## TL;DR
Scott Feld wrote this paper when he was professor at Ston...
Scott L. Feld is a mathematical sociology professor at Purdue Unive...
It is actually possible for Facebook users to download their social...
The average number of friends of a person is given by the average o...
The friendship paradox was used in 2009 by Nicholas Christakis and ...
It's interesting to study other situations in which you can observe...
Why
Your
Friends Have More
Friends
than
You
Do1
Scott
L.
Feld
State
University
of
New York at
Stony
Brook
It is reasonable
to
suppose
that individuals use the
number of
friends that their
friends
have
as
one basis
for
determining
whether
they,
themselves,
have an
adequate
number of
friends.
This article
shows
that,
if
individuals
compare
themselves with their
friends,
it
is
likely
that most
of
them
will feel
relatively inadequate.
Data on
friendship
drawn
from
James
Coleman's (1961) classic
study The
Adolescent
Society are used to
illustrate the phenomenon
that most
people
have
fewer friends than
their
friends have. The
logic under-
lying the phenomenon
is mathematically explored, showing
that the
mean
number
of friends of friends is
always greater than
the mean
number of
friends
of
individuals.
Further analysis shows
that the
proportion
of individuals who
have fewer friends than
the mean
number of
friends their
own friends have is affected by
the exact
arrangement
of
friendships
in a
social network. This
disproportion-
ate
experiencing
of
friends with
many
friends
is
related to a
set
of
abstractly
similar "class size
paradoxes"
that includes such diverse
phenomena
as the
tendencies for
college
students to experience the
mean class
size as
larger
than
it
actually
is
and
for
people
to
experi-
ence beaches
and
parks
as more crowded than
they
usually
are.
Friendship
is
not
only
a source of satisfaction and
security;
it
is also
a
way
that individuals
evaluate
themselves
and others.
People expect
themselves
and others
to
have
friends
and wonder about the
normality
of those
individuals
who
appear
to
have
few or no friends. There has
been little
study
of how
people
determine what
is
an
adequate
number
of
friends,
but
it
is
reasonable
to
suppose
that individuals use the number
of
friends
that their friends
have as one
basis
of
comparison.
This article
shows
that,
if individuals make this
type
of
comparison,
it is
likely
that
1
A previous version of this paper
was presented at the 1986 Sunbelt
Social Network
Conference
in
Santa
Barbara,
Calif.
I
appreciate
the
helpful suggestions
of Bernard
Grofman,
Guillermo
Owen,
and
Jill
Suitor.
Requests
for
reprints
should be sent
to
Scott
Feld, Department
of
Sociology,
State
University
of New York, Stony
Brook,
New York 11794-4356.
?) 1991
by The
University of
Chicago.
All
rights
reserved.
0002-9602/91/9606-0005$01
.50
1464
AJS
Volume
96
Number
6
(May
1991):
1464-77
Friends
most of them
will feel
relatively inadequate.
I
use data on
friendship
drawn
from
James
Coleman's
(1961)
classic
study
The Adolescent
Society
to
illustrate
the
phenomenon
that most
people
have
fewer
friends than
their friends have.
I
will explore
mathematically the logic underlying the phenomenon,
showing
that
the mean number of friends of friends is
always greater
than
the
mean
number of friends
of
individuals. Further
analysis
shows
that
the
proportion
of
individuals
who have
fewer friends than the mean
number of friends
their own friends have
is
affected
by
the
exact arrange-
ment of
friendships
in
a social network. While it is not a mathematical
necessity
that
each
individual will have fewer friends than the
mean
of
her or his own friends,
it
is
likely
that most
people
will find
themselves
in
this situation.
The basic logic can be described
simply.
If
there are some
people with
many friendship
ties and others
with
few,
those
with
many
ties show
up
disproportionately
in
sets of friends. For
example,
those with 40
friends
show
up
in
each
of 40 individual
friendship
networks
and
thus
can make
40
people
feel
relatively
deprived,
while those with
only
one friend show
up
in
only
one
friendship
network and can
make
only
that one
person
feel
relatively advantaged. Thus,
it is
inevitable that individual
friendship
networks disproportionately
include those with the most
friends.
EMPIRICAL
EXAMPLES
Friendship
is
usually thought
to be a
symmetric relationship, as indicated
by
the common
phrase, "They
are friends."
One
way
to
operationalize
friendship
is to
consider
a
friendship
to
be
one
that
is so
regarded by
both of the individuals.
In The
Adolescent
Society,
Coleman
(1961)
col-
lected data
on
friendships
among
the
students
in
12
high
schools.
