In it, @squiggle7 wonders about the number of connections able to be made by people in a 'room', be that virtually or physically.
Let's keep this up to date and use connecting with like-minded individuals on twitter as an example:
Imagine there's only one person on twitter who shares a particular interest of yours- how many connections can be made? By 'connections' I'm meaning mutual followships (to keep it simple), so in this case it's obvious: only one connection can be made.
Spotting the pattern
How about two other like- minded people? That's three altogether. I'll call them A, B & C (because they're cool names, right?). Because we're looking at mutual followships only, we can assume that if A is linked with B, then B is linked with A, so we can have the following links:
- A with B
- A with C
- B with C
- 3 connections altogether.
O.k, now how about four people? A, B, C & D this time:
- A with B
- A with C
- A with D
- B with C
- B with D
- C with D
- 6 connections available to be made.
[I'm being systematic when listing these possibilities- can you see the system I'm using?]
And five people (A, B, C, D & E):
- A with B
- A with C
- A with D
- A with E
- B with C
- B with D
- B with E
- C with D
- C with E
- D with E
- 10 connections possible.
We could carry on, but it's fairly clear by now that it's going to get a lot more complicated a lot more quickly, so can we do anything to simplify things? Can we find a pattern to follow?
Connections 1 3 6 10
Look at the numbers in that table: as the number of people goes up, the number of connections increases too. But by how much? It's fairly easy to see that as the number of people increases by 1, the number of connections increases by 2, then 3, then 4... so we can be confident that the number of connections for 6 people would be 10 +5= 15, and the number of connections for 7 people would be 15+6= 21 and so on. If we wanted to find out how many connections would be possible for, say, 20 people, we could just carry on the sequence. But what about 100 people? 150 people? 2000? 6000? 132,947 people?
Using algebra
Time to think about it a different way... Imagine there's a large number of people all with the same interest who all connect with each other. How many connections will there be? How many people exactly doesn't matter, so we'll call it "n" (so it can stand for any number).
- Person A can make a connection with everybody (except him/herself), so that's n-1 connections.
- Everyone else can make the same number of connections, so that's n-1 connections, n times: n(n-1)
- But we're forgetting that once a person makes a connection, they're connected both ways, so we've actually counted each connection twice. That's easy to fix, though, we just need to halve what we've got so far:
And that's that. In simple(ish) terms, this means that to find the number of connections that can be made with any number of people in this situation you have to multiply the number of people by one less than the number of people, and then halve the answer.
So, for 132,947 people, there are (132947 x 132946) / 2 = 8,837,385,931 connections that can be made. That's well over eight billion!
I wanted to use twitter as my example but thought it was a bit too complicated as you may follow someone but they might not necessarily follow you back. Can you explain that or does it get a bit too complicated?
ReplyDeleteThat's why I restricted it to mutual followings, but you can use the same ideas to start from:
ReplyDeleteWhat I've done above is to find the greatest number of connections that can occur, and assumed that for a connection to be functional it needs to be mutual. If you want to look at one-way connections, i.e. you may follow someone else with the same interest, or they may follow you, but not necessarily both happen at the same time, then you just miss out the last step: dividing by two (i.e. every person can follow every other person, regardless of whether they've already been followed by them or not).
In this case the number of possible one-way connections would just be n(n-1), where n is the number of people.
But what if some were one way and some were two way?
ReplyDeleteErm... The absolute max one-way connections would still be n(n-1). Any two-way connection would just be two one-way connections.
ReplyDeleteI don't think it's possible to be any more specific when you're using words like 'some'!
Interesting...and I really suck at math. I liked how you explained it though.
ReplyDeleteThanks! I bet you don't suck as much at it as you think you do. Most of the time it's about confidence... most people use some fairly in-depth maths every day without even realising they're doing it! Feel free to ask any questions you come up with :-D
ReplyDelete