Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Homophily and Search
Sergio Currarini1
Fernando Vega Redondo2
1 Universita’
2 European
di Venezia
University Institute
June 17, 2010
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Definition and Discussion
Measuring Homophily
Empirical Patterns of Homophily
Homophily: the tendency of social actors to form ties with
other actors similar to themselves
Pervasive feature of many social networks:
Type of similarity: race, gender, religion, preferences, age,...
Type of relation: friendship, marriage, acquaintance,
professional ties, job contacts,..
Distance in characteristics translates into distance in social
network
The flow of information, opinions, etc... becomes localized if
characteritics matter
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Definition and Discussion
Measuring Homophily
Empirical Patterns of Homophily
Homophily at work
At U.S. National level, only 8 percent of people discuss
important matters with people from other race (Marsden
1987) - 1/7 of expected figure.
Interracial Marriages in U.S.: 1 percent of White, 5 percent of
Black, 14 percent of Asian (Fryer 2006)
Middle school U.S.: only 1/10 of expected cross-race
friendships are observed (Shrum et al. 1988)
Closest friend: 10 percent of men name a woman, 32 percent
of women name a man (about 50 percent expected)
(Verbrugge 1977).
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Definition and Discussion
Measuring Homophily
Empirical Patterns of Homophily
U.S. High School Frienships Network
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Definition and Discussion
Measuring Homophily
Empirical Patterns of Homophily
Sociological literature: Lazarsfeld and Merton (1954), Blau
(1974, 1977), Shrum (1988), Marsden (1988), Moody (2001),
McPherson (1991).
Recent contributions in economics:
Currarini, Jackson, Pin (2009a, 2010): model of friendship
formation - roles of choice and opportunities
Bramouille and Rogers (2009), Currarini, Jackson, Pin
(2009b): random growing network with heterogeneous agents scientific citations
Baccara and Yariv (2009): polarization and similarity in groups.
Galeotti, Squintani, Giglino (2009): cheap talk games played
on networks
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Definition and Discussion
Measuring Homophily
Empirical Patterns of Homophily
Measures of Homophily
si = average number of same type ties in group i
ti = average total number of ties in group i
dji = average number of type j friends made by agents of type i.
di = ti − si
wi = population share of group i.
Excess Representation of type j
dji
− wj
ti
(1)
si
− wi
ti
(2)
Excess homophily of type i
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Definition and Discussion
Measuring Homophily
Empirical Patterns of Homophily
Coleman Homophily Index
IHi =
si
ti
− wi
1 − wi
(1 − wi ) is the maximal potential excess homophily of group i.
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
(3)
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Definition and Discussion
Measuring Homophily
Empirical Patterns of Homophily
−.2
0
Coleman Index
.2
.4
.6
.8
High School Friendships
0
.2
.4
.6
Relative Group Size
Coleman
.8
1
Fitted values
AddHealth Data. Each dot is one ethnic group in one High School.
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Definition and Discussion
Measuring Homophily
Empirical Patterns of Homophily
−.2
0
Coleman Index
.2
.4
.6
.8
Inter-ethnic Marriages
0
.2
.4
.6
Relative Group Size
Coleman
.8
1
Fitted values
IPUMS (Integrated Public Use Microdata Series). Each dot is one
ethnic group in one marriage market.
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Search
Monotonicity Axiom
The Model
n agents.
η units (intensity) of search for matches with other agents
Θ of cardinality θ ≤ n: search pool.
Search profile (η, θ) gives access to potential matches, given
by the random variable ν(η, θ), with distribution Pη,θ .
Each potential match is enjoyed with independent probability
p.
Each agent has a utility function U (y) defined on the number
of enjoyed matches y. We will require that U (0) = 0,
U (1) > 0, U non decreasing .
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Search
Monotonicity Axiom
Monotonicity Axioms
Strong Monotonicity
Let (η, θ) ≥ (η 0 , θ0 ). Then ν(η, θ) FSD ν(η 0 , θ0 ).
Moreover, ν(η, ∞) ≡ lim ν(η, θ) has a well defined distribution
θ→∞
for all values of η.
Weak Monotonicity
Let (η, θ) ≥ (η 0 , θ0 ). Then E[ν(η, θ) ≥ E[ν(η 0 , θ0 ].
Moreover, E[ν(η, ∞)] ≡ lim E[ν(η, θ] is well defined for all values
θ→∞
of η.
Continuity
Let S ⊂ Θ with size s. Let θs → 0. Then the random variable
ν(η, θ) selects matches out of S with probability 1.
