Graph ti Since were only considering opposite-gender relationships, every edge connects an M vertex on the left to a w vertex on the right. So the sum of the degrees of the M vertices must equal the sum of the degrees of the w vertices x∈M y∈W Now suppose we divide both sides of this equation by the product of the sizes of the two sets,|M|·|W ∑ IEM deg(a).1/∑ Yew deg().1 The terms above in parentheses are the average degree of an M vertex and the average degree ofa w vertex. So we know: Avg deg in M Avg deg in w avg deg in w\Avg deg in w M Now the Census Bureau reports that there are slightly more women than men in Amer- ica; in particular WI/M is about 1.035. So- assuming the Census Bureau is correct- weve just proved that the University of Chicago study got bad data! On average, men have 3.5% more opposite-gender partners. Furthermore, this is totally unaffected by dif- ferences in sexual practices between men and women; rather, it is completely determined by the relative number of men and women 1.3 Graph variations There are many variations on the basic notion of a graph. Three particularly common variations are described below. In a multigraph, there may be more than one edge be- tween a pair of vertices. Here is an example� � Graph Theory 3 M W Since we’re only considering opposite­gender relationships, every edge connects an M vertex on the left to a W vertex on the right. So the sum of the degrees of the M vertices must equal the sum of the degrees of the W vertices: deg(x) = deg(y) x∈M y∈W Now suppose we divide both sides of this equation by the product of the sizes of the two sets, |M| · |W|: �� � �� � x∈M deg(x) 1 y∈W deg(y) 1 · = W| · |M| | |W| |M| The terms above in parentheses are the average degree of an M vertex and the average degree of a W vertex. So we know: Avg. deg in M Avg. deg in W = |W| |M| W Avg. deg in M = | | · Avg. deg in W |M| Now the Census Bureau reports that there are slightly more women than men in Amer￾ica; in particular |W / M | | | is about 1.035. So— assuming the Census Bureau is correct— we’ve just proved that the University of Chicago study got bad data! On average, men have 3.5% more opposite­gender partners. Furthermore, this is totally unaffected by dif￾ferences in sexual practices between men and women; rather, it is completely determined by the relative number of men and women! 1.3 Graph Variations There are many variations on the basic notion of a graph. Three particularly common variations are described below. In a multigraph, there may be more than one edge be￾tween a pair of vertices. Here is an example: