The predominant component structure consisted of a single, large component with multiple smaller ones (Table 3), but this structure varied with the underlying sampling scheme. Flagstaff and Atlanta used a chain link sampling design that produced a single connected component. The scheme was replicated once in Flagstaff and in 3 separate neighborhoods in Atlanta. The latter, therefore, had only 3 components of approximately equal size. The Urban2 project in Atlanta used a snowball design that produced a single connected component. For the studies that used contact tracing procedures to identify contacts, the component structure reflected the extent of interrelatedness of the subjects. The Rockdale study uncovered a single, highly connected group. The PPNG study performed contact tracing on a highly interconnected group, producing a large connected component that contained 90% of the subjects. Manitoba, Chlamydia (Colorado Springs), GC1981 (Colorado Springs), HIV (Colorado Springs), and Syphilis (Atlanta) used contact tracing procedures on more diffuse groups, and though each had a large connected component, it constituted a smaller proportion of the total group. Bushwick, Houston, and Project 90 (Colorado Springs) all used a purposive chain-linked design and each produced a dominant connected component, substantially larger than the second largest component. The Antiviral (Atlanta) and Baltimore projects recruited individuals, used egocentric interviews, and attempted some post hoc sociometric analysis. Their largest connected components were relatively small, and constituted a small proportion of the total population.
All of the networks examined exhibited small world characteristics (Table 4). Visual inspection of the data makes it clear that larger networks have larger mean geodesic paths, a phenomenon readily explained by the constraint on path length imposed by small network size. The corrected mean geodesic reduces the length of all the geodesics, but proportionally more for those in the larger networks, and demonstrates that the corrected mean geodesics for all 15 studies were similar (mean, 2.45, 95% C.I. 0.85–4.05, range 1.3–4.3). Thus, though there appears to be considerable variability in the mean length of geodesics in these 15 studies, with several that are greater than 9.0, such variability may be a distortion induced by variable network size. If mean geodesic is plotted against network size, the R2 is 0.15. When plotted against the log of network size, the R2 is 0.60.
Respondents in most of these networks reported concurrent relationships for both sexual activity and, in those networks with injection drug use, for needle-sharing as well (Table 5). In general, those networks ascertained through contact tracing (for example, GC1981 and Manitoba, which used unmodified STD program approaches to finding contacts) tended to have lower levels of concurrency. Those obtained through purposive designs (for example, Project 90, Houston) had higher levels. There were several important exceptions, however. Most notably, the highest level of sexual concurrency was observed as a result of contact tracing among the 98 persons who constituted the Rockdale network, a group of teenagers who were involved in frequent parties with individual and group sex.23 Paradoxically, the highest levels of needle-sharing concurrency was observed in the Project 90 network—an area with low HIV prevalence.
On average, networks demonstrated lower levels of assortative mixing by degree (32%) and by age (28%) than by ethnicity (45%). These average values hide considerable diversity among the networks however (Table 6). Stratification by degree revealed no systematic differences in assortativity by age and ethnicity, suggesting that a person’s level of activity may be determined independently of his or her choice of partner (data not shown). Sociometric and egocentric assessments of assortativity produced almost identical results (variations of less than 1%), lending credence to the use of egocentric information for estimating assortativity (data not shown).
Unlike large networks such as the World Wide Web, wherein transitivity is highly prevalent, transitivity is much less common in sexual networks, in part because it cannot exist, by definition, in heterosexual relationships. Transitivity can be observed when social or drug-using (noninjecting) relationships are included (Table 6, “All contacts,” largest component), but its presence in sexual networks implies some degree of same-sex activity, and such transitivity was present in only 5 of the 15 networks examined here. On the other hand, needle sharing among 3 persons is fairly common, and transitivity was as high as 26% in 1 network (Table 6, “Houston,” second largest component).
Perhaps the clearest feature to emerge from this consideration of 15 completed network studies is their variability. Because their primary research questions varied as well, it is difficult to associate specific network properties with specific network outcomes, such as the incidence or prevalence of HIV or other STDs and BBIs. Some general hypotheses about network formation can be extracted from these data, however, most notably a sense of “fixed” factors and variable factors.
The term “fixed” is used advisedly in this context because results are not uniform and invariate, but certain patterns emerge nonetheless. With only several exceptions, the long right tail of the degree distribution in these studies may be fitted by a power law curve with an exponent between 2.0 and 3.0, the region for which the distribution is scale-free. If all the dyads from these studies are combined and the low degree persons removed, there is a clear straight-line relationship between the log (cumulative probability of degree) and the log (degree), with an exponent of 2.077, also in the scale-free distribution range (Fig. 1). On the other hand, several of the distributions (Flagstaff, Manitoba) are well outside this range, and the left side of many of these distributions (the portion not included in the calculation of the exponent) tended to be bumpy and irregular. It is likely that small network size permits substantial variation from what may be the basic underlying pattern.