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Redundancy in Search
Taste for Variety
The Inbreeding-Outbreeding Game
Random Draws With Replacement: One-way Flow
Each agent in a population of size n makes a given number
η > 1 of independent draws with replacement out of a
population of size θ < n
”One-way flow” payoffs: matches only come from own draws.
Relevant potential matches are the distinct draws νa (η, θ).
Theorem
The random variable νa (η, θ) satisfies axiom SM. In particular,
when the pool size grows large the distribution converges to
νa (η, ∞) = η.
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Redundancy in Search
Taste for Variety
The Inbreeding-Outbreeding Game
Random Draws With Replacement: Two-way flow
One can easily envisage search situations in agents find
matches and are found.
Let i make η random draws with replacement in N \ {i};
let each j ∈ N \ {i} make η draws with replacement in
N \ {j}.
A match is when an agent either finds or is found.
Theorem
Let (η, θ) ≥ (η 0 , θ0 ). Then the profile (η, θ) induces a larger
expected number of matches than (η 0 , θ0 ), and WM is satisfied.
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Redundancy in Search
Taste for Variety
The Inbreeding-Outbreeding Game
Meeting Friends of Friends
Each agent is endowed with a set of ”friends”.
These friends can be used to search for additional friends.
If average degree is independent of the size of the population,
then the clustering of the friendships network is inversely
related to size.
This implies that the probability of meeting new friends is
larger in larger groups.
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Redundancy in Search
Taste for Variety
The Inbreeding-Outbreeding Game
Complementarity in Characteristics
Agents are located on a circle, with a randomly assigned
characteristic.
Similarity is negatively correlated with distance on the circle.
A match is effective if and only if two agents have different
characteristics (Dixit (2003)).
Search in the whole population yields better odds of an
effective match than search around one’s location.
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Redundancy in Search
Taste for Variety
The Inbreeding-Outbreeding Game
The Inbreeding-Outbreeding Game
Agents are organized into homogeneous groups.
Their search can target the whole population (Outbreeding ),
or just their own group (Inbreeding ).
Outbreeding implies a fixed cost c.
The profile s of I/O strategies of all agents determines θs.
One Sided Search
θl (s) =
nl if sl = I
n if sl = O
(4)
Two Sided Search
θ̃l (s) =



Pnl
nl 0
if sl = I
if sl = O
{l0 :sl0 =O}
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
(5)
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Threshold Equilibrium
Equilibrium Patterns of Coleman Index
Threshold Equilibrium - One-sided
Theorem
Consider a One-Sided search context where either axiom SM holds
or WM holds and the U is linear. Assume also that outbreeding
costs are low in the following sense:
n
X
Pη,∞ (νi )[1 − (1 − p)νi ]U (1) > c.
(6)
νi =0
Then, there exists some given (finite) τ ≥ 2 and n̂ such that if
n ≥ n̂, the equilibrium strategy profile s∗ satisfies, for all types q:
s∗q = I ⇔ nq ≥ τ
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
(7)
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Threshold Equilibrium
Equilibrium Patterns of Coleman Index
Threshold Equilibrium - Two-sided
A similar results holds for the two-sided model, under the condition
that the mass of small groups is non negligible
Theorem
Consider a Two-Sided search context where either axiom SM holds
or WM holds and U is linear. If outbreeding costs satisfy condition
(6), then there exists some given (finite) τ ≥ 2 and n̂ satisfying
the following property
P for all n ≥ n̂: if the distribution of groups’
nq > αn for some α > 0, then the strategy
sizes is such that
l:nq <τ
profile s∗ such that s∗q = I ⇔ nq > τ is an equilibrium.
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Threshold Equilibrium
Equilibrium Patterns of Coleman Index
Some Empirical Evidence of Threshold Equilibrium
Two-sided model predicts:
Small outbreeding groups uniformly search in restricted pool
of outbreeders.
Inbreeding groups only search inwards.
This implies three patterns:
Small outbreeding groups find other outbreeding groups at
rates that exceed population shares
All groups have excess homophily.
Excess homophily should jump up at the threshold size, and
then decrease linearly.
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
1
Excess Representation
.4
.6
0
0
.2
.2
Excess Representation
.4
.6
.8
.8
1
Threshold Equilibrium
Equilibrium Patterns of Coleman Index
0
.2
.4
.6
Relative Group Size
.8
1
0
.2
.4
.6
Relative Group Size
.8
Excess representation of small (< 5 percent) groups in the ties of
groups of various relative sizes (x-axis)
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
1
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Excess Homophily
.4
0
0
.2
Excess Homophily
.5
.6
1
.8
Threshold Equilibrium
Equilibrium Patterns of Coleman Index
0
.2
.4
.6
Relative Group Size
.8
1
0
.2
Excess homophily by relative group size.