Individ-
uals were asked
to
name their
friends,
and
pairs
of
individuals who
named one another were given
particular attention. It is these "friend-
ships"
that
will
be used as
examples.
To illustrate
the
phenomenon
under
study here, consider the set
of
relationships depicted
in
figure
1,
found
among eight girls
in
"Mar-
ketville,"
one
of
the
high
schools included in
the
study. The names are
fictitious.
In
this example, Betty's only
friend, Sue, has more friends than Betty
has; Jane's two friends,
Dale and
Alice, average
more
friends than Jane
has; Dale's three friends, Sue,
Alice, and Jane, average more friends
than
Dale;
and so forth. Of the
eight girls,
five
(Betty, Jane, Pam, Dale,
and
Tina)
have fewer friends than the
average among
their
friends, while
only
two
(Sue
and
Alice)
have
more friends
than the average among their
friends;
one
(Carol)
has as
many
as the
average among
her
friends. Table
1465
American Journal of Sociology
1(4)
-
4(2.75)
-
4(3)
-
2(3.5)
Betty Sue
Alice
Jane
3(3.3)Pam
??(3.3)
Dale
2(2)
Carol
1
1(2)
Tima
The number beside each name
is
her
number of friends. The number
in
parentheses
beside each name
is
the
mean number of friends of her friends.
FIG. 1.-Friendships
among eight girls
at Marketville High
School
1 shows each girl's number
of friends in the first column and,
in the third
column,
the mean
number of friends
her
friends have.
Twice as
many
(5:2) have fewer
than
average
as have more
than the
average
among
their
friends.
The
complete
network of all of the
girls
in
Marketville shows
the same
pattern. Figure
2
reproduces
the entire
sociogram
of mutual choices. Of
the
146
girls
who
have
any
mutual
friends,
80 have
fewer friends than
the
mean
among
their friends while
41 have
more;
25 have
the same
as
the mean among
their
friends.
Thus, nearly
twice as
many
have fewer
as
have
more
than
the mean
among
their friends. The
same
pattern
TABLE
1
A
SUMMARY
OF THE NUMBERS OF FRIENDS AND
THE MEAN
NUMBERS
OF FRIENDS
OF
FRIENDS
FOR EACH
OF
THE
GIRLS
IN
FIGURE
1
Total Number of Mean Number
of
Number
of Friends of
Friends
of
Friends Her Friends Her Friends
(xt)
(Y-x,)
(-x,l/xt)
Betty
.1
4
4
Sue
.4 11
2.75
Alice
.4 12
3
Jane
.2 7
3.5
Pam
.3
10
3.3
Dale
.3
10 3.3
Carol
.2
4
2
Tina
.1 2
2
Total
.20 60
23.92
Mean
2.5*
3t
2.99*
*
For
eight girls.
t
For 20 friends.
1466
016
018
033
X
girls
X
W
V
~~~~~~~shown
in
047 03
4
50
FIG.
2. -Network
of
reciprocated
friendships
among
Marketville
girls;
the
triangle
at
right
indicates
friendships
illustrated
in
fig.
1.
(From
The
Adolescent
Society
by James
S.
Coleman.
(C
1961
by
the Free
Press,
a division
of
Macmillan,
Inc.
Used with
permission.)
American Journal of Sociology
appears among
the
boys
in
Marketville and
among the girls and boys of
the other
high
schools
reported
in
the
Coleman
study.2
TWO
DISTRIBUTIONS: FRIENDS OF
INDIVIDUALS
AND
FRIENDS
OF
FRIENDS
The
phenomenon
of
people
finding
that
their friends have more
friends
than
they
do can be
partially
understood
by recognizing
the
difference
between
the distribution
of
numbers
of
friends of
individuals and the
distribution
of
the numbers
of friends of friends. The
distribution of
friends
of individuals
is
just
the usual
distribution of numbers of
friends
that we would
usually examine,
but
the
distribution of friends
of friends
includes
some of the same
individuals
over and over. This
complexity
may become
more
understandable
in
reconsidering
the case of
the
eight
girls
in
figure
1. In
the situation described
in
figure
1
and
table
1,
the
distribution
of
friends of individuals would be for the
eight girls having
a
total
of
20
friends,
with
a
mean
of
2.5
friends
per
individual.
However,
the distribution
of friends of friends
includes more
cases.
For
example,
Jane's friends,
Dale and
Alice,
contribute their
numbers
of
friends
to
this distribution,
and
Sue's
friends, Dale, Alice, Pam,
and
Betty,
also
contribute
their numbers to
this distribution.
