The Modeling of Human Sexual Contact Networks

I recently reviewed F. Roger Devlin’s book, “Sexual Utopia in Power.” Devlin conjectures in his book,  that the “total amount of sex” available has unchanged since the 1960’s, but the “distribution” of sex in society has changed. This is mathematically more complicated to model than I think Devlin has considered. How do you mathematically define the “the distribution of sex”? Does he mean the “distribution of the number of partners”? Or the distribution of the frequency of sexual intercourse? After all, a happily married male having sex 5 times a week with his wife is probably far happier than an unmarried male that has sex perhaps once or twice a month with different women. A more precise mathematical description is necessary to model this “distribution of sex.”

An Overview of the Sexual Contact Network Model

One way of modeling human sexual networks is as a graph, with subsequent analysis performed using concepts from graph theory and network theory. This framework allows us to mathematically described the behavior that is allegedly occurring in Western society. Human sexual behavior has already been modeled as an unweighted, undirected, bipartite graph by Ergun. The number of partners in human sexual networks have been observed to follow a power-law degree distribution by Liljeros et al. This is evidence (though not proof) that “preferential attachment” is the mechanism at work in the network (see all the Barabasi-Albert model page on Wikipedia). An example of such a network is shown below:

An example human sexual network. Males are blue circles, and females are pink squares. The numbers indicate sexual market value. The thickness of the connecting lines indicates the frequency of sexual intercourse.

The prior model by Ergun is unweighted and undirected. A more complete model of human sexual behavior can be obtained with several modifications. We will ignore the situation where the gender distribution diverges significantly from 50/50. Each node has two labels: its sexual market value (SMV), and its amorousness. I have chosen integer values for SMV, though this is not necessary in practice; SMV can be a continuous function. From the standpoint of disease prevention, it is readily obvious why a married, monogamous sexual relationship is the safest; the married couple is disconnected (“disjoint”) from the (vastly larger) sexual network of people who had had sex with someone who has had sex with someone else. We have not considered in this work the dynamics of a sexual contact network. To match experience, amorousness would decay with age, as older people are simply less amorous than young people. Tastes in the opposite sex also changes due to time and chance.

The extensions to the model are as follows:

• The edges of the graph can be weighted according to the frequency of sexual intercourse within the given time frame. For example if a male is having sex with a female 5 times a week, then their corresponding entry in the adjacency matrix would be a “5.” The edge weights are only positive integers, since it is not possible to have sex “1/2” times (though I’m sure some unsatisfied females often feel that way after a premature ejaculation). Reasonable bounds on the frequency of sexual intercourse in the past year depend on the style of the relationship, as well as how amorous the couple is. A deeply infatuated, young, married couple might have sex two or three times a day, maybe even more (average is about 109 times a year for young couples). There are limits to male virility and sperm supply, so simply because a man has more partners, does not mean he can produce more sperm and perform vastly more sex acts than the married male of similar physical fitness. A young man with his wife might have sex 100 times a year; a cad with his 10 lovers probably cannot exceed that value by much, simply due to constraints on time and sperm supply.
• A person’s amount of sexual desire could be theoretically modeled with an “amorousness” score for each node. This is the number of times the person desires sex within a given time frame. For example, a male with an amorousness of 5 would desire sex 5 times a week (or whatever the time frame was). If this male is wed to a female with an amorousness of 1, or an amorousness of 15, the marriage is likely to be strained. There is probably some positive correlation between amorousness and SMV, but without actual data, it is hard to guess how strong the correlation is likely to be. Theoretically, there is no reason why a handsome, muscular, wealthy male must have a high amorousness level; he might be perfectly content to have sex only once a week. A broke, overweight man might desire sex up to 3 or 4 times a week. Couples with high and similar levels of amorousness will have greater frequency of sexual intercourse; couples with low and similar levels will have less sexual intercourse. It becomes tricky when the values are different; who wins the argument? I would argue that since the woman gets to choose when sex is had, that if the woman’s amorousness is less than the man’s, then the couple has less sex. If the woman has higher amorousness than the male, the male will likely cave in and have sex a little more often than he would prefer. This is because the woman can always threaten to leave the relationship, thus cutting the male off from all further access to her body.
• Humans engage in a variety of sexual behavior. The most important sexual behaviors are vaginal, anal, and oral sex. In order to model these behaviors, three separate adjacency matrices are used: one for vaginal sex, one for anal sex, and lastly one for oral sex. The graphs for vaginal and anal sex are undirected, since it is impossible for a woman to perform these activities on a man; only the man can perform them on the woman. However, oral sex is a directed graph, since both partners have mouths. An arrow pointing into a node means oral sex was performed on that person. An arrow pointing out of a node means that person performed oral sex on someone else.
• Humans have preferences based on sexual market value (SMV). Each node is described by an SMV score. The higher the score, the most sexually attractive the person is. The likelihood of sexual intercourse declines rapidly as the difference between male and female SMV increases.

