Preferential attachment in sexual networks

Abstract

Many social networks are characterized by a highly uneven distribution of links. The observed skewed distributions have in several cases been attributed to preferential attachment (PA), a tendency among nodes in a growing network to form new links preferentially to nodes with high numbers of links. We test the PA conjecture in sexual contact networks. A maximum likelihood estimation-based expectation-maximization fitting technique is used to model new partners over a 1-year period based on the number of partners in foregoing periods of 2 years, 4 years, and lifetime. The PA model is modified to account for individual heterogeneity in the inclination to find new partners and fitted to Norwegian survey data on heterosexual men and women. Results show evidence of nonrandom, sublinear PA when comparing the growth in 3- to 5-year periods. The potential implications of these findings are discussed.

Fig. 1.

Schematics of a sample path showing the interarrival (waiting) times T_i1, T_i2. The observation period consists of two time intervals, the initial period, and the study period. The partner numbers in the first period j_i are used to model the number of new partners Δj_i in the study period (shaded region).

Fig. 2.

Histograms showing the distribution of total partners in the observation periods for women (white) and men (shaded) in the Norwegian survey: 3-year period (A) and lifetime (B). (A Inset and B Inset) Double-logarithmic plots of the cumulative average numbers of new partners in the 1-year study period for women (circles) and men (stars) plotted as a function of the number of partners j in the foregoing period. Thus, the mean values are group averages among all individuals having exactly j partners in the initial part of the observation period.

Fig. 3.

The cumulative probability of new partners P_cum(Δj) = Σ_i≥j P(Δi) as function of the new partners Δj in the 1-year study period. The probabilities were calculated by using the model parameter (Table 1) and by conditioning on the initial numbers of partners j₁, j₂, … j_n in the study populations. (A) Results for women in the 3-year observation period. (B) Results for men in the 3-year observation period.

Fig. 4.

The cumulative probability of new partners P_cum(Δj) = Σ_i≥j P(Δi) as a function of the new partners Δj in the 1-year study period. The probabilities are calculated by using the model parameter (Table 2). The individuals are partitioned based on their age or civil status, and the expected numbers of new partners in each strata were calculated separately based on the initial numbers of partners j₁, j₂, … j_n. These numbers were then summed to provide the pmf of the entire population. (A) Results for women in the 3-year observation period. (B) Results for men in the 3-year observation period.