Note that
Dale
and Alice
contribute to
this distribution more
than
once;
in
fact,
each friend con-
tributes
to
the
distribution
of the numbers
of
friends
of
friends as
many
times
as she
has friends. There are
a
total of 20 friends
(obviously
count-
ing
some of
the
eight girls
more than
once) having
a total
of
60
friends,
with
a mean
of 3.0 friends
per
friend. When
each
individual
compares
him-
or
herself
with the
average
number
of friends of
his
or her
friends,
the
comparison
is
with a
sample
from the
numbers of friends of
friends,
which is
a different distribution
from that of numbers of friends
among
individuals.
For the entire set
of
girls
in
Marketville
(from fig. 2),
the distribution
of friends
among
individuals
is skewed to the
right,
as shown
in
figure
3
a,
which indicates
that a few individuals have
many friends;
this
appears
to
2
Nearly
identical results are obtained when
individuals
compare themselves with the
"median" among their friends. The
phenomenon that most individuals would feel
relatively deprived
if
they compared
themselves with the mean of their friends is
essentially replicated
if
they compare
themselves
with the median
(which is equivalent
to
determining
whether a
majority
of their
friends have more
friends than
they have).
This can
be
seen
in
the
example
of
fig. 1,
where
exactly
the
same girls who are below
(above)
the mean
of
their friends
are
also
below
(above)
the
median
of
their
friends.
Although comparison
with the
majority
of
one's friends
may be as important as
comparing
with
the
mean,
the
present
discussion is
limited
to
discussion of the mean
to
simplify
the
discussion-also,
the
mathematics
concerning the mean is somewhat
more straightforward
than that
of the
median.
1468
Friends
55
53
50
45
40
35
30
29
2
25
20
20
15
14
10
1 2 3
4
5 6
7
a) The mean is 2.7.
360
350
330
320
300
270
240
212
234
210
180
150
120 108
90
60
4
30
29
1
2 3
4
5 6
7
b)
The mean is 3.4
FIG. 3.
-(a)
Distribution
of
numbers of
friends
for
Marketville
girls;
(b)
distri-
bution
of
number of
friends' friends
for
Marketville
girls.
be a
typical
distribution
of
numbers of
friends
among
individuals.
The
distribution of friends
among
friends
is
a
weighted
version
of the
original
distribution, weighting
those with
many friends
especially heavily; this
weighting
counteracts
the
original skew,
as
shown
in
figure 3b. The
important characteristic of the distribution of
numbers of friends of
friends is that it
inevitably
has a
higher mean than the
distribution of
friends of individuals.
In
this situation,
if
individuals compared their
numbers of friends with
the mean number of
friends of their own
friends,
and their friends were
a
representative sample
of
friends
(mean number
of
friends of friends was
3.4),
then
74%
of
the
individuals
would find
themselves to be relatively
deprived.3
3In
a discussion
of this
phenomenon,
Guillermo
Owen and Bernard
Grofman
noted
that,
even
if
individuals are
accurately
informed
of
the
distribution
of
friends among
individuals,
a
majority
of
the
individuals
will
generally
be below the
mean, because
the
median is
below
the mean
in
distributions that are skewed to the
right.
In
this
1469
American Journal of Sociology
VARIANCE AFFECTS
THE
MEAN NUMBER OF
FRIENDS OF FRIENDS
In general, there
is
a
simple
relationship
between the
original
distribution
of
friends
among
individuals and the distribution of
friends of
friends.
If
the original
distribution has
n individuals with
xi
ties
apiece,
the mean
can
be
determined as
Exiln.
However,
the distribution
of friends has
Ex
cases (for
all of the
friends)
and
they
have a total of Ex? friends, since
each
individual is counted
as
many
times as she
or he
has
friends,
xi,
and
that
individual has
Exi
friends.
Thus,
the
mean
number
of
friends
among
the friends is
(Ex?)/(Exi).
This can
be shown to be a
simple
func-
tion
of
the
mean and variance
in
the
original
distribution of
ties.4
That
is: mean
number
of friends
of
friends
=
(Ex?)/(Exi)
=
mean(x)
+
vari-
ance(x)/mean(x).
The
expression
above shows
that
the mean
among
friends is
always
at
least as great
as the mean
among individuals,
and the mean
among
friends increases
with
the variance
among individuals,
with
a
given
mean
among
individuals.
The
mean
among
friends is much
greater
than the
mean
among
individuals if
there is much variation in the
population.
