PATTERNS OF SEXUAL CONTACT within populations are central to the epidemiology and control of sexually transmitted infections.^{1,2} Consequently, empirical studies have been undertaken to describe patterns of sexual mixing between persons classified according to numbers of sexual partners^{3–6} and according to a range of geographic, social, and demographic variables.^{4,7–10} In addition, sexual partnership networks have been reconstructed through the connection of egocentric data^{11} and the tracing of sexual partners.^{12,13} These studies have provided data on the structure of sexual networks within specific risk groups. However, the extent to which such finely stratified data on patterns of sexual interactions are capable of improving our understanding of the dynamics of sexually transmitted disease (STD) transmission has not been examined. In particular, we have little idea which measures of sexual network structure best predict important epidemiologic factors. What kinds of sampling and analysis are most helpful for understanding the establishment of infection? Are these also predictive of endemic prevalence? We address these questions using a mathematical model describing gonorrhea transmission.

Mathematical models have been used extensively to explore the processes determining observed patterns of infection in STD epidemiology. Most studies have used simple deterministic models of populations categorized according to the stage of infection and stratified by sexual activity level.^{14–17} Some have explicitly included sexual partnerships,^{18–20} allowing concurrent partnerships to be represented in the models. Our current goal is a more general exploration of the significance of the complex network structures that could exist within populations. To generate these structures, we use a stochastic individual-based model in which the explicit representation of individuals and their partnerships allows great flexibility in the definition of the pair formation and dissolution process. The model explicitly generates different partnership networks from different underlying processes, because this is the most rational way to sample from the range of possible networks. We are not seeking to explain how network structures emerge from underlying biologic mechanisms. Instead, from a variety of such structures, we examine the role they play in the transmission dynamics of STDs.

We choose to model the spread of gonococcal infection in the network to represent a typical sexually transmitted infection. To examine the role of network structures, we focus on two areas: the establishment and endemic prevalence of infection in the population and the risk of infection in individuals. Logistic regression is used to investigate how the time over which the network is formed, sexual activity levels, mixing patterns, and network attributes influence these outcomes.

**Methods**

Sexual partnership networks were generated by simulating the formation and dissolution of heterosexual partnerships in a closed population over time. Infection then was introduced into the simulated partnership networks. Measures of sexual activity, sexual mixing patterns, and the network structure were assessed to determine the extent to which they influence the establishment of infection, the prevalence of infection, and a person's risk of infection.

*The Simulation Models*

Individuals in the model are characterized by gender, a preferred number of sexual partners, and disease status. The structure of the network at any point in time can be characterized by an “incidence” matrix I(z) (Figure 1). Each matrix entry I_{AB}(z) contains an indicator vector z = (z_{1}, z_{2}, z_{3}) in which the z_{i} take values 0 and 1. We let I_{AB}(z_{1}) denote the presence-absence of a partnership between male A and female B, I_{AB}(z_{2}) denote the disease status of male A, and I_{AB}(z_{3}) denote the disease status of female B.

Events occur in discrete time. At each timestep, three events may occur:

- A new partnership may form between a man and woman.
- An existing partnership may dissolve.
- Disease may be transmitted from an infected person to a susceptible person within an existing or new partnership.

All three events occur as the realization of stochastic processes. The details are given in Appendix 1.

*Parameters*

The parameters required by the simulation model can be categorized into two types: biologic parameters that determine how gonorrhea is transmitted over the network and behavioral parameters that determine how the network of partnerships evolves. A summary of these parameters and their ranges is presented in Table 1. To investigate a wide range of scenarios, parameter sets were drawn uniformly from within specified ranges, as described below. Latin hypercube sampling was used to ensure that each parameter was sampled independently.^{21}

*Biologic Parameters.* We have chosen to define the transmission probability as the per-day probability. This is equivalent to assuming that sexual contact occurs on a regular basis within partnerships and that the risk of transmission is constant per contact. Thus, on average, each person has a constant risk of transmission from an infected partner per day, although in practice, sexual contact will not occur every day. Studies suggest that the transmission probability per partnership through penile-vaginal intercourse from men to women is in the range of 0.5 to 0.9,^{22,23} whereas that from women to men is in the range of 0.6 to 0.8.^{22} Here, we set the values for the transmission probability per day from either gender to be in the range of 0.05 to 0.2, which gives the probability of transmission a range of 0.4 to 0.9 after 10 days in a susceptible-infected partnership.

The duration of infectivity is distributed exponentially with a mean of 1/ν days and a maximum of 50 days. For gonorrhea, this duration depends on both the incubation period of the infection and the probability that infected persons are symptomatic and treated. The incubation period for urethral infection in men is estimated as between 1 and 14 days,^{24} whereas that for urogenital infection in women is estimated as approximately 10 days.^{25,26} The majority (>80%) of men become symptomatic within 2 to 5 days of infection,^{27} whereas in women, this proportion may be much lower (20–80%).^{28} The duration of infectivity is complex and dependent on available health services. In this work, we chose to sample 1/ν between 10 and 50 days, a sufficiently wide range to cover most possible scenarios.

*Behavioral Parameters.* Several behavioral parameters are included in our model to specify sexual activity levels, patterns of mixing, and the duration of partnerships. Sexual activity levels depend on a “desired” number of partners, which is drawn from a negative binomial distribution, with a mean ranging from one to four partners, representing a high-sexual-activity population with a substantial risk of an STD epidemic. To generate typical levels of diversity, the dispersion parameter is drawn from the range 0.1 to 1.0. Recent studies of the distribution of sexual behaviors within random samples of the population indicate that the behaviors we represent are found in a small fraction of the population.^{4,29,30} Patterns of mixing in the model are set to vary from mildly disassortative through proportionate to mildly assortative, the patterns observed in the few empirical studies of mixing to date.^{3,5,6}

The duration of partnerships is based on early observations from a study of sexual mixing at Imperial College School of Medicine at St. Mary's, London, in which the date of first sex and date of last sex are recorded for each partner named by patients who have gonorrhea. The mean duration's τ_{1} and τ_{2} are taken from the ranges of 1 to 10 days and of 30 to 300 days, respectively. This allows a fraction of all partnerships to be significantly shorter (e.g., casual relationships) than the remainder. This fraction (α) is varied between 0.4 and 0.8.

*Simulations*

The simulation was run for three population sizes (200, 500, and 1,000 persons) with equal numbers of men and women. Forty sets of initial parameters were obtained for each population size using hypercube sampling.^{21} Five stochastic simulations were run within each parameter sample. Disease was introduced into the population 20 days after the start of the simulation. The simulation was run for 2 years (730 days), by which time either an endemic state had been reached or disease had died out.

*Network Measures and Statistical Methods*

Network measures that are considered in the analysis are outlined in the following sections. The network terminology is defined in Appendix 2. Each measure was calculated for two different networks, the first the network of partnerships that exists at the end of the simulation (the static network), and the second the network built up over the last 90 days of the simulation (the cumulative network) (Figure 2).

*Establishment and Prevalence of Infection.* The extent to which sexual activity levels, sexual mixing patterns, and the structure of the partnership network influence both the establishment of infection and the prevalence of infection in the population was assessed using three statistical regression models. Logistic regression was carried out using SAS (Statistical Analysis Software, Version 6, SAS Institute, Inc., Cary, NC) controlling for differences between the networks in the transmission probabilities, duration of partnerships, and infectiousness. Only first-order interactions were included in each model. A summary of the models used is given in Table 2. The best fitting model (defined by the deviance statistic) was chosen at each stage. The procedure was carried out for the network present at the end of the simulation (the static network) and that built up over the preceding 90 days (the cumulative network). Model outcomes were the presence-absence of infection for all 600 networks and the prevalence of infection in the cases where infection is established. Three sets of model inputs were used.

The first model (regression model I) assumes that only the distribution of the number of sexual partners is known. This is characterized in the regression model by the mean and standard deviation in number of partners.

The second model (regression model II) includes measures of the sexual mixing pattern in the population. In transmission dynamics models, the patterns of mixing between different sexual activity groups often are characterized by the mixing matrix,^{31,32} in which the entries p_{ij} are the probabilities that a person in activity group i forms a partnership with a person in activity group j. The trace of this matrix (sum of the values on the diagonal from top left to bottom right) provides a measure of the pattern of mixing in the population. This ranges from N (the number of activity groups) in the extreme case of assortative mixing through 1 for proportionate mixing, to 0 at the other extreme, disassortative mixing. For the analysis of results in this model, the population was divided into four sexual activity levels (1, 2, 3–4, and 5 or more partners over the period that the network was analyzed), and the mean trace of the two (male and female) mixing matrices between these groups was calculated.

Another measure of the mixing pattern examined in the model is the mixing pattern of the core group (i.e., those of the highest sexual activity level). This is measured as the degree of within-group mixing in this subgroup of the population. Highly sexually active persons were defined as those with five or more partners over the period that the network was analyzed.

A final measure of the mixing pattern included in the analysis is based on possible paths of transmission. Three variables were assessed; the proportion of mutually monogamous pairs, the proportion of pairs with one monogamous person, and the proportion of mutually non-monogamous pairs (Figure 3). In terms of transmission, the first measure represents pairs who have no possible source of infection, the second represents pairs into which infection may enter but cannot be transmitted on, and the third represents pairs which form a path along which transmission may occur.

The third model (regression model III) includes measures of the network structure. Analytic network measures have been formulated in the study of social networks to characterize this structure.^{33} Initially, we consider characterizations of the component distribution, which provides a measure of the pattern of connectedness in the population. Two measures are considered: the mean and standard deviation in size of components and the number of components. Next, we introduce the concept of a k-cohesion (commonly referred to as a k-core in social network literature, but changed here to avoid confusion with the concept of a core group in STD research). A k-cohesion is a group of vertices with edges to at least k other vertices.^{34} This group is a component in the network that arises when only those with k or more partners are considered. Because there can be more than one such group in a network, we include all persons in such groups when we measure the size of the k-cohesion. As k increases, persons within the cohesion become more densely connected. The distribution of connections through a population is reflected by the proportion of persons who fall out of the k-cohesion as the value of k is increased. The k-cohesion-collapse n_{k} is the number of persons in the k cohesion but not in the k + 1 cohesion for each k. We summarize this information by two measures: the mean k-cohesion-collapse (which we shall refer to as the mean cohesion),Equation 5

where N is the population size, and the size of the high-activity cohesion, n_{5}.

*Individual Risk of Infection*

The risk of infection with an STD is traditionally linked to number of partners, so that those with more partners are at higher risk.^{35–37} We assessed how further knowledge of a person's position within the network structure aids in the identification of those at highest risk from infection. We measured the number of partners each person had at path lengths 1, 2, and 3. Thus, those at path length 1 were that person's partners, those at path length 2 were their partner's partners, and so on.

In addition, a measure of global centrality, closeness, was used to assess the extent to which a person's risk was dependent on the whole network structure. Closeness is defined as the sum of the shortest path lengths from each person to all other persons in the network.^{38} This was calculated within components using a minimum path algorithm.

The explanatory variables were added in ascending order to assess the additional contribution that knowledge of the network structure gave in predicting those at risk from infection. The closeness measure was added as a separate regression model. Because closeness was calculated within components, interactions between closeness and component size also were included in this model.

A summary of the models used is given in Table 3. Each of the models was run within simulations on the networks that were infected. The procedure was carried out for the network present at the end of the simulation (the static network) and that built up over the preceding 90 days (the cumulative network).

**Results**

*Establishment*

The increased information carried by measures of sexual activity, sexual mixing patterns, and the structure of sexual partnership networks add, in turn, to the ability of the regression model to predict whether infection will spread (Table 4). In addition, for all three levels of behavioral detail, the information accumulated over 3 months' time provides a better indication of potential spread than does a simple static picture (Table 4). In the most complete model (model 3), the most significant variable was the population size (Table 4). Minor epidemics are likely to be sustained within small sections of the population. Thus, the larger the population, the more likely gonorrhea is to find a niche.

The establishment of infection is made more likely by an increased number of partners (although this is not significant in the final model), but, contrary to expectation, the chances of establishment decrease as the standard deviation in numbers of sexual partners increases. This illustrates one of the main difficulties in interpreting regression analyses when many variables are included. For a fixed mean number of sexual partnerships, an increased standard deviation should generate a “core” highly sexually active group within which infection can persist. Indeed, when the relation between the standard deviation and establishment are simply plotted, a weak positive correlation is suggested. This indicates that other variables in the regression model, such as component size, must correlate with the standard deviation in the number of partners. This multicollinearity between variables means that other variables indicate more effectively the influence of heterogeneity.

Just as it has been well established that behavioral heterogeneity makes the existence of a core group more likely,^{14,15} so too is it accepted that mixing patterns can influence the establishment of infection.^{32} However, it is unclear how mixing should best be measured. In the case of the establishment of infection, the proportion of mutually nonmonogamous pairs best captures the influence of mixing. These pairs form paths along which the infection can spread.

The most influential network measure is the mean component size. As components increase in size, establishment of infection becomes more likely. An increase in the standard deviation in component size also results in an increased likelihood of infection establishing, although this is not a significant effect. The other significant network measure is the high-activity cohesion. A large high-activity cohesion indicates that those with many partners are forming relationships with each other, resulting in a core group, as well as reflecting how assortative mixing is. This has been seen in simpler models to increase the likelihood of infection persisting.^{32} However, because it provides more information on the density of contacts within the high-activity group, the high-activity cohesion is a better predictor of establishment than are simple measures of mixing patterns.

*Prevalence*

The cumulative network is better at capturing the factors influencing the prevalence of infection than is the static network (Table 5). Once again, the most significant variable is the size of the population, but in this case, increased population size causes a decrease in the prevalence of infection. This is a sampling effect resulting from only networks in which infection established being included in the analysis. As the population size is increased, low-prevalence endemic infections are more likely to be sustained, causing a wider range of prevalence to be included in the regression model for larger population sizes.

As individuals have more sexual partners, the prevalence of infection in the population increases. However, in the final regression model (model 3), this variable is not significant, and a negative coefficient is observed (Table 5). This appears to be because of a correlation between the mean number of partners and the mean collapse, the latter being a better predictor. Despite our expectation of increased heterogeneity limiting the spread of infection, the standard deviation in the number of sexual partners also has no significant influence in the final model.

The influence of sexual mixing patterns on the prevalence of infection is best captured by the behavior of the high-activity persons. As the mixing of this group becomes more disassortative, infection spreads to other groups, and the overall prevalence of infection increases.

As the number of components increases, the prevalence of infection decreases. However, this measure is not significant in the final model, suggesting component distribution has little influence on the endemic prevalence. The only significant network measure is the mean cohesion. As this increases, high-activity persons become more connected, preventing the spread of infection from the group and thus decreasing the overall prevalence. The significance of this variable in the final model suggests that this measure of the explicit connectivity of persons of varying sexual activity levels provides additional information on the mixing patterns in the population that is not captured by pair-based measures.

*Individual Risk*

The number of partners at distances 1, 2, and 3 and the closeness measure were calculated for the static and cumulative networks for the 70 simulations with 500 persons in whom infection was established. As in the previous sections, the dynamic network model gave lower deviance statistics than did the static model (Figures 4a and 4b). To interpret these results, we look for a significant change in deviance between the baseline and model I, model I and model II, model II and model III, and model III and model IV. The change in deviance between the regression models takes a chi-square distribution with degrees of freedom equal to the change in degrees of freedom between the models. For example, to assess the significance of the number of partners at path length 1, we subtract the deviance in model I from that in the baseline model. This value has a chi-square distribution on 1 (2,499–2,488) degree of freedom. For the static network, there was a significantly improved fit of the model with the inclusion of the number of partners at distance 1 and 2, but little improvement occurred when the number of partners at distance 3 was included (Figure 4a). The results for the dynamic model show a significantly improved fit of the model with the inclusion of number of partners at distances 1, 2, and 3 (Figure 4b). In terms of identifying those at risk for gonorrhea, it appears that knowledge of an individual's partner's number of partners and their number of partners in turn all play a role in risk.

The final model, model IV, uses the global centrality measure, closeness, as a measure of risk. The model gives a higher deviance statistic than any of the other models. On this evidence, it appears that individual risk is more dependent on local centrality than on position in the wider network structure.

**Discussion**

In a departure from the methods used to construct simple models of disease transmission, we have used an individual-based simulation model to explore how information on the detailed structure of a network could help in understanding patterns of disease incidence. This makes the interpretation of results more complex, but also allows us to identify the most informative measures of sexual mixing and network structure from the range of measures included in our evaluation.

The analyses suggest that the most important aspects of mixing are the proportion of the population in mutually nonmonogamous pairs and the pattern of partnerships formed by those with the highest levels of sexual activity. The former was the measure that best predicted the establishment of infection. Mutually nonmonogamous pairs are the only pairs that provide a path for transmission. Thus, in the absence of such pairs, infection cannot establish. The latter, the sexual mixing pattern of the group of highly sexually active persons, best predicts the prevalence of infection. It was more influential than was the mixing pattern in the whole population, suggesting that data collection on sexual mixing patterns within this group could be more efficient than collection within the general population. This is a convenient result, given the relative accessibility of those attending STD clinics.

Our results add weight to the view that the detailed network structure plays a role in the epidemiology of STDs. However, different aspects of this structure play a role in the establishment of infection, the prevalence of infection, and a person's risk for infection. For example, the greater the mean size of components, the more likely an infection was to establish, whereas the component distribution had little influence on the prevalence of infection once established. Both the establishment and the prevalence were influenced by the cohesion. This provides an explicit measure of the connectivity of persons with varying levels of activity. It is a more informative measure of the pattern of mixing than are the mixing matrices derived from interviews with persons who seek to determine their partners' number of partners.

In addition to looking at the risk of gonorrhea spreading in the population as a whole, we also assessed the influence of the network structure on a person's risk for infection. The cumulative network provided more information than did the static network, and the number of partners at path lengths 2 and 3 was significant. This suggests that the identification of a partner's number of partners will increase the accuracy with which persons at high risk are identified. Our measure of global centrality, closeness, was not useful in identifying those at risk from infection. In our model, a person's risk for infection appears to be related to localized network structure rather than to position in the complete network structure.

When two periods over which data could be collected were compared, the cumulative network built up over 90 days gave a lower deviance statistic in the regression models than did the static network. The period of 3 months is a similar length of time to the duration of infectivity, and the period over which partners usually are traced in contact tracing protocols for gonorrhea,^{39} making it comparable to empirical data collected on sexual partner networks. The difference in the results illustrates how the time over which measures are ascertained affects their interpretation. Measures that are time dependent, such as the duration of partnerships, will be affected more than those that remain almost constant over time. For example, a sequence of serially monogamous partnerships will appear as concurrent partnerships in a network built up over time. In addition, the chronology in which partnerships are formed and the time at which infection occurs will be significant in determining the epidemiology of infection. More detailed investigations of the influence of the pattern and timing of partnerships may help to identify measures that could be estimated empirically from emerging data.

It is clear that the potential for sexually transmitted infection epidemics and their magnitude can best be gauged if studies of sexual behavior include a detailed description of the sexual partnership network. In Colorado Springs, data on the sexual and social networks of persons at high risk for human immunodeficiency virus were obtained.^{12} Studying the network as a whole, the Colorado Springs data suggested that the fragmentation of sexual partner networks acts as a barrier to transmission. This is in agreement with our findings that an increased component size made the establishment of infection more likely. However, we also found component size to have little effect on the endemic prevalence, suggesting that once infection has established within some components of the network, it is the properties of those components (such as their cohesion) that influence the endemic prevalence. In addition, the Colorado Springs data discuss the position of infected persons and how this relates to their position in the network, finding those infected to be both locally and globally noncentral within the largest component.^{40} In contrast, our findings suggest that both locally (measured by number of partners at different path lengths) and globally (measured by closeness) central persons are more likely to be infected (results not shown). Possible reasons for this discrepancy include the different periods over which the networks were constructed in relation to the infectious period of the disease, and the nonrandom sampling procedure used to collect the empirical data.

In empirical studies of sexual partner networks such as that undertaken in Colorado Springs, nonrandom and incomplete samples are a major problem. This is true of all of the studies undertaken so far, including those based on retrospective human immunodeficiency virus partner notification data^{11,41} and prospective studies of particular risk groups.^{12} The incompleteness of the network is most commonly the result of either a cutoff in contact tracing when a noninfected person is reached or the appearance of untraceable contacts. The methods used to collect network data will affect the way in which they are incomplete. For example, ego-centered networks may be complete locally but incomplete globally, whereas contact tracing will provide a more global picture. It is unlikely that a complete network within a closed population will ever be traced. To interpret partial data on partnership networks, methods for generalizing from incomplete data need to be formulated. Mathematical models can be used to generate complete networks, from which incomplete networks can be derived through different sampling processes. The corresponding complete and incomplete networks then can be compared to estimate errors in measures of sexual mixing patterns and partnership networks. This article provides a framework for further work in this area.

Recent years have seen a rapid accumulation of quantitative data on sexual behavior from surveys of large populations.^{4,29,30} Our understanding of the behavioral milieu in which STDs spread has been enhanced greatly by these studies. The work presented here illustrates how different layers of quantitative information on sexual behavior can be added to generate a more complete picture. The challenge in drawing up a map of the detailed structure of sexual partner networks is great. However, perhaps the most comforting conclusion to be drawn from our analysis is that the patterns of mixing and cohesion in the highest activity section of the population have the greatest impact on the prevalence and, therefore, the potential control (but not eradication) of an STD. It is this group that is most likely to attend STD clinics.^{42} Hence, one conclusion that is perhaps common sense, but nonetheless welcome, is that the majority of studies of sexual behavior related to STDs are best carried out among persons with STDs.

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**Appendix 1. Mathematical Details of the Simulation Model**

**Pair Formation**

Let X_{ij} denote an individual of gender i and sexual activity preference level j. The sexual activity preference level represents a “desired” number of partners.

At each point in time, an individual may form a new partnership if his or her current number of partners is less than his or her desired number of partners. We denote this desired number of partners by j, where the distribution of j in the population is given by the negative binomial distribution. (1)

where m is the mean and k the dispersion parameter. The tail of the distribution is truncated so that f(j) > 1/S, where S is the population size. This distribution was chosen as an over dispersed distribution, so that sexual activity levels were representative of patterns observed in the United Kingdom population (refer to Parameters section).

Let N(X_{ij}, t) denote the actual number of partners of individual X_{ij} at time t. The set of “available” individuals at time t is therefore given by Equation 6

This is the set of individuals from which new partners may be chosen.

Partnership formation within this set takes place according to a random process weighted by a mixing preference. Mixing preferences are relative preferences based on the desired sexual activity level of each possible partner. The preference of an individual with activity level j for an individual of the opposite gender with activity level k is given by φ_{jk}, where φ_{jk} is (2)

normalized so that Σφ_{jk} = 1, N_{max} is the maximum number of partners an individual may have at any point in time, and Θ_{1}, Θ_{2}, and Θ_{3} are parameters defining preference in partner choice. This function is a combination of a preference for assortative-disassortative mixing, and a preference for partners with higher-lower sexual activity levels. The preference for assortative-disassortative mixing is given by the value of Θ_{1} and Θ_{2}. The value Θ_{2} = 0 gives random (proportionate) mixing, Θ_{2} > 0 introduces assortative mixing, and Θ_{2} < 0 introduces disassortative mixing. The magnitude of Θ_{1} controls the strength of the assortative-disassortative effects. Thus, when Θ_{1} is large, extreme patterns of assortative-disassortative mixing are achieved, whereas for smaller values of Θ_{1}, the patterns are moderate. The preference for partners with higher-lower sexual activity levels is given by the value of Θ_{3}.

In random order, for each man, all possible partnerships with available women are examined successively. The probability that a partnership will form between X_{j} and X_{k} is the product ρ_{jk} = φ_{jk}φ_{kj}. This process is repeated until a partnership is created or until the set of available partners is exhausted. Each individual forms at most one new partnership at each timestep.

**Partnership Dissolution**

Once formed, the duration of a partnership is distributed exponentially with mean τ days. Individuals are allocated a “preferred” duration of partnership, τ′, which is taken from a step function (3)

where α, τ_{1}, and τ_{2} are parameters, and x∼U[0,1]. The preferred duration of a partnership is negatively correlated with the sexual activity preference level. This allows those with lower sexual activity levels to participate in longer partnerships, whereas those with higher sexual activity levels have a preference for short partnerships, reflecting a crude classification of types of partnerships. The duration of a partnership, τ, is set as the minimum of the two individuals' preferred durations. Thus, (4)

**Disease Transmission**

Disease may be transmitted between a susceptible individual and an infected partner with probability β per unit time. Individuals recover and return to the susceptible population at rate v. In addition, all individuals return to the susceptible population after 50 days of infection.

**Appendix 2. Network Definitions**

A network (graph) G(V,E) is a collection of vertices v ∈ V and *edges* (v_{1}, v_{2}) = e ∈ E. Here the vertices represent the individuals, and the edges denote a partnership between individuals. A sequence of vertices v_{0}v_{1}…v_{k} in a network is known as a *walk,* and one in which all the vertices are distinct is known as a *path.* The length of a path is the number of edges it contains.

Two vertices, v_{1} and v_{2}, are *connected* if there is a walk from v_{1} to v_{2}. The *distance* between v_{1} and v_{2} is the length of the shortest path connecting them. A *component* is a maximal set of connected vertices.