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
.4
.6
Relative Group Size
.8
1
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Threshold Equilibrium
Equilibrium Patterns of Coleman Index
Moody 2001
”Tipping point” theory: race becomes salient for tie formation
only after a given size of group
Friendship segregation has non linear relation with school
heterogeneity.
Segrega3on
School’s
Racial
Heterogeneity
Sergio Currarini, Fernando Vega Redondo
0.6
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Threshold Equilibrium
Equilibrium Patterns of Coleman Index
Coleman Index: Equilibrium Predictions
Can we generate inverted bell trend of Coleman Index?
Need to introduce frictions in search.
Some small amounts of search intensity are independent of
inbreeding/outbreeding decisions.
rI = extra intensity directed only within own group.
rO = extra intensity directed only outside own group.
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Threshold Equilibrium
Equilibrium Patterns of Coleman Index
Small Groups
Proposition: small Coleman Index
Let nq < τ . There exists n̂ such that if n ≥ n̂, then IHq is
m(rI ,nl )
bounded above by the term m(η+r
.
O ,∞)
Sketch of Proof
sq = m(rI , nl )
dq = m(η + rO , ∞)
IHq =
n
m(rI ,nl )
− nq
m(rI ,nl )+m(η+rO ,∞)
nq
1− n
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Threshold Equilibrium
Equilibrium Patterns of Coleman Index
Small Groups
Proposition: increasing Coleman Index
Let nq < τ . There exists n̂ such that if n ≥ n̂, then nq0 < nq
implies IHq0 > IHq .
Sketch of Proof
sq = m(rI , nl )
dq = m(η + rO , ∞)
IHq =
n
m(rI ,nl )
− nq
m(rI ,nl )+m(η+rO ,∞)
nq
1− n
IHq0 − IHq > 0 ←
m(rI ,nq0 )
m(rI ,nq0 )+m(η+rO ,∞)
Sergio Currarini, Fernando Vega Redondo
−
m(rI ,nq )
m(rI ,nq )+m(η+rO ,∞)
Homophily and Search
>
1
n
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Threshold Equilibrium
Equilibrium Patterns of Coleman Index
Groups of Intermediate Size
Relative size bounded away from zero and one.
Proposition: Large Coleman Index
n
Take ε > 0, and let nq be bounded away from both zero and one.
m(rO ,∞)
≤ δ1
There exist δ1 > 0 and n̂ such that if n ≥ n̂ and m(η+r
I ,∞)
then IHq ≥ 1 − ε
Sketch of Proof
When group q makes up for a non negligible share of population,
then
n
IHq '
m(η+rI ,∞)
− nq
m(η+rI ,∞)+m(rO ,∞)
nq
1− n
.
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Threshold Equilibrium
Equilibrium Patterns of Coleman Index
Groups of Intermediate Size(2)
Proposition: Deacreasing Coleman index
Let q and q 0 have relative sizes bounded away from both zero and
1. Then, there exists n̂ such that n0q > nq implies Cq0 < Cq , for all
n > n̂.
Sketch of Proof
IHq0 − IHq '
n 0
m(η+rI ,∞)
− nq
m(η+rI ,∞)+m(rO ,∞)
n 0
1− nq
Sergio Currarini, Fernando Vega Redondo
−
n
m(η+rI ,∞)
− nq
m(η+rI ,∞)+m(rO ,∞)
nq
1− n
Homophily and Search
.
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Threshold Equilibrium
Equilibrium Patterns of Coleman Index
Large Groups
For groups that cover almost the whole population, we show that
the Coleman index becomes negative.
Proposition
Fix rO and rI . There exist some δ2 > 0 and n̂ such that if n ≥ n̂,
n
then any group q with relative size nq ≥ 1 − δ2 has IHq < 0.
Sergio Currarini, Fernando Vega Redondo
Homophily and Search
Homophily
A Search Model of Network Formation
Monotonicity Axioms: Illustrations
Results
Threshold Equilibrium
Equilibrium Patterns of Coleman Index
Conclusions
Some patterns of inter- and intra-group ties are common to
diverse social networks.
Simple stylized model to apply to diverse search situations.
Threshold equilibrium explains some patterns of inter-group
ties: small groups interact with small groups.
Frictions in search and threshold equilibrium help explain
patter of intra-group ties (bell shaped curve).
Our Inbreeding/Outbreeding choices induce biases in meeting
opportunities as the outcome of strategic behaviour in the
presence of incentives for homophily.
Sergio Currarini, Fernando Vega Redondo
Homophily and Search