The Arrangement
of
Individual
Friends
While
the
mean of
the distribution of number of friends of friends is
completely
determined
by
the
distribution
of friends of
individuals,
the
particular samples
of
friends of friends
may vary among
individuals.
If
each
individual's
friends are
approximately representative
of
all
friends
(in terms of their
numbers
of
friends),
then individuals
comparing
them-
selves with
their own
friends
are
essentially comparing
themselves with
the overall
distribution
among
friends. The calculations
in
the
previous
section
indicate
that the mean number of
friends
of
friends
is
higher
case, 56%
of the individuals were below
the
mean number
of
friends
of
individuals;
the
higher
mean of numbers
of friends' friends
increases
the
proportion
of
individuals
who
experience
themselves as below the
mean
of their
friends'
friends-74%
of the
individuals
are
below
that mean.
4
The mean
number
of friends' friends
is
just
the
total number of
friends' friends
divided
by
the
number
of friends.
To
determine
the
total number of
friends'
friends,
consider that each individual
is a friend
x; times
and has
x;
friends,
so that
individual
contributes
xi
friends'
friends
x,
times,
a total of
x'
friends' friends.
Thus,
the
total
number
of
friends'
friends
(the
numerator)
is
simply
this
quantity
summed over all
individuals,
Ex?.
The total number
of
friends
(the denominator)
is
simply
the number
of friends of each individual,
xi,
summed over all
individuals,
Exi.
Thus,
the
mean
number
of
friends'
friends
is
just
(Ex?)/(Ext). Some relatively simple
algebra shows
that this can be
expressed
as a function
of
the mean
(Ext/n)
and
the
variance
(Yx?In
-
mean2). Thus, the
mean
number
of
friends'
friends is:
(Yx')/(fxt)
=
mean(x)
+
variance(x)/mean (x).
1470
Friends
than the
mean
number
of friends of
individuals;
consequently,
a
higher
proportion
of
individuals
will
be below the mean number
of friends
of
friends
than below
the mean number of friends
of individuals
(the
more
appropriate
comparison).5
However,
individuals
may
have
unrepresenta-
tive sets
of
friends,
and the
following
two sections
consider some
of
the
possible
consequences of such
unrepresentativeness.
First,
it is
important
to
recognize
yet
another distribution and
another
mean. Refer
again
to the situation
in
figure
1
as
summarized
in
table 1.
The eight
Marketville
girls
have
a
total of
20
friendships,
with
a mean
of 2.5.
The friends
have a total of 60
friends,
with a mean
of
3.0.
At
the
same
time,
each
girl
has a
mean
among
her
friends,
and
the means for
all the
girls
have
a
mean of 2.99. This last mean
differs
from
the
mean
number
of
friends
of friends
(only slightly
in
this
case)
because the two-
step
averaging process
weights
each of the friends
differently:
each
of
the
means of Sue's
four friends are
averaged,
and
that
average
counts
equally
with
Betty's average
based
on
her
only
friend.
Thus,
the
particular
ar-
rangement
of
the
friendships
affects this last
average.
Correlations
between
Individuals and
Their Friends
There
may
be
situations
in
which
individuals are
disproportionately
friends
with others with similar
friendship
volumes
(similars
attract)
and
other situations
in
which
people
are
disproportionately
friends
with those
with different
friendship
volumes
(e.g.,
where
some
individuals
"collect"
several
otherwise-isolated individuals as
their
friends).
The
implications
of these
types
of correlations
can be seen
in
some
hypothetical
examples.
Figure
4
shows
four
possible
ways
that the same distribution of
friendship
volumes can
be
arranged
in
networks;
this
distribution
includes
four
individuals
having
three friends
each and
12
individuals
having
one
friend
each.
The
possibilities
are
arranged
from the one with a
perfect
positive
correlation
between
individuals
and
their
friends
(fig. 4a)
to one with a
5Note
that, even though
the mean
number of
friends of an
individual's friends
is
unrepresentatively high
as an estimate
of the mean
number of
friends of
individuals,
an
individual who
understands the
nature of this
problem can use
the
information
derived from
his or her
friends to
estimate the mean
number of
friends for all
people.
The
information from
friends should
be weighted to
take account
of the
frequency
with
which
that
particular individual is
included in
various people's
experience, so
a
friend with
x; friends
should be
weighted by
1/xZ
so
that friend is
not
overcounted.
Thus,
the
individual's
appropriate
estimate of the mean
number of
friends of individu-
als is
given
by
1xj(1Ix,)IE(1Ixj)
=
xjIE(1Ix,).
