Hey Maths guys help preez !!!

[Tab]

I've been thinking about network connection configurations, and nodal count.

I doodled on a bit of paper. A connection between any two points can either be on ______ ie a solid line, or off -------- a dotted line.



It seems the total number of configurations, as you add nodes, goes 2 -> 8 -> 16 if you only draw connections between the outer nodes ie a power law based on 2 where:

total number of (outer) configurations = 2 to the power (total number of nodes)

BUT: if you also connect within the shape the configuration number jumps to 2->8->64

ie 2x2x2x2x2x2 (2 power 6)

I only drew as far as 4 nodes. Because when I got to 5 nodes - a pentagon - the number of internal connections (imagine a 5-pointed star) - made me decide to say "screw this".

Two questions for the maths people.

1) How does the sequence continue, with internal connections ie 2,8,64,__ ?.
2) What's the law..?

I'm guessing that for five nodes the total will be 2power 10 ie 2x2x2x2x2x2x2x2x2x2

and the rule is TC = 2 to the power (number of possible connection routes between nodes)

Am I right..?

One final thing. As the above stands, there are only two connection states - on or off - if I added a third, let's say 'half-on' would that make the law follow a 3x3x3x3x3x3 ad infinitum sequence..?

ie TC = (number of connection states) to the power (number of possible connection routes between nodes)

Answers on a postcard please.

[AnitaS]

Tab, this is the best I can do right now, gotta go make dinner. Here's the equation (I think, went over this really quickly):


I'll come back and check on you later!

[Tab]

Thankyou.

What do all the 'n' s mean..?

(Sorry - I only took maths till 16, and that was back when we still used our fingers)

[Liteninbolt]

I had access to a scientific calculator. I took off my shoes and socks so I had twelve more possible calculating tools.

[AnitaS]

What do all the 'n' s mean..?

"N" would be the number of nodes. Sorry, I should've put an equal sign in there to complete the equation. If you substitute the number of nodes for "n," it will equal the number of possible configurations.

Make sense? [I'm not very good at explaining...]

[Tab]

So:

TC = 2 to the power node x ((node x (node-1)) / 2)

ie say node = 3

TC = 8 x ((2x1)/2)) -> 8 x 1 = 8 ok

node 4

TC = 16 x ((4 x 3)/2) -> 16 x 6 = 96 ??

Hmm..?

Something's gone runny. Either I'm missing something at 4 nodes, or I've done my sums wrong..?

[AnitaS]

Yep, you're right, something's off. [Sigh.] Let me take a little more time to look at it a bit closer.

[AnitaS]

How about this:


Getting closer?

[Tab]

Okay let's see.

node 4

TC = 2 to the power ((4x(3))/2) >> 2 p 6 = 2x2x2x2x2x2 = 64 yay.

logically, it checks out I think. Let's see. I said:

ie TC = (number of connection states) to the power (number of possible connection routes between nodes)

2 = connection states

node 4 = four outer connects

and on it's own - disregarding connections within the shape - follows the strict rule: (number of connection states) to the power (total number of nodes)

the inner is the bastard though. Basically - when you get to 4, that gives you 2 extra pathways - equalling 4 extra inner configurations for each of the outer configurations. Leaving us with:

((number of connection states) to the power (total number of nodes)) x (possible number of inner configurations)

ie. in the case of 4 nodes:

(2 power 4)x 4 = 64.

Yay.

the hard bit though is the rule for possible inner config.

Okay, each extra node adds how many paths..?


4 nodes - 2 inner paths - and extra 2 for one increase in nodes.

5 nodes - 5 inner paths - an extra 3, for one increase in nodes.

6 nodes - 9 inner paths - an extra 4, for one increase in nodes.

Aha. Ze pattern forms. Kinda. Er.

hmm. Time for a fag.

Okay. I'm doing this the hard way. Inner/outer.

in a four node network - each node has 3 connections - but only the first node makes 3 new connections, the second (already connected to the first) makes 2 new connections, the third only 1, and the last 0.

ie 3+2+1+0 = 6 total.

Argh. okay. Google.

Wiki answers says - Where n = number of nodes

The number of connections in a full mesh = n(n - 1) / 2 (Way to go Anita!!! You are as clever as Wiki)

okay, so we had from earlier, from a three node:

TC = (number of connection states) to the power (total number of nodes)

But this is wrong. I fucked up at three because the number of nodes at that point equals the number of connections.

it should be:

(number of connection states) to the power (total number of connections)

ie: 2 to the power n x (n - 1) / 2

node 1 - 2power1x0/2 = 0 config
node 2 - 2pwr2x1/2 = 2 config
nd3 - 2pwr3x2/2 = 8 config
nd4 - 2pwr4x3/2=64 config
nd5 - 2pwr5x4/2=2pwr10 config.

I'm guessing that for five nodes the total will be 2power 10 ie 2x2x2x2x2x2x2x2x2x2

Actually, looking back, I can't remember why I guessed this figure - but hey, I was right..! The important thing though, now I know why.

So, final rule:

The Maximum number of possible configurations within a network of N nodes, where each connection has S states - TC - is:

TC = (S) to the power (n x (n - 1) / 2)

Great big thanks Anita. You rock, and everybody else at ILP who didn't help, they suck.

[AnitaS]

You're welcome, Tab, but by the time I got back here, you'd done all of the heavy lifting yourself! So let me just ask, is this how you spend all of your vacations?

Talk about power, man, if we could only use your brain to fuel our cars, we wouldn't have any energy problems!