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PROFESSOR: So now we start on
another topic in graph theory,

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namely the topic
of simple graphs.

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So last week we were talking
about directed graphs

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where the arrows have
a beginning and an end,

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as shown here.

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But simple graphs are simpler.

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The edges don't have direction.

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They just correspond to a mutual
connection, which is symmetric.

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So there's a picture
of a simple graph,

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and the edges are shown
without an arrowhead.

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A special thing
about directed graphs

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is that it's possible
to have an arrow going

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in each direction
between two vertices.

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But when we have
undirected edges like this,

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that doesn't happen.

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So there's only one
edge between a pair

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of vertices in a simple graph.

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In addition, a
directed graph might

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have a self loop, an edge
that starts and begins

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at the-- starts and
ends at the same vertex.

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And those are also
disallowed in simple graphs.

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Now, you could
allow those things.

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There's a thing
called multi-graphs

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where there are multiple
edges between vertices.

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And there could also be self
loops, but we don't need those.

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Let's not complicate matters.

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We're talking about
simple graphs.

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OK, so the formal
definition of a simple graph

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is that it's an object G
that has a bunch of parts.

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Namely it has a nonempty
set, v of vertices,

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just like directed graphs.

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And it has a set E of
edges, but the edges now

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are somewhat different since
they don't have beginnings

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and ends.

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An edge just has two
endpoints that are in V,

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and we don't distinguish
the endpoints.

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So let's just draw a picture.

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Here's a case where there are
six vertices V shown in blue,

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and there are these undirected
edges shown in green.

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In this case I see
seven edges in E.

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There is an example of an edge
that goes between two vertices

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that I've highlighted
in yellow and red.

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And we've made that
particular edge dark green.

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An edge like that can
formally be represented

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as just the set of its end
points, a set of two things,

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red and yellow.

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In text, we'll often indicate
it as the two vertices connected

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by a horizontal bar.

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But you have to remember
that the order in which

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the red and the
yellow occur don't

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matter, because it's
really a set consisting

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of red and yellow.

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When two vertices are
connected by an edge,

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they are said to be adjacent.

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And the edge is
said to be incident

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to its end points, just a little
vocabulary that we use here.

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A basic concept in
graph theory, which

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is what we're going
to make a little bit

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of in this video segment, is the
idea of the degree of a vertex.

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The degree of a vertex is simply
the number of incident edges,

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the number of edges,
that touch it,

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the number of edges for
which its an end point.

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So let's look at the red vertex.

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There are two edges
incident to the red vertex,

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so its degree is 2.

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OK, let's look at
the yellow vertex.

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Here there are four edges
incident to the yellow vertex,

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so its degree is 4.

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No surprises yet.

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So let's examine some
properties of vertex degrees

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that are motivated
by a simple example.

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Suppose I asked
the question, is it

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possible to have a graph with
vertex degrees of 2, 2, and 1.

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So implicitly it's
a 3 vertex graph.

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And one vertex has degree
2, another has degree 2,

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and one has degree 1.

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Well let's see
what it looks like.

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If I'm going to have
a vertex of degree 1,

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then I know what it looks like.

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There's the vertex.

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It's got one edge out of it.

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It's going to some other vertex.

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Now this other vertex
must have degree 2,

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so it's connected
to something else.

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And the something else must be
another vertex with degree 2,

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because those are the only
possible spectrum of degrees,

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1, 2, and 2.

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And that means
that this last guy

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has to have an edge out of
it, because it's degree 2.

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And it can't go back to
2, because there's already

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an edge between these two.

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And it can't go back to 1,
because that has degree 1.

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So we're stuck.

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And by this ad hoc
reasoning we figured out

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that there can't be a degree
3 graph with this spectrum

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of degrees 2, 2, 1.

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It's impossible.

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Well, we could have
reasoned more generally.

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And there's a very elementary
property of degrees

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that we're going to actually
make something of in a minute.

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And it's called the
handshaking lemma.

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It says that the
sum of the degrees

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summed over all the
vertices is equal to twice

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the number of edges.

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There it is written
as a formula,

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twice the number of edges.

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So that's the
cardinality symbol.

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Absolute value of a set
means the size of the set.

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E here is finite.

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Twice the number of
edges is equal to the sum

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over all the vertices of
the degree of the vertices.

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Why is that true?

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Well, if you think about
it, in the sum on the right,

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every edge is counted
twice, once for each vertex

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that it's the end of.

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So we're really just
count-- and this

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is a way of summing up
over all of the vertices

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in which each vertex
gets numerated twice.

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So the sum is twice
the number of vertices.

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And the proof is trivial, but
let's make something of this.