This more
realistic
estimate is always
less
than
the
mean number
of friends of
that individual's
friends and
so will make
the
individual
feel
appropriately better
relative to others.
1471
American
Journal
of
Sociology
a)
Perfect Positive
Correlation
b)
Zero
Correlation
5> .E-F
A
.B
Dl- G-H// J
D
B
I
-J
D
c
K-L
4 A
K-L
M-
N
H,
\
M-N
0
-
pG
E
F
Mean
individuals' mean number of
Mean individuals' mean
number
of
friends'
friends
is
1.5.
friends'
friends
is 2.0.
c) Negative Correlation
d)
Perfect
Negative
Correlation
KA
D
c
H/
/\
M-N
D\ A\
G
E
F
0PK
LM
NO0P
Mean individuals' mean
number of
Mean individuals'
mean
number of
friends' friends
is 2.17.
friends' friends
is 2.5.
FIG. 4.-Four
arrangements
of
the same distribution of
individual numbers
of friends
(A, B,
C,
and
D
have
three friends
each,
and
E, F,
G, H,
I, J, K,
L,
M, N,
0,
and
P
have one
friend
each).
perfect
negative
correlation
(fig.
4d).
It
should
be
apparent
in
all cases
that the
mean
among
the 16 individuals is 1.5 friends each and
the mean
among
the 24 friends is 2.0 friends each.
However,
the mean
individual
mean
number of
friends' friends
increases
from
figure
4a to
figure
4d;
the
more
negative
the correlation between the individuals
and their
friends,
the
greater
the mean individual
mean number
of friends' friends
and
the
greater
the
proportion
of individuals below that mean. For
fig-
ures
4b,
4c,
and
4d,
where some individuals
differ
from their
friends,
the
proportions
of those who have fewer
friends than
the mean of their
friends
are
60%, 67%,
and
75%,
respectively.
The
mean individual mean number of
friends'
friends can
be calcu-
lated as
follows:6
mean
individual
mean
number of
friends'
friends
-
1(xj/x)1n,
for all
i
and
j
who
are
friends
with
one
another.
6
It is
apparent
that for
each
individual, i,
with friends
designated
by j's,
the total
number
of
friends' friends is
YX1
and
the mean is
(Yx,)/xx
=
>2(x_/x).
The total of
these
1472
Friends
For
a given set
of
friendship volumes,
this
expression
is
minimized
when
xi
=
Xj
in
all
cases;
in
that
situation,
the
mean
individual
mean
number
of friends' friends
is
just
the
mean
number
of
friends
of individu-
als. On
the
other
hand,
the maximum value is
achieved when
individuals
with the
fewest friends are friends
of those
with
the most friends-in
that
case,
the mean
can be
considerably
larger
than the
mean number of
friends'
friends.
It
may
be as
high
as:7
mean(x)
+ 2
x
[variance(x)/mean
(x)].
Further
Implications
of the
Exact
Arrangement
of
Friendships
Even with a
given
correlation between individuals and
their
friends,
there can
be variation
in
the distribution
of
friendships.
The exact ar-
rangement of
friendship
among
individuals will determine the
number
of individuals
with more friends than the
mean of
their
friends
(e.g.,
there
could be a few individuals whose friends
have
many
more friends
means
is
just
the
sum of these
expressions
over
all
i's
and
their
corresponding j's,
and
the mean of these
means is
just
this total
divided
by
the number of
individuals,
n. That
is,
YY(x?/x,)/n.This
can be
illustrated with
three
individuals, A, B,
and
C,
and
two
ties,
A-B-C,
and four
friendships,
AB, BA, BC,
and CB. Individual
A
has one
friend, B,
who has two
friends
(a
mean of
2);
B
has
two
friends,
A
and
C,
with one friend each
(a
mean of
1);
and
C
has one
friend, C,
with two
friends
(a
mean
of
2).
The
three individuals have
a
mean
of
their
means
of
5/3.
7The
mean
individual's mean number of
friends'
friends
is
given
by
this
expression
in
the case of the
"wheel"
pattern
of
friendships,
in
which one
person
is
friends with
everyone
else and
they
are
not
friends with one another.
Some
equations
show the
values
of
the
parameters
in
the case of a
wheel
composed
of
n
individuals (1 hub
and
n
-
1
spokes). They
are: mean
number
of
friends of
individuals
=
2(n
-
1)/n;
variance
in
number
of
friends
=
(n
-
2)2(n
-
1)/(n2);
mean number of
friends' friends
=
n/2;
and
mean
individual
mean
number
of
friends' friends
=
[(n
-
1)2
+
1]/n.
