Multiplex crack smoking and sexual networks, 1
Online supplement
Title: Multiplex Crack Smoking and Sexual Networks: Associations between Network Members’ Incarceration and HIV Risks among High-Risk MSM
Mathematical definition of networks
Our study specified two separate adjacency matrices, based on crack smoking relationships, and an adjacency matrix, , based on sexual relationships. The adjacency matrix records if a respondent i (ego) ever smoked crack with partner j (alter), and otherwise. The adjacency matrix records if a respondent i (ego) ever had sex with partner j (alter), and otherwise. We converted these matrices into bi-directional graphs by maximally symmetrizing them (i.e., adding any ties that were not reciprocated) due to the symmetric natures of these relationships and to address potential recollection issues and discordant ties.
Decomposed network exposure model
This study employed a decomposed network exposure model (DNEM) (Fujimoto, 2012; Fujimoto & Valente, 2013) to measure the level of risk exposure to incarcerated partners (or network members) who are connected through (1) crack-and-sexual relationships and through (2) crack-only relationships (no sex involved). DNEM is built on the network exposure model(Valente, 1995, 2005), which is designed to measure the extent to which a respondent (ego) is exposed to contacts (alters) with specific behavioral attributes. It assesses the degree to which crack-smoking relationships () overlap or do not overlap (non-overlap) with sexual relationships (). DNEM is for a given actor iis defined as follows:
where is an adjacency matrix based on crack-smoking relationships, is an adjacency matrix based on sexual relationships, and is partners’ incarceration measured separately by (i) lifetime arrest, and (ii) total months of lifetime incarceration. Mathematically, (1) was computed by element-wise product of the matrix by a matrix (crack-smoking relationships overlapped with sexual relationships), and then row-normalized by dividing the total number of relationships (i.e., outdegrees), and matrix-multiplying by the partners’ incarceration measures of . To compute (2), we subtracted the adjacency matrix from a unit matrix (where all elements are ones), and everything else as identical to the computation of (1), which is defined in the following formula:
where , , and were defined above. Essentially, the matrix based on crack-smoking relationships has been partitioned by the sex matrix.
Network proximity measures: computation of geodesic distance matrix
We computed the shortest paths, which is called the geodesic distance in social network analysis (Wasserman & Faust, 1994), between all MSM. We created geodesic distance matrices by taking the powers of the matrices. The number of powers reflects the length of a shortest path (degrees of separation) between actor i and partnerj(Wasserman & Faust, 1994). For instance, with power =1, the geodesic distance matrix is equivalent to an adjacency matrix for drug-use relationships, with power=2, the geodesic distance matrix becomes . Therefore, geodesic distance matrix of less than or equal to two degrees of separation can be generated by + , meaning that the number of shortest paths of length up to two from actor i to partner j is given by the entry (i, j) of matrix + .
Statistical Model
Our specifications of generalized linear models were definedin the following:
where is an intercept term, representing control variables and are the corresponding coefficients, representing two incarceration-related variables of frequency of lifetime arrest and incarceration months respectively and are the corresponding coefficients, represents two terms of decomposed exposure to incarcerated partners connected via “crack-and-sexual relationships” and are the corresponding coefficients, represents two terms of decomposed exposure to incarcerated partners connected via “crack-only relationships” and are the corresponding coefficients.
Reference
Fujimoto, K. (2012). Using mixed-mode networks to disentangle multiple sources of social
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Valente, T. W. (1995). Network Models of the Diffusion of Innovations. Cresskill, NJ,
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Valente, T. W. (2005). Network models and methods for studying the diffusion of
innovations.Models and Methods in Social Network Analysis: Structural Analysis in the Social Sciences.P. J. Carrington, J. Scott and S. Wasserman. New York, Cambridge University Press: 98–116.
Wasserman, S. and K. Faust (1994).Social Network Analysis: Methods and Applications.
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