Brief Insight on Homosexual Activity in the Network

An example sexual contact network of gay men. Unlike the heterosexual network, this network is a directed graph. Arrows pointing into a node indicate the number of times this male was the receiving partner (“the catcher”). Arrows leaving a node indicate this man was the active partner (“the pitcher”).

If we include homosexual men into the sexual contact network, the graph becomes a directed graph. This is because gay men can be either a receptive of active partner with regards to oral and anal sex; with straight couples, the female is always the receiver (except for oral sex), so adding direction to those graph edges adds no new descriptive power to the model. An arrow pointing at a node for a gay male would indicate that partner has been the passive partner in the sex act; an arrow pointing out from a gay male would indicate that gay male is the active partner. As far as the modeling of disease goes, lesbians are virtually irrelevant to the discussion, so whether their graph is directed or not (or if they are even included in the model at all) is not really important.

Based on a cursory examination of the statistical and anecdotal evidence, I conjecture that the sexual contact network of gay men is predominantly composed of “bottoms” (men who are never the active partner), with comparatively fewer “tops” (men who are always the active partner), with a minority of “versatiles” (men who are either active or passive, depending on mood and circumstance). This structure comports with the rates of HIV infection among gay men, since passive anal sex partners are at far greater risk of HIV transmission than active partners. This is also why HIV is far more dangerous to straight women than straight men, and why HIV tends to “pool up” in straight women and passive gay men.

I conjecture that, as far as the spread of HIV goes, the most dangerous individuals in the sex contact network are bisexual men who have receptive anal sex with men but also have vaginal sex with women, since these males form links between a high-risk HIV population and the straight population at large. Further discussion of homosexual male sex contact networks is a little beyond our scope here, though it is an interesting topic to cover in another blog post.

Sexual Inequality and the Gini Coefficient

An interesting concept from economics that describes the inequality of sexual intercourse is the Gini coefficient. The Gini coefficient is mathematically abstruse to define, but its limits are easy to understand. For the incomebased Gini coefficient, a Gini coefficient of zero means that every person in the sample population earns the exact same income; a Gini coefficient of 1 means one person is acquiring all of the income in the population. Likewise for the wealth-based Gini coefficient; a Gini coefficient of zero means everyone in the sample population has the same amount of wealth; a Gini coefficient of 1 means one person possesses all the wealth. According to the CIA, a society becomes increasingly unstable as its income Gini coefficient rises above 0.5.

A sexual Gini coefficient would measure the distribution of sexual intercourse in the society. However, it is not possible to fully describe intercourse inequality with a single number, because sex has so many more facets than money and wealth. In order to fully describe the inequality of sex in a network of human sexual contacts, we need several Gini coefficients.