It
can be seen that the
value
of
the mean individual
mean number
of
friends'
friends
is
the
specified
function of the
mean and variance. For
example,
in
the case
of 10
individuals
(1 hub and 9
spokes),
the
mean
number of
friends of
individuals
is 1.8
with a variance of
5.76,
the mean number of
friends'
friends is
5,
and the
mean
individual mean number of
friends'
friends is 8.2.
An
example with
three
individuals
(1 hub and
2
spokes)
is
presented
in n.
6. Since
the wheel
appears to be an
extreme
type
of
distribution,
it
is
conjectured
that
this
expression gives the
maximum
value of
the mean
individual's
mean number of
friends'
friends for a
specified
distribution of
friendship volumes.
An
additional
basis for this
conjecture is the
apparent
symmetry.
The minimum
value
of
mean
individual mean
number of friends'
friends is
achieved
when all
individuals
have the
same
number
of
friends as
each of their
friends,
and
that mean is
just
the mean
number of
friends.
When particular
friends are
"ran-
domly"
assigned,
the mean individual mean
number
of
friends' friends is
equal
to
the
mean number of
friends'
friends,
which is
the
variance
divided by the
mean
greater
than
the mean
number
of
friends
(as given
by
the
expression
in
the text). Since
the minimum value is
just
the
variance
divided
by
the mean
less than the
random
case,
the
conjecture
is that
the maximum
value is
just
the
variance
divided
by the
mean more
than
the
random case.
1473
American Journal
of Sociology
A
c
D
EF
A
and
B
each have
fewer
friends
(2) than
the mean of their friends
(3);
but
C,D,E
and
F each have more friends (3)
than the
mean of their friends (2.67).
FIG. 5.-An
exceptional situation
in
which
a majority
of individuals have
more
friends
than
the mean
of
their friends.
than they
have,
while there
are
many
others whose friends
have a few
more
friends than
they
have). Thus,
while the means of the
various
distributions
are
determined,
the number of individuals who have fewer
friends
than the
mean of
their
friends'
friends will depend on
the exact
arrangement
of
friendships.
As
shown
in
figure 5,
it is
even
possible,
under
very
carefully
contrived
conditions,
for
a
majority of
individuals
to
have more friends
than the mean of
their
friends.
However,
very
few
arrangements
of
friendships
have this
consequence,
and
there are
no theoretical
reasons to
expect
these
exceptional
situations.
If the mean
number
of friends' friends and the mean individual
mean
number of friends'
friends are much
higher
than the mean among
individ-
uals,
we can
expect
that a
high proportion
of individuals
will
have fewer
friends than
the
mean among their friends,
as is true among
the Mar-
ketville
girls
and the
boys
and
girls
of the other
high
schools included
in
The Adolescent Society (Coleman
1961).8
8
Further
research
might
explore how various
systematic
processes
in
the
construction
of social networks
might
lead to
particular
types
of
patterns
of
friendships
with
partic-
ular
consequences
for the
experiences of friends' friends.
For
example,
if
friendships
are
primarily established
through
one focus
or
a
few foci of
activity (Feld
1981),
then
individuals
might
have numbers of
friends similar
to
the numbers
their
own
friends
have
(i.e., people who
draw many
friends
from a
large focus of
activity
will
have
friends
who also
have
many friends
from the
same
large focus
of
activity), and
the
experience
of relative
deprivation
may
be
minimized. On the
other
hand, if
individuals
disproportionately make friends with a few
individuals with
particular
desirable char-
acteristics
(see Feld
and
Elmore
1982),
there
may be
large
amounts of
variation
in
friendship
volumes that
lead to the
widespread
experience of
relative
deprivation
by
individuals. Actual
patterns
of
friendships reflect several
underlying
processes
by
which
friendships
are
developed
and
maintained and
can
become
very
complex.
For-
mally,
the
proportion
of
individuals who
experience
relative
deprivation
is
determined
by
the
probability
that the mean
(median)
number of an
individual's
friends' friends
is greater
than the
individual's own
number of
friends
under a
particular
specifiable
set
of
conditions.
1474
Friends
Asymmetric Relationships
An analogous phenomenon
occurs
in
situations
with
asymmetric
relation-
ships,
the
type
that are
directly
revealed
in sociometric-choice
data.
In
that
case,
most
individuals choose
people
who are more
popular
than
they
are. The
logic employed
when individuals
compare
their own
popu-
larity with
that of
the
people they
choose
is
identical to
that
described
above.