The scale-free characteristic is of importance because it describes a portion of the curve for which the variance is theoretically infinite and produces a network with no epidemic threshhold.24 Several large scale studies have demonstrated such skewness. Morris used it to account for the discrepancy between mean partnerships reported by men and women but did not assess the characteristics of the distribution per se.25 Liljeros et al. examined the results of a Swedish national sex survey and were able to fit a scale-free distribution with an exponent of 2.54 for women and 2.31 for men. Similarly, Schneeberger examined 4 populations from large studies in Africa and England and found a similar scale-free distribution.26 Jones and Handcock24 challenged this approach for a number of statistical and operational reasons: sequential correlation of data points; heteroschedasticity along the power curve; exclusion of small values in order to make the curve fit; sensitivity to misreporting in the high-degree portion of the curve. They applied a maximum likelihood approach to similar data and found that all but 1 of 6 distributions fit a power law curve with the exponent in the 2-3 range. Liljeros et al.27 point out that because these measures apply to finite populations, the issue of infinite variance is moot, and the ratio of variance to mean may be more important in determining real world epidemic thresholds. At the very least, a “fat tail” to the right appears to be a constant and important feature of these small transmitting networks, and a direct correspondence with the phenomena observed in large networks may be of lesser importance.
However these concerns are ultimately resolved, the power law fit to degree distribution (or at least, the long tail to the right) is a critical element in the development of a large component within a network. In the giant social networks, the mathematical basis for the development of a large component from a random graph (Poisson) distribution has been explored.28 It is of considerable interest that most of the networks in the compilation have a single large component, and many smaller fragments. Study design clearly modifies the relative size of the largest component (from comparatively small in routine contact tracing data conducted in isolated communities [Manitoba] to a single giant component in an outbreak investigation of a closely knit group [Rockdale]). But the existence of a large component in most cases suggests that the underlying degree distribution has a predictable connection to the formation of such a group. In addition, a right-skewed degree distribution will produce nodes of high centrality that act as “short circuits” within a network and produce small world phenomena. In these data, the shortest paths and the diameters of the network are both evidence of small world characteristics, particularly when prorated for the overall size of the connected component in which they are being measured.
The right-skewed degree distribution, large component, and small world features may be thought of as the infrastructure for disease transmission. In keeping with Morris’ concept of local choices, individuals will decide on the number of sex partners they wish to have, and this number will have a distribution. In community situations of relatively monogamous (or serially monogamous) behavior, persons with high degree will be underrepresented or perhaps even absent, and the predominant network picture will be 1 of dyads, with occasional triads or longer dendritic connections. In situations, such as those that predominate among groups in which significant transmission has been detected, the degree distribution will include a long tail to the right, providing opportunities for the formation of large components and short circuits. Thus, the personal (“local”) choices on level of sexual activity within a community can lead to the network substrate for transmission. Specific behavioral acts, with differing transmission probabilities for HIV, STDs or BBIs, may also be thought of as personal choices that are part of the infrastructure for transmission. Though not considered here because of data limitations, we have proposed elsewhere that geographic distance is another of the local choices that may determine disease propagation.29
The actual amount of transmission may then be determined by other local choices, and the variability of such choices is reflected in these data. Concurrency, assortativity, and transitivity—which encompass the notions of partner sequencing and mixing that Morris describes11—are individual choices that, when taken in the aggregate, “fill in” the network structure and can provide the epidemiologic basis for transmission intensity. Concurrency and transitivity are obvious mechanisms for the amplification of transmission, as are higher-order structures in a network. Assortativeness will play a major role in determining the disease prevalence that a particular group may face. Other choices may be available as well, but those described here may be especially important as the conduits from micro- to macro-social phenomena.7 Taken together, fixed and variable factors are rooted in local choices and produce a global picture.
Perhaps the primary public health impact, at this point, is the ability to recognize networks in which substantial transmission can take place. Techniques for rapid network assessment have not been a prime focus of network research, but they can easily be extracted from the long experience with partner notification that is embedded in STD/HIV control programs. Rapid identification is a forerunner, in turn, to the use of networks as an element for disease surveillance, both to monitor known sites of transmission and detect new ones. Such applications will also serve as an empirical basis for detection of other network configurations that support transmission.
The data from these studies do not provide direct answers to the relationship of these network phenomena to disease transmission, but they do speak to the dynamics, and provide research agendas for further empirical, theoretical, and blended investigation. Blending is probably the critical approach, since empirical studies have certain important limitations. Interestingly, as this group of investigations demonstrates, one of the limitations is not our ability to get information from individuals about highly personal activity. Rather, the number of replications and the quantity of data required for a comprehensive, planned approach is daunting. Similarly, theoretical development in the absence of empirical verification is problematic. Blending these approaches is the current creative challenge in network research.
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