Communicating Mathematics -

Assesment 2

By George Stevens13811875

Why Do Your Friends Have More Friends ThanYou Do?

Do you ever feel like all your friends are more popular than you are? That theyhave more friends than you do? Well you’re probably right but you’re not alonebecause this is true for nearly everyone.The phenomenon that on average, most people have fewer friends than theirfriends 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 thinkinglogically you’d assume that this is only true for those with a lower than averagenumber of friends, however this is not the case.The following example, adapted from the article ‘Why are your friends morepopular than you?’ in The Economist Newspaper[1], will help to visualise thisconcept in action.

Four Friend Network Example

Figure 1: The diagram above shows a simple network of four people, where afriendship between two people is denoted by a line connecting them directly toeachother.

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 hasthree friends. Person C and Person D are friends with each other and are bothfriends with Person B, so they each have two friends.From that we can easily deduce that the average number of friend that a personin this particular network has, is two: 1 + 2 + 2 + 3 = 8 (total number of friendseach person has), 8÷ 4 = 2 (total number of friends each person has divided bythe number of people in the network).From that it would appear that because the average number of friends is two,

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the average number of friends that each persons friend has will also be two, butas we said earlier, this is not the case. The following table summarizes the mainfigures involved in this particular example and should go some way towardsexplaining why this is not the case.

Person No. of Friends No. Friends of Friends Average Friends of FriendsA 1 3 3 ÷ 1 = 3B 3 1 + 2 + 2 = 5 5 ÷ 3 = 1.67C 2 3 + 2 = 5 5 ÷ 2 = 2.5D 2 3 + 2 = 5 5 ÷ 2 = 2.5

Total 8 18Mean 2 18 ÷ 8 = 2.25

Figure 2: ‘No. Friends of Friends’ is the total number of friends that the personsfriends have. This is calculated by adding together the number of friends eachone of that persons friends have. The ‘Average Friends of Friends’ is the averagenumber 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 fromPerson 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 atthe overall means for ‘No. of Friends’ and ‘Average Friends of Friends’ we cansee that on average, a person in this network will have 2 friends, however, theirfriends will have 2.25 friends on average. This shows that on average a personin 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 thisworks, but a comprehensive study by the Pew Research Center [3] found thatthis was true for Facebook users. They found that the average user has 245friends whereas each of their friends have an average of 359 friends.So why does this phenomenon occur? It’s actually down to selection bias. Peoplelike Person B, who are the most well connected in the first place, are also goingto be counted most when people are looking at ‘Friends of Friends’, so are goingto raise the overall average.

The Maths Behind All This

The maths involved in the friendship paradox isn’t overly complicated and thefollowing proof, adapted from the article ‘Why your friends have more friendsthan 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 entirenetwork 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 haveto add up the total number of friendships each person in the network has, and

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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 wewill say that person i has xi friends, that there is n people in the network, andwe will call the average number of friends µ. If we use this notation in theformula that we just derived, we get that:

µ =

∑xin

.

Now we need to calculate the average number of friends each persons friendhas. We would start by looking at each person in the network, and add up thenumber of friends that each of their friends has. If we use Person B from thesame four person network as earlier as an example, we would see that Person Bis friends with Person A, Person C and Person D, so we would add together thenumber of friends that each of Person A, Person C and Person D have. Now, ifwe 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 personi, the term xi will appear in the sum.If you think about it, the only time you will ever need to count the number offriends for a given person i, is when we are counting the number of friends thatone of person i’s friends has. From this we can deduce that each one of personi’s friends will contribute the number xi to the total sum, so, to get the totalnumber of ‘friends of friends’ we will just have to work out xi×xi for each personin the network. We can then take this figure and divide it by the total numberof 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 Friendsof Friends’ will always be bigger than than the ‘Average No. Friends’. We canhowever use a few substitutions that will show this.

Firstly, µ =

∑xin

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, (theaverage 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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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 whythey don’t have many friends, but the applications could be a lot more usefulthan they appear. The paper ‘Social Network Sensors for Early Detection ofContagious Outbreaks’ by Nicholas Christakis and James Fowler [5], lookedat two groups, students at Harvard University and friends of those students.Those who were named as friends of the students, contracted the flu two weeksearlier on average than the students themselves. If more studies like this wereconducted you could potentially give vaccinations to people named as friendsand reduce the outbreak of the virus.

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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 EarlyDetection of Contagious Outbreaks, Harvard Medical School, September15th, 2010

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