Individuals
who are
popular
are chosen
by many
others and
so
can lead
many
others to feel
relatively deprived;
individuals
who
are
unpopular
are
rarely
chosen and so can make few
people
feel
advantaged.
The distributions
of
popularity among
individuals
and
among
those
they
choose
can
be shown
to have
the same characteristics
as the
various
distributions
of numbers of friends described
above.
Related Phenomena
The tendency for individuals to experience a biased sample of numbers
of
friends of others
is
one
of
a large set of related phenomena. Feld and
Grofman
(1977)
called one such
phenomenon
the "class size
paradox";
they showed that,
if
there is
any variation
in
college class sizes, then
students experience
the
average
class size
as being larger than it is. They
experience
a
higher average
class size than
exists for the college because
many
students
experience
the
large classes, while few students experience
the
small classes.
Hemenway (1982) noted the same phenomenon
in
terms
of
college
class size and
remarked
on
several other similar
phenomena;
specifically,
he
suggested
that
people disproportionately experience the
most
crowded times
in
public places (including restaurants, beaches, and
highways) and so experience these places as being more crowded than
they usually
are.9
It
should
be
noted
that
class size paradoxes are often experienced in
situations
in
which
they
are
not seen as paradoxical. For example, most
cities are
small,
but most
people
live
in
large cities; while most organiza-
tions
are
small,
a
disproportionate number of individuals work for large
organizations (Granovetter 1984).
Whether
paradoxical
or
not,
it
is
important
to
recognize that the experi-
ences
of class sizes
have a
reality of their own. The fact that many
9
A
class
size paradox
arises when
individuals
disproportionately
experience
classes
containing
more
people.
This
idea can be extended to include
an "observer
class size
paradox,"
whereby
individuals observing
classes of objects are
more likely
to observe
and
therefore be aware
of the larger
classes of objects.
For example, Good
(1983)
suggested
that
galaxies
with more
planets
are more
likely
to be observed
than smaller
galaxies;
consequently,
the
average
size of
galaxies
that
are
observed is
larger than
the average
size
of
galaxies.
1475
American
Journal of Sociology
individuals
experience
disproportionately
large average
class sizes (large
college
classes,
crowded expressways, populous
cities,
large families,
etc.)
may be
more sociologically
and
practically
significant
than
the object
average;
for example,
it may not matter
so much that
roads are usually
empty
if
most
people
are
caught
in rush-hour
traffic.
Furthermore,
the recognition
of
the different ways
that people experi-
ence the
same
objective
situation can
help
us understand some conflicts
of interest.
For
example,
Feld
and Grofman
(1980)
consider
that college
faculty
members
experience
the actual average
class size,
while their
students disproportionately
experience
the
larger
classes;
as a
result,
even
though
faculty
and students
have similar
preferences
for smaller classes,
students
have
an interest
in
minimizing
variation
in
class
size,
while
faculty
have
an interest
in
maximizing
that variation.
CONCLUSIONS
The term
"class size
paradox"
can be considered
a
generic
term for
all
phenomena
that arise where
classes are
of varied sizes,
members of those
classes
disproportionately
experience
the
larger classes,
and most individ-
uals therefore
experience
the
average
class size as
larger
than
it
is.
Such
phenomena
are often
more than
mathematical
curiosities; they have
im-
plications
for how
people
experience
and
respond to
various aspects
of
their
environments.
The tendency
for most
people to have
fewer friends
than their friends
have is one
sociologically
significant
class size
paradox.
Individuals
who
find themselves
associated
with
people
with more friends than they
have
may
conclude
that
they
themselves
are
below
average
and
somehow
inad-
equate.
The
analysis presented
in
this
paper
indicates that
most
individu-
als have
friends who
have
more
friends
than
average
and so
provide
an
unfair basis
for
comparison.
Understanding
the
nature of
a
class
size
paradox
should
help people
to
understand that their
position
is
relatively
much better
than their
personal
experiences
have led them to believe.
REFERENCES
Coleman, James
S. 1961. The Adolescent
Society.
New
York:
Free
Press.
Feld, Scott L. 1981. "The
Focused Organization of
Social Ties." American Journal
of Sociology 86:1015-35.
Feld, Scott L.,
and
Richard Elmore.
1982.
"Patterns of
Sociometric Choices:
Transi-
tivity Reconsidered." Social Psychology
Quarterly
45:77-85.
Feld, Scott L.,
and
Bernard Grofman. 1977.