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You might wonder why it's
called the handshaking lemma.

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That will emerge
in some problems

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that we're going to have you do.

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But let's go on and apply
the handshaking lemma

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in an interesting way.

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And by the way, of course,
since 2 plus 2 plus 1 is odd,

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we could have without that
ad hoc analysis figured out

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that the sum of the
degrees can't be odd,

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because it's twice something.

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All right, so here's
the applications

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designed to get your attention.

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It is an application
of graph theory to sex.

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And we ask the question, are
men more promiscuous than women?

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And there have been
repeated studies

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that are cited in the notes that
show again and again that when

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they survey collections
of men and women

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and ask them how many
sexual partners they have,

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it's consistently the case that
the men are assessed to have

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30% more, 75% more,
sometimes 2 and 1/2 times,

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3 times as many
partners as the women.

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And there's got to be
something wacky about this.

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We're going to come up with
a very elementary graph

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theoretic argument that says
that this is complete nonsense.

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By the way, the most recent
study that we could find

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was one that's mentioned
in the notes in 2007

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by the US Department of Health.

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And the statistician
who collected the data

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knew that the results
were impossible.

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But her job was to
report the data, not

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to explain it or interpret it.

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And the men reported 30%
more partners than the women.

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And we're going to show
that somebody is lying.

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Here's how we're going to do it.

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We're going to model the
relationships between men

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and women by having a graph
that comes in two parts.

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It's going to be called a
so-called bipartite graph.

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So there's going to be
one set of vertices called

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M and another set of
vertices called F. M for men

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and f for women, or females.

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And we're going to have edges
going between men and women,

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between M's and F's, precisely
when they have been involved

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in a sexual liaison.

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So looking back at this graph,
this edge from that blue M

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to that orange F indicates
that they had a sexual liaison.

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They were partners.

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OK, so this is a
simple graph structure

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that we can use to
represent who got together

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with whom in any given
population of men and women.

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Now, if you think
about the same argument

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that we use for handshaking, if
you sum the degrees of the men,

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you're counting each
edge exactly once.

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And so the sum of the
degrees of the men

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is equal to the number
of edges in this graph.

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And likewise, if you
sum over the females,

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you're counting each edge once.

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And so the sum of
the female degrees

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is also equal to
the number of edges.

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In particular, the sum over
the degrees of the males

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is equal to the sum over
the degrees of the females.

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Because every time
there's a liaison,

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it involves one
male, one female.

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All right, now let's
just do a little bit

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of elementary arithmetic.

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I'm going to divide both
sides of this equality

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by the size of the
male population,

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by the number of men.

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And if I do that,
I get this formula.

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The left hand side
is the number of-- is

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the sum of the degrees of men
divided by the size of the M

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population.

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And here I'm doing
a little trick.

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Notice that the F's cancel out.

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But I've expressed some of
the female degrees divided

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by M as the sum of the female
degrees divided by F times

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this factor F over M, which is
the ratio of the populations

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of women to men.

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Now the reason I'm doing
this is that if you

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look at this thing
on the left, this is

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the average degree of the men.

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This is the sum of
all the degrees of men

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divided by the number of men.

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So it's the average number
of partners that men have.

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And likewise, now you
can recognize over here

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that I've got the average
number of partners

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that each woman has.

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And what we've just
figured out then

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is that there's a
fixed relationship

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between the average
number of partners of men,

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the average degree
of the M vertices

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and the average degree
of the F vertices.

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And these two average
degree, these average numbers

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of partners, is simply
related by the ratio

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of the populations.

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The men degree is
the female population

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divided by the M population
times the average degree

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of the females.

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Now, what this tells us
is that these wild figures

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of twice as many, and
30% more, and so on

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are completely absurd.

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Because we know a lot
about the ratio of females

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to males in the population.

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As a matter of fact,
in the US overall

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there are slightly
more women than men.

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There is 1.035 women for each
man in the US population.

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And that tells us then
that if you surveyed

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the population of all the
men and women in the country,

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you would discover that the men
looked 3 and 1/2 percent more--

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had 3 and 1/2 percent more
partners than women per man.

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But this has nothing to do with
their behavior, or promiscuity,

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or lack of it.

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It's simply a reflection of
the ratio of the populations.

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Which gets us to
the question of,

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where do these crazy
numbers come from?

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And the answer seems to
be that people are lying.

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One explanation would
be that men exaggerate

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their number of partners,
and women understate

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their number of partners.

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But the truth is
that nobody knows

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exactly why we get these
consistently false numbers.

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But we do get them consistently
in one survey after another.

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You will no longer be
fooled by such nonsense.