Hello welcome to the first tutorial on networks and we're going to start off by
talking about what is a network a network is basically a collection of
objects connected in some way and in the case of this topic we're going to
represent the objects as little circles which we're going to call them nodes so
these things here we're going to call them nodes and they're going to be
connected by some lines that we're going to call arcs
okay so networks are joined by sorry networks are made up of nodes joined by
arcs and usually the arcs will have some kind of value on them indicating the
distance between the nodes so for example I can say the arc length here is
three arc length here is four five seven and so on but for our purposes right now
we don't have to worry about arc length okay each node can then be classified as
an odd node or an even node so if you have a look at this node here this is an
even node
okay but if you have a look at I'm just going to join these two nodes with
another arc if you have a look at this node here this is an odd note so what is
the difference between even and odd how can we tell and even node has an even
number of Arcs joined to it
so you can see we have one two two arcs joined to it odd node has an odd number of
Arcs
joined to it
so we have one two three three Arcs jointed to it and you can also say that this
node is a two node and this is a three node all right because there's two arcs
there three arcs here and another definition you might be you might find
useful is path path can be a series of arcs so if we define a path as let's say
from this circle here to this circle here we have to use two arcs to get to
it so
okay now we're going to talk about whether a network is singly traceable
and to test whether is singly traceable I'm going to start here at this circle
and go to the left down up right down so I have just proven to you that this
network is singly traceable because I went through all the arcs without
repeating the same arc now something that might not be singly traceable let's
try and see if that's singly traceable so I'm going to start here one down the
right all right
as you can see it's not singly traceable because I can't go through the same arc
again so I've arrived at this point I can't go anywhere so that leaves this
one not traced so it's not singly traceable now a quick way to test
whether to test whether a network is singly traceable is to see how many odd
nodes and even nodes that it has okay so how do we determine if the network is
singly traceable without trial and error by drawing all over it
or here are the strategies for determining whether it's singly
traceable it will be singly traceable when number one it has no odd vertices
so for example in our first network if we just had
four nodes 1 2 4 arcs each of these nodes are even each of these nodes are
all even nodes so there are no odd vertices and you can see that it's very
easily singly traceable if it has two odd vertices it's also singly traceable
as the case here we have this one is odd and this one is also odd so we can start
an odd singly trace it and end up at the other odd and all the other cases it
will not be singly traceable okay so all other cases all other cases not singly
traceable
so hopefully that that gives you a quick way of determining whether a network is
singly traceable without without having to do it by trial and error okay now
we're going to talk about how to draw a matrix from the network and this is
actually pretty easy even though when people talk about matrices you think of
something complicated but it's actually pretty easy so let's have a look at one
of the networks and this is the network and what we're going to do is create a
numerical representation of this by drawing a matrix for this network okay
so to draw a matrix basically there are four nodes here so I'm going to write
the each node as a heading both the column headings and the row headings and
I'm going to draw a big bracket
I'm going to draw big brackets okay now from A to A if you have a look at my
note there is no connection going from no arc going from A to A so that would
be zero now from A to B there is one arc so I'm going to put one there is nothing
connecting A to C directly and there is no arc between A and D so zero so
basically a matrix is simply saying how many arcs are connecting from one node
to another from B to A there's one from B to C there is two from B to B
there is zero from B to C there is two from B to D there is one from C to B
there is two C to A there's zero C to B is two C to C is zero C to D is 1 and lastly
for D D to A is zero D to B is one D to C is one D to D is zero so this is our matrix and
usually when we write the matrix we don't write the headings so this part is
usually omitted so when you see a matrix like this
and from this matrix you can draw this network without knowing that the nodes
are called ABCD ABCD and on the left-hand side is from on the top is 2
and so on you are given this network and you have to you're given this matrix and
you can be asked to redraw the network so for example if we didn't have any of
this information so if i blank out all of this and I asked you by giving you
this matrix draw a network for the above matrix
then you should be able to just do the ABCD yourself ABCD and figure out that
we have four notes so A it doesn't even matter how you draw it A B C D A to B is
1 B to C is 2 B to D is 1 and C to B is 2 C to D is 1 D to B is 1 D to C is 1 and
you can see that this is pretty much exactly the same as this network
even though B and C have been switched around
it still works exactly the same way this network has the same matrix as that
Network
okay now let's talk about loops now for example in this same network if I add a
loop at a particular node okay let's say on D I now have a loop going from D back
to D now what happens is sorry that might make it a bit confusing I'm just
going to write the word D somewhere else
so what I'm going to do is on the matrix now corresponding to this network I will
then have to change the number zero to two now why is it two and not one
because from D back to D there is two ways you can do it you can either go
this way or you can go this way so because there are two ways of going from
D to D you have to put the number two there in a diagonal so you will see with
all of these matrices by the way matrices is the plural for matrix that
they are all symmetrical across the diagonal so if this is a diagonal okay
this is a diagonal and we have the top right section here and the top bottom
left section here these two sections are symmetrical okay so you can see we have
1 2 1 and a 1 1 2 1 and a 1 so this side is a mirror image of this side and that
is a character characteristic of networks of this type of network which
we call non directed networks so this network here is a non directed network
so when does the network become directed well a network becomes directed when we
start drawing arrows on the networks so using this network here I'm going to
draw some arrows on them so I'm going to say A can only go to B I'm going to
draw an arrow there draw an arrow here now as soon as you start drawing arrows
on the network then it becomes a directed network okay so for directed
Network what happens is the matrix will no longer be symmetrical so we're going
to have to make some changes to our network here so from A to B it's 1 but
from B back to A is 0 so I'm going to change this number to 0 now right the
reason is for this arc here you only allow to move in one direction so you
can't go from B back to A and same thing here we go from B to C is 2 right
because there is no there is no restriction on the direction of this arc
but going from C to B there is only one line so C to B is only 1 okay and then D
to B is only 1 D to B is 1 but B to D is 0 so B to D is 0 B to D is 0
A to B zero oops B to A is zero A to B is one yep D to B is zero B to D
is hang on a second D to B is 1 this should still say one B to D is zero
so when you're doing the matrix for directed Network it's very important
that you remember these are the From nodes and these are the To notes so
from B to A is 0 from B to D is 0 from C to B is 1 from B to C is 2 ok so once
you have drawn the matrix for the directed Network you will see that it's
no longer symmetrical so this is a diagonal and we can see that these
numbers are no longer symmetrical to these numbers so directed Network they
are not symmetrical across the diagonal and if we have a loop okay if we have a
loop at one of the nodes let's put a loop around B and let's put a direction
on that so from B to B we are allowed to have one on the diagonal whereas for a
non directed Network on the diagonal we're only allowed to have even numbers
on a directed Network we're allowed to have odd numbers and that is because for
a non directed network each time you draw each time you draw a loop that's
two that's two ways whereas if you draw a loop on a directed Network then you
and you put a Direction restriction on that then you can only have you could
get a 1 or 3 on the diagonal
okay so hopefully from this tutorial you've learned a little bit about
networks and how to draw matrices for them see you in the next tutorial thanks
for watching.