During the high school, I have experienced an embarrassing game during a health education workshop:
a) $N$ boys and $N$ girls, each player shakes his/her hand with (average) $d$ players of the opposite sex.
b) Then the teacher selected a player and asked him/her to raise the hand, and said, Aha, this guy has AIDS and hand-shaking means having a corporal relationship. So everyone who has shaked with someone who is raising hand please raise your hand.
Clearly, the teacher is proposed to tells us the merit of monogam: If $d>1$ then everyone should be infected. Now, let's suspend the ethical dispute and talk about the math:
Is it true that the critical value is $d=1$, and are there an analytic proof?
I was very confused at that time because I thought the order of having relationships is ignored. So, if one becomes infectious only after he/she have shaked with an infectious player, will there be any significant change on the model?
It may be offensive but let's face the science: If anyone can shanke with anyone, will there be any significant change on the model?
You can't draw that conclusion from the model. Suppose there are $4$ men and $4$ women each divided into two groups of $2$. Pair the groups and have each mixed sex group of $4$ shake hands all around their group. Then everyone shakes twice so the average is $2$ but hlaf the people are clearly disease free. Of course that distribution of handshakes is not random.
I think the point the teacher was trying to make is that random graphs are more connected than you might think.