: Before looking at the tally, can you find every edge joining a high-wealth family to a low-wealth family?
Use the H/L labels and dashed cross-type edges in Figure 4. Count each marriage once.

The 15-family Florentine marriage network with high and low wealth labels. Five high-high, three low-low, and twelve cross-type edges sum to twenty marriages. — Figure 4:The wealth groups use the same threshold as our Julia code: high means strictly above the median of 42. Cross-type marriages account for 12 of the 20 edges.

Let’s compare the marriage network with the benchmark for high_wealth
There are 7 high-wealth families among the 15 families in our graph
Counting edge types gives:
- High–high edges: 5
- Low–low edges: 3
- Cross edges: 12
Total: $5 + 3 + 12 = 20$ edges
The observed cross-edge proportion is $12/20 = 0.60$
The high-wealth share is $p_x = 7/15 \approx 0.467$

E = ne(g1)
Exy = 12  # cross edges
n_high = 7
N = nv(g1)
px = n_high / N

# compare with the benchmark
2 * px * (1-px), Exy/E

Our independent-endpoint benchmark is $2(7/15)(8/15) \approx 0.498$
The observed cross-edge share is 0.60, about 0.10 higher
This points in the direction of inverse homophily: more cross-wealth marriages than the benchmark predicts
We haven’t established statistical significance, and the comparison alone doesn’t tell us why these families married

Exercise¶

Repeat the counting exercise, but for the high_power characteristic
What do you find? Do you see homophily in this characteristic?