1. Communicating Mathematics -
Assesment 2
By George Stevens
13811875

2. Why Do Your Friends Have More Friends Than
You Do?
Do you ever feel like all your friends are more popular than you are? That they
have more friends than you do? Well you’re probably right but you’re not alone
because this is true for nearly everyone.
The phenomenon that on average, most people have fewer friends than their
friends have, dubbed ‘The Friendship Paradox’, was first observed by the soci-
ologist Scott L. Feld [2]. It might be quite hard to imagine, if you were thinking
logically you’d assume that this is only true for those with a lower than average
number of friends, however this is not the case.
The following example, adapted from the article ‘Why are your friends more
popular than you?’ in The Economist Newspaper[1], will help to visualise this
concept in action.
Four Friend Network Example
Figure 1: The diagram above shows a simple network of four people, where a
friendship between two people is denoted by a line connecting them directly to
eachother.
Consider the friendship network in Figure 1, Person A only has one friend,
Person B. Person B is friends with everyone in the network and therefore has
three friends. Person C and Person D are friends with each other and are both
friends with Person B, so they each have two friends.
From that we can easily deduce that the average number of friend that a person
in this particular network has, is two: 1 + 2 + 2 + 3 = 8 (total number of friends
each person has), 8 ÷ 4 = 2 (total number of friends each person has divided by
the number of people in the network).
From that it would appear that because the average number of friends is two,
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3. the average number of friends that each persons friend has will also be two, but
as we said earlier, this is not the case. The following table summarizes the main
figures involved in this particular example and should go some way towards
explaining why this is not the case.
Person No. of Friends No. Friends of Friends Average Friends of Friends
A 1 3 3 ÷ 1 = 3
B 3 1 + 2 + 2 = 5 5 ÷ 3 = 1.67
C 2 3 + 2 = 5 5 ÷ 2 = 2.5
D 2 3 + 2 = 5 5 ÷ 2 = 2.5
Total 8 18
Mean 2 18 ÷ 8 = 2.25
Figure 2: ‘No. Friends of Friends’ is the total number of friends that the persons
friends have. This is calculated by adding together the number of friends each
one of that persons friends have. The ‘Average Friends of Friends’ is the average
number of friends each persons friend has. This is calculated by dividing the
‘No. Friends of Friends’ column by the ‘No. of Friends’ column.
The main part of the table to focus on is the ‘Average Friends of Friends’
column. If we compare this to ‘No.of Friends’ column, for everyone apart from
Person B has a greater ‘Average Friends of Friends’ than they do actual friends,
which supports the idea of the friendship paradox. Furthermore if we look at
the overall means for ‘No. of Friends’ and ‘Average Friends of Friends’ we can
see that on average, a person in this network will have 2 friends, however, their
friends will have 2.25 friends on average. This shows that on average a person
in the networks friends will have more friends than the person has themself.
You might be thinking that its all well and good using an example where this
works, but a comprehensive study by the Pew Research Center [3] found that
this was true for Facebook users. They found that the average user has 245
friends whereas each of their friends have an average of 359 friends.
So why does this phenomenon occur? It’s actually down to selection bias. People
like Person B, who are the most well connected in the first place, are also going
to be counted most when people are looking at ‘Friends of Friends’, so are going
to raise the overall average.
The Maths Behind All This
The maths involved in the friendship paradox isn’t overly complicated and the
following proof, adapted from the article ‘Why your friends have more friends
than you’ by Presh Talwalkar[4], shows the maths behind all this.
The two things we need to know are the average number of friends for the entire
network and the average number of friends that any person in the network has.
The first part is the same as it was in the example from earlier, we just have
to add up the total number of friendships each person in the network has, and
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4. divide it by the total number of people in the network.
Average No. Friends =
Total No. Friendships
Total in Network
.
For the purpose of this proof, it will be easier to use some notation, so we
will say that person i has xi friends, that there is n people in the network, and
we will call the average number of friends µ. If we use this notation in the
formula that we just derived, we get that:
µ =
xi
n
.
Now we need to calculate the average number of friends each persons friend
has. We would start by looking at each person in the network, and add up the
number of friends that each of their friends has. If we use Person B from the
same four person network as earlier as an example, we would see that Person B
is friends with Person A, Person C and Person D, so we would add together the
number of friends that each of Person A, Person C and Person D have. Now, if
we had a much bigger network, it would obviously be a very long sum, however,
as we’re working out an average, we only need to worry about the total number,
not each individual term.
To get the total number we need to know how many times for any given person
i, the term xi will appear in the sum.
If you think about it, the only time you will ever need to count the number of
friends for a given person i, is when we are counting the number of friends that
one of person i’s friends has. From this we can deduce that each one of person
i’s friends will contribute the number xi to the total sum, so, to get the total
number of ‘friends of friends’ we will just have to work out xi×xi for each person
in the network. We can then take this figure and divide it by the total number
of friends each person in the network has. We can represent this in a formula by:
Average Friends of Friends =
(xi)2
xi
.
Looking at this formula, we can’t necessarily tell that the ‘Average Friends
of Friends’ will always be bigger than than the ‘Average No. Friends’. We can
however use a few substitutions that will show this.
Firstly, µ =
xi
n
can be rearranged to give xi=µn, so we can substitute µn
into the formula.
We can also use a rearranged version of the formula for the variance, σ2
, (the
average number each term is away from the mean) to get:
(xi)2
= (µ2
+ σ2
)n.
If we substitute these values of (xi) and (xi)2
into the ‘Average Friends
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5. of Friends’ formula we get:
Average Friends of Friends =
(µ2
+ σ2
)n
µn
.
This simplifies to µ+
σ2
µ
. Now if we compare the final results, for both ‘Average
No. Friends’ and ‘Average Friends of Friends’ we can see that:
µ ≤ µ +
σ2
µ
,
Which shows that the ‘Average Friends of Friends’ will always be greater than
‘Average No. Friends’.
How Can This be Applied In The Real World?
Research into the friendship paradox may look like mathematicians proving why
they don’t have many friends, but the applications could be a lot more useful
than they appear. The paper ‘Social Network Sensors for Early Detection of
Contagious Outbreaks’ by Nicholas Christakis and James Fowler [5], looked
at two groups, students at Harvard University and friends of those students.
Those who were named as friends of the students, contracted the flu two weeks
earlier on average than the students themselves. If more studies like this were
conducted you could potentially give vaccinations to people named as friends
and reduce the outbreak of the virus.
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6. References
1. J.P, Why are your friends more popular than you?, The Economist News-
paper LTD, April 22nd
2013
2. Scott L. Feld, Why Your Friends Have More Friends Than You Do, Amer-
ican Journal of Sociology, May 1991
3. Keith Hampton, Lauren Sessions Goulet, Cameron Marlow and Lee Rainie,
Why most Facebook users get more than they give, Pew Research Center,
February 3rd
2012
4. Presh Talwalkar, Why your friends have more friends than you: the friend-
ship paradox, www.mindyourdecisions.com, Published September 4th
2012,
Accessed 26th
November 2015
5. Nicholas Christakis and James Fowler, Social Network Sensors for Early
Detection of Contagious Outbreaks, Harvard Medical School, September
15th
, 2010
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