"Variations
in
Class Size, the
Class
Size
Paradox,
and
Consequences
for Students."
Research
in
Higher
Education
6:215-22.
1476
Friends
.
1980. "Conflict
of Interest
between Faculty, Students, and
Administrators:
Consequences
of the Class Size Paradox." Frontiers
of
Economics 3:111-16.
Good,
I.
J.
1983. Good
Thinking:
The
Foundations
of Probability
and Its
Applica-
tions. Minneapolis: University
of
Minnesota Press.
Granovetter,
Mark. 1984. "Small Is
Bountiful: Labor Markets
and Establishment
Size."
American
Sociological
Review 49:323-34.
Hemenway,
David. 1982.
"Why
Your
Classes
Are
Larger than 'Average.'
"
Mathe-
matics Magazine 55:162-64.
1477
## TL;DR
Scott Feld wrote this paper when he was professor at Stony Brook in 1991. In this paper, Scott explores the following counterintuitive idea from a mathematical perspective
#### Your friends (probably) have more friends than you
Most people think that they have more friends than their friends [(this effect was well documented by a NYU report on undergraduate students at the University of Chicago)](https://nyuscholars.nyu.edu/en/publications/what-makes-you-think-youre-so-popular-self-evaluation-maintenance). However, you are actually more likely to be friends with somebody with lots of friends and consequently your average friend will have more friends than you.
Scott L. Feld is a mathematical sociology professor at Purdue University, focusing on understanding the causes and consequences of social patterns in such diverse contexts as social networks, collective decision-making, interpersonal violence and the structure of science.
It's interesting to study other situations in which you can observe a similar paradoxical effect. For instance, it turns out the friendship paradox can be generalized to scientific collaboration: [your co-authors will have more co-authors and citations than you](https://core.ac.uk/download/pdf/80688293.pdf)! It can also be generalized to wealth and happiness.
Young-Ho Eom at the University of Toulouse in France and Hang-Hyun Jo at Aalto University in Finland studied the so called "generalized friendship paradox" and derived the mathematical conditions in which it occurs - you can check [their paper here](https://arxiv.org/pdf/1401.1458.pdf).
It is actually possible for Facebook users to download their social graph and use a simple Python script to verify the Friendship Paradox.
Khuyen Tran shared the code [here](
https://towardsdatascience.com/observe-the-friend-paradox-in-facebook-data-using-python-314c23fd49e4).
The friendship paradox was used in 2009 by Nicholas Christakis and James Fowler to test if they could use the paradox for early flu detection. To do that, they contacted 744 Harvard undergrads and split them in 2 groups: 319 randomly chosen students and 425 friends they nominated. By monitoring the 2 groups they found that the friend group tended to get the flu before the random group.
The average number of friends of a person is given by the average of the degrees of the vertices in the graph: if vertex $v$ has $d(v)$ edges touching it represents a person who has $d(v)$ friends and the average number $μ$ of friends of a random person in the graph is
$
\mu=\frac{\sum_{v\in V} d(v)}{|V|}=\frac{2|E|}{|V|}
$
The average number of friends of a friend can be modeled by choosing a random person and then calculating how many friends their friends have on average. This amounts to choosing, uniformly at random, an edge of the graph (representing a pair of friends) and an endpoint of that edge (one of the friends), and again calculating the degree of the selected endpoint. The probability of a certain vertex $v$ to be chosen is
$
\frac{d(v)}{|E|}\frac{1}{2}
$
The first factor corresponds to how likely it is that the chosen edge contains the vertex, which increases when the vertex has more friends. The halving factor simply comes from the fact that each edge has two vertices. So the expected value of the number of friends of a (randomly chosen) friend is
$
\sum_{v} \left ( \frac{d(v)}{|E|}\frac{1}{2} \right )d(v) = \frac{\sum_v d(v)^2 }{2|E|}
$
We know from the definition of variance that
$
\frac{\sum_v d(v)^2 }{|V|} = \mu ^2 + \sigma^2
$
where $\sigma^2$ is the variance of the degrees in the graph. This allows us to compute the desired expected value as
$
\frac{\sum_v d(v)^2 }{2|E|} = \frac{|V|}{2|E|} (\mu^2 + \sigma^2) = \frac{\mu^2+\sigma^2}{\mu} = \mu + \frac{\sigma^2}{\mu}
$
For a graph that has vertices of varying degrees (as is typical for social networks), ${\sigma}^{2}$ is strictly positive, which implies that the average degree of a friend is strictly greater than the average degree of a random node.