1. A “number of partners” Gini coefficient, $G_{NOP}$. For the case of perfect equality, each male node has the same degree; that is, each male has the same number of female sex partners. Likewise with the females.
2. A “quality of partners” Gini coefficient, $G_{QOP}$. For a given person, their “quality of partners” score is the average SMV of all their partners. This distribution is likely to be wracked by inequality, since only a small minority of males are capable of having sex with higher-quality women. No matter how hard they sell, females of score 3 are not reasonably capable of having sex with males of score 8. Since this score is an average, the occasional low-quality male mating with a high-quality female is unlikely to change his overall score much.
3. A “frequency” Gini coefficient, which is simply the number of times a man has had sex, $G_F$. This is the sum of the number of times a man has had sex with all of his partners. In a society of monks, no male is having sex (logically also, no women are having sex). In a society of egalitarian cads, each man is having sex with a multitude of women, but the frequency of intercourse is equal for each man.
4. A “sexual quantity satisfaction” Gini coefficient, $G_{SQS}$. For a particular male node, this is the absolute value of the male’s amorousness score minus the number of times he has had sex. Ideally, each male is having sex exactly equal to the number of times he wants to; no more no less.

I have listed four Gini coefficients, but since the distributions of these quantities for women and men are different, we need these four Gini coefficients for each gender, for a total of eight. There are likely some mathematical constraints on these Gini coefficients imposed by the fact that it takes one woman and one man to create a sexual union, but they are not readily apparent to me. This framework also ignores the fact that some people are unfit for monogamy, and require sex with different people to stay happy. Thus, a more “number of partners satisfaction” Gini coefficient may be necessary, but defining it is beyond the scope of this essay.

Substantial Data Requirements

Unfortunately, the data requirements for accurately identifying the Sexual Market Value distribution of American women are absolutely monstrous, since the Pareto surface is composed of such an enormous number of variables, and beauty is somewhat subjective. However, Hot-or-Not.com and data analyses from OkCupid show that beauty is at least semi-quantifiable based on user-submitted beauty scores and number of messages received.

A Network Composed Solely of Husbands and Wives

Let us consider a “sexually ideal” society, run presumably on some type of religious principles, which possesses monogamous married couples. We will ignore the nuns and monks, and the corner case of “Lucy and Ricky” where the husband and wife sleep in separate beds; we will always assume for a married couple a minimum sexual contact frequency of 1. The network would look something like this (the different boldness of the connecting lines indicates the frequency of sexual intercourse between the husbands and wives):

Sexual network composed exclusively of married couples.

There are many advantages to such a network over the current state-of-affairs. For one, the spread of disease is eliminated completely; diseases can only be transmitted from husband to wife or vice versa, but the rest of the network is safe. Furthermore, everyone who wants sex is having sex. Ideally, the married couples would be 100% sexually compatible in terms of frequency; that is, the husband and wife are having sex at a frequency that is satisfactory to both partners. In practice this is likely impossible on a large scale, since the male libido is so much greater than the female’s, and there aren’t enough high-libido women to go around. Husbands usually want more sex than the wives want to provide. In a state of perfect sexual harmony, the married couples would all be of commensurate sexual market value, and also of commensurate amorousness. Commensurate sexual market value is necessary, or the marriage is likely to be strained.

This network does not even require any calculation to show that $G_{NOP}^{male} = G_{NOP}^{female}=0$; everyone has exactly one partner. In a Muslim society, where men can have up to four wives, $G_{NOP}^{male}$ will be higher, while $G_{NOP}^{female}$ still remains zero.

Random Sex Networks vs. Power Law Networks

For a random network (uniformly random pairings, uniformly random SMV), we observe highly unrealistic results. We observe that the Gini coefficients are very nearly zero, corresponding to a sexual utopia. Of course, this is not how the real world works, as people do not have sex with random people off the street.

The type of network that we have today in Western society, and likely what Devlin is referring to, is a power-law network. Power-law networks are characterized by the power-law distribution of the degree; in this case, degree of a node is the number of sex partners. Power-law networks follow the type of mating mechanism that Devlin claims is at work in Western society today; a small number of men are hogging all the women. The mechanism which generates power-law networks is known as “preferential attachment” – women are more likely to have sex with a man who has already had sex with a large number of partners, and the more partners he has, the greater his likelihood of mating with a female he finds attractive. Having a high degree demonstrates higher value, and thus make a female more pliant. Power-law networks are wracked by inequality. Most of the males only have a few sex partners, while a small number of men are deflowering females left and right.