Generating a Heterosexual Bipartite Network Embedded in Social Network
aa r X i v : . [ q - b i o . Q M ] A ug Generating a Heterosexual Bipartite NetworkEmbedded in Social Network
Asma Azizi a, ∗ , Zhuolin Qu b , Bryan Lewis c , James Mac Hyman d a Department of Mathematics; University of California, Irvine, CA 92612 b Department of Mathematics The University of Texas at San Antonio, San Antonio, Texas78249 c Network Dynamics and Simulation Sciences Laboratory, Virginia Bioinformatics Instituteat Virginia Tech, Blacksburg, VA 24061, U.S.A d Department of Mathematics, Tulane University, New Orleans, LA 70118, U.S.A
Abstract
We describe how to generate a heterosexual network with a prescribed joint-degree distribution that is embedded in a prescribed large-scale social contactnetwork. The structure of a sexual network plays an important role in how sex-ually transmitted infections (STIs) spread. Generating an ensemble of networksthat mimics the real-world is crucial to evaluating robust mitigation strategiesfor controling STIs. Most of the current algorithms to generate sexual networksonly use sexual activity data, such as the number of partners per month, togenerate the sexual network. Real-world sexual networks also depend on biasedmixing based on age, location, and social and work activities. We describe anapproach to use a broad range of social activity data to generate possible hetero-sexual networks. We start with a large-scale simulation of thousands of peoplein a city as they go through their daily activities, including work, school, shop-ping, and activities at home. We extract a social network from these activitieswhere the nodes are the people and the edges indicate a social interaction, suchas working in the same location. This social network captures the correlationsbetween people of different ages, living in different locations, their economic sta- ∗ Corresponding author
Email address: [email protected] (Asma Azizi)
Preprint submitted to Journal of L A TEX Templates August 10, 2020 us, and other demographic factors. We use the social contact network to definea bipartite heterosexual network that is embedded within an extended socialnetwork. The resulting sexual network captures the biased mixing inherent inthe social network, and models based on this pairing of networks can be used toinvestigate novel intervention strategies based on the social contacts of infectedpeople. We illustrate the approach in a model for the spread of Chlamydia inthe heterosexual network representing the young sexually active community inNew Orleans.
Keywords: subgraph; B KB KB K network; Bipartite Network; Social contactnetwork; Sexual network; Joint degree distribution.
1. Introduction
The structure of heterosexual networks plays an important role in the spreadof sexually transmitted infections (STIs). These networks are captured in com-puter simulations by a bipartite graph where the nodes represent the peopleand the edges are sexual partnerships between nodes of different sexes. De-termining what is predictable in STI models requires an algorithm to generatean ensemble of random graphs that resembles real-world sexual activities, in-cluding the distribution for the number of sexual partners people have (theirdegree distribution) and the number of partners their partners have (the joint-degree, or degree-degree, distribution). The existing algorithms that generatebipartite random graphs preserving degree and joint-degree distributions of thenodes are strictly based on the number of partners people have, and not otherdemographic factors, such as age or location [23, 15, 9].The degree and joint-degree distributions are just two of many propertiesfor a heterosexual network that can affect its structure and the validity of anepidemic model. The heterosexual network is also correlated to an underlying social contact network of acquaintances connected by interpersonal relation-ships. A person’s sexual activity depends on age, race, sociodemographic, andsocioeconomic features of the environment that can be captured by a social2ontact network [4, 2, 24, 16, 21]. In other words, using the extended socialnetwork of a person as a source of sexual partner selection when generating aheterosexual network, enables the network to capture the bias in heterogeneousmixing based on age, race, economic status, and geographic location [19].Although it is widely accepted that social contact (non-sexual partners) andheterosexual (sexual partners) networks are related, there are few studies onhow a population’s social contact network impacts the spread of heterosexualSTIs. The social network can affect the structure of the heterosexual network byproviding the pool of sexual partners and can transmit information and culturalnorms regarding STI test and safer sex, but there is no mechanistic approachthat addresses and uses this capability of social networks. We will describe anew approach that fills this gap by applying social contact networks to generatethe heterosexual network while preserving the joint-degree distribution of data.Our new network generation approach uses the underlying extended socialnetwork of a population to extend these previous algorithms for generatingbipartite heterosexual networks with prescribed joint-degree distribution [9].Many sexual partnerships are formed from within a person’s social circle, de-fined by the people they have regular social contact with, and the contacts oftheir contacts (their extended social network). These social circles have beenmodeled through large-scale simulations of thousands of people in a city as theygo through their daily activities. We start with a network that mimics the socialactivity of the population [12], as generated by a complex social network sim-ulation. We use this simulated data to create an extended social network andthen identify a bipartite network of men and women to define our heterosexualnetwork. We then create a virtual heterosexual network as a subgraph of thisbipartite social network that captures a prescribed joint-degree distribution.As a case study, we construct a heterosexual network that is embedded inthe social contact network of New Orleans population and mimics the sexualbehavior obtained from a sexual behavior survey of the young adult AfricanAmerican population in New Orleans [17, 14].3 . Materials and method
People often find their sexual partners within their extended social network,the individuals they come in contact with each day at work, school, or othersocial activities. There are sophisticated simulations of these social networksthat can be used to produce a sexual network, which is more realistic thanbasing partnerships on just the sexual activity of different individuals.The social contact network is a graph where the nodes are synthetic people,labeled by their demographics (sex, age, income, location, etc.), and the edgesbetween the nodes represent contacts determined in which each synthetic personis deemed to have made contact with a subset of other synthetic people throughsome
Activity types. Each edge of the network is labeled with one of theseactivity locations and is weighted by the time spent on these contacts per day.For example edge ( i , j ) labeled by the activity A = W ork and weighted by T Wi,j means two persons i and j have a contact for T Wi,j fraction of their totaltime spent at work. We base our algorithm on a social contact network, called
SocNet , generated by Eubank et al. [12] with activity at different locations(e.g. home, work, school, shopping, or other activity).We introduce an algorithm that embeds a heterosexual network within asocial network and also meets the joint-degree distribution of the sexually ac-tive population. The heterosexual network preserves the bipartite joint degree(
BJD ) distribution matrix that represents the correlations between then num-ber of partners a person has and the number of partners their partners have [9].The algorithm has three stages:(i) Generate an extended social contact network,
ESocNet : The original so-cial contact network is a simple graph, whose nodes are synthetic people,and neighboring nodes are their social contacts during a typical day. Weassume that most sexual partnerships come from a person’s social con-tacts, or the social contacts of their social contacts, e.g. the neighbors ofthe neighbors of a node. We extend the social contact network to create anew network, the extended social network,
ESocNet , where some of the4eighbors (social contacts) of an individual’s neighbors in this network areadded to his/her social contacts.(ii) Generate a reduced social bipartite network,
BSocNet : The
ESocNet in-cludes all the individuals in the region being modeled. Our sexual networkis based on individuals within a prescribed age range. In this step, weremove all nodes where the associated individuals are outside this agerange. The extended social network is a simple graph where nodes havesome neighbors that are the same sex. We identify the embedded bipar-tite subgraph of this network by removing all edges between individualsof the same sex. The resulting bipartite graph is a social network wheremale nodes are only connected to female nodes and vice versa. Finally,we assume siblings are not sexual partners and therefore, we remove alledges between individuals living in the same household, which is the edgelabeled activity H for home. We call this reduced social bipartite networkas
BSocNet .(iii) Generate an embedded heterosexual bipartite network,
SexNet : We thenuse the
BSocNet to define a heterosexual network of sexual partnerships,the
SexNet , with a prescribed
BJD based on survey data [9]. That is,we preserve the correlations between the number of partners a personhas and the distribution for the number of partners their partners have.We assume that most of a person’s sexual partners are neighbors in the
BSocNet and a few of the partners are randomly selected from elsewherein the population where they might have met through social media or atany other event.
ESocNet ) In the first stage of our algorithm, we create an extended social contactnetwork,
ESocNet , so that an individual’s social contacts include some of thecontacts of their contacts. That is,
ESocNet will add potential sexual partnersby including some of the social contacts of an individual’s social contacts.Consider two people (nodes) i and j who are not currently connected, but5ave k A ( i, j ) > A . We define p Aij as the probability that they will meet through a single contact. Therefore, theprobability that they will meet and be connected in the
ESocNet after k A ( i, j )contacts is 1 − (1 − p Aij ) k A ( i,j ) .The probability p Aij is a function of the time that i and j spend in an activity A in SocNet . From the data in
SocNet , we can define τ Ak as the averagefraction of time person k spends with each social contact, when engaged inactivity A : τ Ak = P l ∈ N A ( k ) T Ak,l | N A ( k ) | , (1)where T Ak,l is the fraction of time two contacts k and l spend together in activity A , and N A ( k ) is set of all social contacts for person k through an activitylocation A . We then define p Aij = τ Ai τ Aj . Figure (1) describes an schematic ofthis algorithm for a simple network. BSocNet
In our heterosexual network, we only consider the sexually active populationwithin a prescribed age range α = [ α , α ]. That is, we trim the ESocNet byremoving all people with ages outside this range to not including any edges(sexual contacts) with people outside this range. We then remove all edgesbetween people of the same sex to create a bipartite heterosexual social network.Finally, to avoid including siblings as potential sexual partners, we remove alledges between two individuals living in the same household by removing alledges labeled activity H as home, to define the
BSocNet . SexNet ) The
Soc2Sex algorithm uses
BSocNet to generates a heterosexual net-work,
SexNet , that mimics the heterogeneous mixing of the real population.We assume that we have an estimate for the distribution for the number of part-ners of men and women (the degree distributions for their associated nodes) and6 k k k j A i k , T A i k A ik , T Aik A i k , T A i k A j k , T A j k A jk , T Ajk A j k , T A j k Figure 1: Suppose the persons i and j are not currently social contacts in SocNet but havethree different common social contacts k , k , and k through different activities. They mightbe connected in the extended social network, ESocNet , when at least one of their commonsocial contacts meet them within the same activity location. Suppose A ik = A jk = A ik = A jk = A = A ik = A ik , that is, k meets i and j at the same location, similarly k meets i and j at the same location to k ’s, however, k meets them in different places. To compute p Ai and p Aj we only count the social contacts who meet them at the same location A, therefore, p Ai = ( T Aik + T Aik ) /
2, and p Aj = ( T Ajk + T Ajk ) /
2. Finally, the probability that i and j makean edge- are connected in the extended social network- is 1 − (1 − p Ai p Aj ) . ij , between two persons i and j in SexNet represents a sexualpartnership. The degree of a person i , is defined by the number of his/hersexual partners. The degree distribution { d k } defines the number of peoplewith degree k . The joint-degree distribution ( k, j ) is the number of partnershipsbetween a man with degree j and a woman with degree k . This distributioncan be represented by the Bipartite Joint Degree or BJD matrix:
BJD
SexNet = e e e · · · e m e e e · · · e m ... e w e w e w . . . e wm , where, w is the maximum degree in women nodes, and m is the maximum degreein men nodes, each element e ij is the number of edges between women with i partners and men with j partners.The degree distribution of the number of women nodes, d wk , and men nodes, d mk , with k partners can be obtained from BJD
SexNet : d wk = P mj =1 e kj k , and d mk = P wi =1 e ik k . (2)Though the heterosexual network SexNet is a subgraph of
BSocNet , wealso consider some sexual partners that are within a person’s extended socialcircle. That is, for the general case,
SexNet is partially embedded in
BSoc-Net .The
Soc2Sex algorithm first generates an initial heterosexual network thatclosely agrees with the desired
BJD matrix and is partially embedded in
BSoc-Net . Usually, this network satisfies the desired
BJD , but can fail when theaverage degree (number of partners people have) becomes large. When thishappens, a second fix-up algorithm, based on rewiring the network, is used torepair any discrepancies so the final
SexNet has the desired
BJD matrix.8 .3.1. Generating the bipartite network
The
Soc2Sex algorithm starts with the
SocNet , the
BJD matrix corre-sponding to
SexNet , and the fraction p ∈ [0 ,
1] of partners that are chosenrandomly from the extended social contacts in the
BSocNet . The remainingfraction, (1 − p ), of a person’s sexual partners are randomly selected from else-where in the population. These partnerships might have formed by meetingthrough social media or a social event not captured by the original SocNet .The
Soc2Sex algorithm then generates a heterosexual network that is a partialsubgraph of
BSocNet and has a joint-degree distribution given by the
BJD matrix. Note that p is approximately the percentage of SexNet that is asubgraph of
BSocNet .The algorithm starts with an empty set of nodes
SexNet and then buildsa network guided by the
BJD matrix. The nodes with the smallest degreehave the least flexibility, so we start building
SexNet by randomly selecting aman node of
BSocNet with the highest (social) degree and assign its desiredsexual degree to be column size of
BJD matrix. This is represented by stubs ,or unconnected edges, associated with this node.We repeat the following process until all edges in
SexNet -that are equalto the summation of elements in
BJD matrix- are placed: at any step, we finda node with the highest stub in
SexNet and then with probability of p , wefind a partner with proper degree defined by BJD for them from their socialcontact in
BSocNet , or with the probability of 1 − p , we find a partner withproper degree defined by BJD from closest people but not their social contact,and then reduce its new partner’s stub by one. If we find all the partners forall nodes in
SexNet , we introduce a new node from highly social active nodesin
BSocNet to SexNet and assign its desired degree and stub equal to thecurrent maximum degree frequency of
SexNet .To keep or remove an edge, we have to calculate the degree of nodes attachedto it for each possible edge in the
SocNet , thus, the full set of experiments runin O( | E | P m P w ) time, where | E | is the number of edges in SocNet , P m number9f its men nodes and P w number of its women nodes. This method is feasibleif the average degree ( | E | P m + P w ) of the network is not high.For completeness, we provide pseudo-code for our Python scripts in Algo-rithms 1 and 2. Table 1 is the table of symbols used in these Algorithms. Notation Description
G.n set of nodes in network
GG.e set of edges in network Gd G degree frequency list for network Gd G ( i ) degree of node i in network GG.N ( i ) set of neighbors of node i in Network G dist ( G, u , v ) distance between two nodes u and v in Network G M.col / M.row column/row size of a matrix M M ( i, :) / M (: , i ) i th row/column of matrix M V ( i ) i th element of vector V V.index ( a ) index of element a in vector V | S | size (the number of elements) of a set S S.remove ( m ) remove member m from a set S S.sample ( P ) randomly select an element with property P (if P= 1 there is no property) from set S urn uniform random number in [0 , Table 1: Table of notation for a conventional network G in algorithms. lgorithm 1: Extracting sexual network,
SexNet , from a given social network,
BSocNetfunction SexNet = Soc2Sex ( BSocNet , BJD , p ) Input:
Revised social network
BSocNet , bipartite joint-degree matrix
BJD , p ∈ [0 ,
1] average fraction ofsexual partners selected from social contacts.
Output:
Sexual network
SexNex . /* From the highest degree men nodes in BSocNet , randomly select one node to add to
SexNet */ SexNet .n = ∅ , SexNet .n ← u = max d SocNet ( k ) { k ∈ BSocNet .n } ; /* The desired degree of the first selected node in SexNet is the column size of BJD */ d SexNet ( u ) := BJD.col, stub ( u ) := d SexNet ( u ) ; /* E is total number of edges in SexNet to be filled */ E := P i P j BJD ( i, j ) ; while | SexNet .e | ≤ E do /* NF includes all the nodes in SexNet who are still looking for partners */ NF := { k ∈ SexNet .n if stub ( k ) > } ; while | NF | ≥ do /* Start with the highest degree nodes in NF */ u = max stub ( k ) { k ∈ NF } ; /* Find partner for u via Algorithm 2 → partner v of degree d ′ */ ( d ′ , v ) = FP ( u , BSocNet , SexNet , BJD, NF, p ); if ( d ′ , v ) = False then /* If could not find partner for u remove it from NF */ NF.remove ( u ) ; else /* If find the partner v , add edge between u and v and reduce both their stubs by one */ Add edge ( u , v ) in SexNet , stub ( u ) ← stub ( u ) − , stub ( v ) ← stub ( v ) − /* If u [ v ] has found all the partners, remove it from the candidates in SexNet */ if stub( u )=0 [stub( v )=0] then d SexNet .remove ( d SexNet ( u )) [ d SexNet .remove ( d SexNet ( v ))]; end /* Update the corresponding entry in BJD */ if u is woman then BJD ( d, d ′ ) ← BJD ( d, d ′ ) − else BJD ( d ′ , d ) ← BJD ( d ′ , d ) − endendend /* From the nodes in BSocNet but not
SexNet , choose one node with highest social degree, add to
SexNet */ SexNet .n ← u = max d SocNet ( k ) { k ∈ BSocNet .n − SexNet .n } ; /* Define its degree and stub equal to maximum value in degree frequencies of SexNet */ d SexNet ( u ) := max { d SexNet } , stub ( u ) := d SexNet ( u ) . endreturn SexNet ; lgorithm 2: Finding partner with proper degree for a given node ( FP ) function ( d ′ , v )=FP( u , SocNet , SexNet , BJD , NF , p ) Input:
Node u in the sexual network SexNet , social network
SocNet , BJD matrix, set NF of nodes who needpartners, p ∈ [0 ,
1] fraction of seuxal partnerships from social network.
Output: ( d ′ , v ): Node v with its desired degree d ′ . d = d SexNet ( u ); /* Depending on sex of node and its degree, take the d th row or column of BJD */ if u is woman then R := BJD ( d, :), else R := BJD (: , d ); for range( | R | ) doif R = 0 then /* Choose one nonzero element of R with index d ′ , which is the degree of partner to be found */ w = R.sample ( R ( d ′ ) = 0), d ′ = R.index ( w ); if urn ≤ p then /* Prepare for choosing from the social contacts: *//* K - set of available nodes in SexNet with degree d ′ , who are also social contact; *//* K - set of nodes that are not in SexNet but the social contacts of u with social degree ≥ d ′ */ K { k ∈ NF : k ∈ SocNet .N ( u ) − SexNet .N ( u ) , d SexNet ( k ) = d ′ } ; K { k ∈ SocNet .N ( u ) − SexNet .n : d SocNet ( k ) ≥ d ′ } ; else /* Prepare for choosing outside the social contacts: *//* K - set of available nodes in SexNet with sexual degree d ′ , who are not social contacts *//* K - set of nodes that are not SexNet and not social contact of u with social degree ≥ d ′ )*/ K { k ∈ NF : k / ∈ SocNet .N ( u ) ∪ SexNet .N ( u ) , d SexNet ( k ) = d ′ } ; K { k ∈ SocNet .n − SexNet .n : d SocNet ( k ) ≥ d ′ } ; endif K = ∅ then /* If K is not empty, select the closest node in SocNet */ v = K .sample ( w : dist ( SocNet , u , w ) = min { dist ( SocNet , u , k ) for k ∈ K } );Break; else if K = ∅ then /* if K is not empty, select the closest node in SocNet , define its sex degree and stub as d ′ */ v = K .sample ( w : dis ( SocNet , u , w ) = min { dis ( SocNet , u , k ) for k ∈ K } ); d SexNet ( v ) := d ′ , stub ( v ) := d ′ ;Break; else /* If both don’t work, move on to a different degree */ R ( d ′ ) := 0; endelse /* If all elements of R are 0, we fail to find a proper partner for u */ ( d ′ , v ) = False ;break; endendReturn ( d ′ , v ); BJD matrix for all of the sparse heterosexual networks we have generated inthis project. However, there are some situations, where the desired network isnot sparse, the algorithm can fail to exactly produce a network with the desired
BJD matrix for the joint-degree distribution. When this happens, a secondalgorithm is used to rewire the network so that it will exactly match the desiredbipartite joint-degree distribution for the number of partners that a person’spartners have.
SexNet for a given
BJD matrix
The rewiring algorithm corrects any mismatch between the joint-degree dis-tribution of generated
SexNet and the desired
BJD . We define the joint-degree distribution of the generated
SexNet as ] BJD and the mismatch errormatrix E = BJD − ] BJD . If the matrix E has nonzero elements, then thenetwork is rewired to eliminate the error. There are three possible cases:(i) If entry E ( i,j ) , ( i, j > k , it means that SexNet needs k more edges between degree i women and degree j men. To create theseedges, we iterate the following process k times:(a) First, identify a woman, w i , in SexNet , where d SexNet ( w i ) = i , andhas as a partner, m , with degree-1, i.e. d SexNet ( m ) = 1.(b) Next, identify another man, m j ∈ SocNet .N ( w i ) − SexNet .N ( w i ),where d SexNet ( m j ) = j and m j has a degree-1 partner, w . That is d SexNet ( w ) = 1.(c) Finally, we rewire the network by removing the edges ( w i , m ) and( m j , w i ) and add edge ( w i , m j ), as illustrated in the Rewiring (1) ofthe Figure (2).(ii) If element E ( i,j ) , ( i, j > k ′ , it means that SexNet haveextra k ′ edges between degree i women and degree j men. To remove theseedges, we iterate following process k ′ times:13a) First, identify a woman, w i , in SexNet , where d SexNet ( w i ) = i , whichhas a degree-j partner like m j , that is d SexNet ( m j ) = j . The nodes w i and m j are selected so that they have at least one social contactwith opposite sex that is not their sexual partner.(b) Next, identify another man, m ∈ SocNet .N ( w i ) − SexNet .N ( w i ),and a woman, w ∈ SocNet .N ( m j ) − SexNet .N ( m j ).(c) Finally, rewire the network by removing the edge ( w i , m j ) and addedges ( w i , m ) and ( w , m j ), as illustrated in the Rewiring (2) of theFigure (2). w i ... m m j ... w w i ... m m j ... w Rewiring (1)Rewiring (2)
Figure 2: Schematic of steps 1 and 2 of Rewiring approach to correct
BJD . (iii) In the previous steps, we pushed back nonzero elements in E to its firstrow and column, which causes new nonzero elements in the first row andcolumn. To remove these nonzero values, we have to add or remove smallcomponents. For example, if the element (i,1) of E is a positive value k , it means that we need a small component of a degree i woman whosepartners are all degree 1 men. Therefore, we simply make this componentfrom the people who are not currently in SexNet .n . If the element (1 , j )of E is a negative value k , it means we have to remove a small componentof a degree j man whose partners are all degree 1 women. Therefore, wesimply look for such a component and remove it from SexNet .We have found that this algorithm almost always converges to the desired
BJD . However, there are rare cases when the desired rewiring nodes maynot exist, and algorithm stalls with
E 6 = 0. When this happens, the rewirednetwork will have a better joint-degree distribution of
SexNet , even though it14oes not exactly match the desired
BJD . In the numerical simulations, all ofthe generated New Orleans heterosexual network had exactly the desired
BJD based on survey data.In the next section, we apply our algorithm to generate and analyze sev-eral random
SexNet corresponding to sexual activity of adolescent and youngadult sexually active African Americans reside in New Orleans. First, we ex-plain the inputs of our approach: New Orleans social network and joint-degreedistribution-
BJD matrix- corresponding to its sexual network. Then we useour algorithm to generate and analyze a bunch of
SexNet s for a subpopulationof people in New Orleans.
3. Simulations
We analyze an ensemble of sexual networks with a prescribed joint-degreedistribution representing sexual activity of young adult African Americans inNew Orleans.
SocNet
The
SocNet is based on the synthetic data generated by Simfrastructure[12, 13] for 130,000 synthetic people residing in New Orleans. Simfrastructureis a high-performance, service-oriented, agent-based simulation system, repre-senting and analyzing interdependent infrastructures. The data for the networkincludes information for each individual, identified by their
PID (personal iden-tifier), and includes their age, gender, household, and other demographic infor-mation. The contact information for each PID is encoded in the contact file inTable 2.The Simfrastructure data was used to generate the original
SocNet , whichwas then used to generate
ESocNet and
BSocNet as described in the previoussection. 15
ID FID A T . . . . . . Table 2: Table input for the social contact network of individuals. The
PID is personal ID,
FID is their social contact ID, A is activity in which PID meet FID, and T is the fraction oftime in a day that two social contacts meet with each other through activity A. We have fivedifferent activities, including H as home, W as work, Sc as school, Sh as shopping, and O asothers. For example, person 43722 stays with person 16981 at the same home for 7 . BJD matrix for New Orleans heterosexual activity
An ongoing community-based pilot study was conducted among sexually ac-tive African Americans ages 15 −
25 in New Orleans [17], to assess the effective-ness of prevention and intervention programs for chlamydia. Socio-demographicinformation including age, race, educational level and, sexual behavior–numberand age of heterosexual partners in the past two months– and history of theirSTI test results were collected from 202 men and 414 women participants. Mean-while, their partners’ information has been collected by asking questions refer-ring to the status of each relationship such as the partner’s age and the possibil-ity that their partner(s) have intercourse with others. The survey results wereused to construct the
BJD matrix of a heterosexual network of individuals inNew Orleans. For a population P = 15 ,
000 sexually active young adult menand their women partners residing in New Orleans, we have16 JD = . The dimension of this
BJD matrix is 6 ×
21, that is, the maximum numberof partners women have is 6 and for men is 21.
SexNet analysis
Using the social network and
BJD matrix provided in the previous subsec-tions and approach described in Section 2, we generated 150
SexNet s of 15000people for p = 0 . , . , . , . SexNet s are 20%, 30are 40%, 30 are 60%, 30 are 80%, and the rest 30 are 100% subgraph of
BSoc-Net . We then compared some descriptive measures of this ensemble of randomnetworks that were not imposed when generating the networks, including thesize of giant components and bi-components, number of connected components,and average redundancy coefficient.First, we evaluated and compared the size of the giant component and bi-component (the first and second biggest connected components of the network)for each group of the networks. Figure (3) shows the box plot of these sizes:there is an increment in the size of giant components when people select most oftheir sexual partners from their social contacts. Because in that case, sexuallyactive people are tighter together within the social contact network. But thereis not a significant difference in the size of giant bi-component.The number of connected components,
N c , is another measure characterizingnetwork toughness. This measure can be, not necessarily, correlated to thecomponent’s size of the network. Figure (4) displays descriptive statistics for
N c in each network group. Note that data distributions are approximatelysymmetrical, and measures of
N c are similar across groups, but, they changeby changing the source of partner selection- changing p .17 % s u bg r a pg % s u bg r a pg % s u bg r a pg % s u bg r a pg % s u bg r a pg % s u bg r a pg % s u bg r a pg % s u bg r a pg % s u bg r a pg % s u bg r a pg Networks050010001500200025003000 S i z e o f G i a n t C o m p o n e n t a n d B i c o m p o n e n t Comparison of Component Size of Different Networks
Size of Giant ComponentSize of Giant Bicomponent * Average Value
Figure 3: Box plot representing thee size of the giant component and bi-component for eachgroup of networks: The size of the giant component becomes bigger when the portion of thesubgraph becomes stronger, however, the social network does not have much impact on thesize of the giant bi-component. % s u bg r a pg % s u bg r a pg % s u bg r a pg % s u bg r a pg % s u bg r a pg Networks305030753100312531503175320032253250 N u m b e r o f C o nn e c t e d C o m p o n e n e t s Comparison of Number of Connected Componenets of Different Networks
Number of Connected Componenets * Average Value
Figure 4: Box plot representing the number of connected components, Nc , for each group ofnetworks: A significant difference is observed in Nc between each group, Nc is lower in largersubgraph of social networks. Definition 1.
For a bipartite network, redundancy for a node is the ratio of itsoverlap to its maximum possible overlap according to its degree. The overlap of anode is the number of pairs of neighbors that have mutual neighbors themselves,other than that node [18]. For a typical node v , the redundancy coefficient of v is defined as Rc ( v ) = |{{ u,w } ⊆ N ( v ) , ∃ v’ = v s.t uv’ ∈ E , wv’ ∈ E }| | N ( v ) | ( | N ( v ) |− , where, N ( v ) is the set of all neighbors of node v , and E is the set of all edgesin the network. We compare this measure for the networks in Figure (5): each data point Rc ( k ) for degree k is obtained by averaging redundancy coefficient over thegroup of people with k partners. In most of the networks, Rc ( k ) decreases with k [22]. Redundancy coefficient Rc is affected by social network BSocNet : whenpeople select more sexual partners from their social contacts the value for Rc increases, which is because of stage one of the algorithm- Generate an extendedsocial network. In that stage, by connecting the social contacts of a person in BSocNet , we increase its clustering coefficient. Therefore, because increasing p SexNet becomes a stronger subgraph of
BSocNet , it inherits more propertiesfrom
BSocNet , that is, by increasing p Rc of the
SexNet , which is correlatedto clustering coefficient of
BSocNet , increases.
4. Discussion
We described a new algorithm to generate an ensemble of heterosexual net-works based on heterosexual behavior surveys for the young adult African Amer-ican population in New Orleans. The prescribed degree and joint-degree distri-bution represented the heterosexual network embedded within a social network19 .5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
Degree k −0.010.000.010.020.03 RC ( k )
20% subgraph40% subgraph60% subgraph80% subgraph100% subgraph
Figure 5: Scatter plot of Rc versus degree for five different networks: Rc for SexNet whichis strong subgraph of
BSocNet (higher p ) is higher, because clustering coefficient for BSoc-Net is high and therefore,
SexNet inherits this property by having higher Rc than the oneswhich are weak subgraph of BSocNet . BJD . When the networks had a more percentage of partners selected from theirextended social contacts, then we observed a tighter distribution in the numberof connected components and the size of giant component and bi-components.When more partners are chosen from the extended social network, instead of ran-domly selected from the population, then the size of giant component increases,and following that the number of connected components decreases, which isbecause of reducing the mixing in generating sexual network: when people se-lect their sexual partners from their social contacts they stand in a tight groupwithin social network. In fact, when being subgraph of the extended social net-work become stronger, the candidate set of sexual partners for each person thatis set of social contacts decreases and becomes local ( this set includes closecontacts and contacts of contacts) compared with when this set is the wholepopulation.As p increases, then more partners are chosen from a person’s extended so-cial network. This also increases the network clustering coefficients (where morepartners of your partner’s partner are also one of your partners). The redun-dancy coefficients for networks increases as the dependence of sexual networkon social one rises when p increases, which is because of the high clusteringcoefficient of the social network due to the first stage of the algorithm, gen-erate an extended social contact network. In that stage, we made some newcontacts between the contacts of each individual, which causes the incrementin the clustering coefficient of the social contact network. On the other hand,when more partners are chosen from the social contact network, more proper-ties of the social network such as the clustering coefficient become inherited by SexNet . Thus, increasing p , we observe increment in the redundancy coefficientof SexNet .We studied the measures of networks because they may affect the spread ofan STI such as chlamydia on the
SexNet s. In our future work when studying21he spread of chlamydia on heterosexual networks, we will measure their impacton the prevalence of chlamydia over
SexNet s generated using different p values.There are still unanswered questions for proving the existence of a hetero-sexual network with a prescribed joint-degree distribution embedded within aprescribed social network. That is, there are no explicit criteria to guaranteethat a heterosexual network with a particular joint-degree distribution can beembedded within a particular social network or not.We are currently simulating a stochastic agent-based network model on SexNet for the spread of chlamydia and comparing different interventionstrategies to control the spread of STIs. These simulations will use the un-derlying social contact network to improve the current intervention models byconsidering the impact of counseling and behavioral changes such as increasingcondom use or social contact notification.
Acknowledgments
The authors thank Achla Marathe, Patricia Kissinger, Stephen Eubank, andNorine Schmidt for their useful comments and suggestions. This work was sup-ported by the endowment for the Evelyn and John G. Phillips DistinguishedChair in Mathematics at Tulane University and grants from the National In-stitutes of Health National Institute of Child Health and Human Development(R01HD086794) and Office of Adolescent Health (TP2AH000013) and the Na-tional Institute of General Medical Sciences program for Models of InfectiousDisease Agent Study (U01GM097658). The content is solely the responsibil-ity of the authors and does not necessarily represent the official views of theNational Institutes of Health.
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