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The analysis of social networks

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Abstract

Many questions about the social organization of medicine and health services involve interdependencies among social actors that may be depicted by networks of relationships. Social network studies have been pursued for some time in social science disciplines, where numerous descriptive methods for analyzing them have been proposed. More recently, interest in the analysis of social network data has grown among statisticians, who have developed more elaborate models and methods for fitting them to network data. This article reviews fundamentals of, and recent innovations in, social network analysis using a physician influence network as an example. After introducing forms of network data, basic network statistics, and common descriptive measures, it describes two distinct types of statistical models for network data: individual-outcome models in which networks enter the construction of explanatory variables, and relational models in which the network itself is a multivariate dependent variable. Complexities in estimating both types of models arise due to the complex correlation structures among outcome measures.

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Notes

  1. To aid readers in applying these methods, we provide some references to network software throughout, but our coverage of software is not comprehensive. Huisman and Van Duijn (2005) review software resources available earlier in this decade.

  2. The extent to which distances in a graphical representation correspond to the data on which they rest—dyadic measurements of social distance or proximity—depends on the objective function that serves as a fitting criterion when the plot is constructed. The most widely-used “nonmetric” multidimensional scaling algorithm requires an ordinal correspondence between data and plotted distances; “metric” scaling uses a stronger (linear) criterion. Objective functions used by many spring-embedder methods involve a “node repulsion” term that simplifies the visual representation by discouraging co-location of vertices within a plot, but simultaneously limits the extent to which data and plotted distances correspond. Moreover, a low (ordinarily 2)-dimensional Cartesian plot may do more or less well in representing data on the relationships among N actors, which may in principle be (N − 1)-dimensional.

  3. Note that some or all of the intermediaries along these geodesic paths may be physicians 11–33.

  4. Recall that the undirected physician network is identical to that shown in Fig. 1, except that ties lack directionality.

  5. We calculated centrality scores using the software package UCINET 6 (Borgatti, Everett, and Freeman, 2002).

  6. Eigenvalue centrality can in principle be calculated for a nonsymmetric matrix, but the routine in UCINET 6 handles only the symmetric case.

  7. Because two actors have outdegrees of 0, the associated rows of W sum to 0 as opposed to 1. Therefore, although these actors contribute to the estimation of β and σ2, they do not directly contribute any information about the autocorrelation parameters α and ρ. We retained these actors in the analysis because they were cited by other physicians as influencing them and so removing them would omit information about how other actors were influenced.

  8. Computations were performed using the StOCNET software package (Boer et al. 2006).

  9. Under p1, the estimate of a receiver parameter is infinitely small for actors with indegree 0; likewise, the estimate of a sender parameter is -∞ when the corresponding outdegree is 0.

  10. The p2 model is closely related to a social relations model developed by Kenny and La Voie (1984) for quantitative network variables.

  11. The large number of terms in κ(θ) complicates the estimation of ERGMs. There are 2N(N−1)possible directed binary-valued networks; for example, with N = 10, the number of possible networks—hence terms in κ(θ)—is 1.238 × 1027.

  12. For example, an actor with degree 3 contributes 1 3-star, 3 2-stars, and 3 1-stars; 1-stars are equivalent to individual edges.

  13. The set of k-star statistics is equivalent to the set of degree statistics (the number of nodes of degree k, k = 1,2,3,…) in that a bijection exists between the two sets of the statistics (Snijders et al. 2006).

  14. An analogous “sender covariate” statistic allows the density effect to depend on an attribute of the sender (i).

  15. \( {\mathbf{y}}_{ij}^{ + } \)is the realization of the complement network with y ij  = 1, while \( {\mathbf{y}}_{ij}^{ - } \) is the realization of the complement network with y ij  = 0.

  16. An equivalent statistic based on the degree distribution itself is known as the “geometrically weighted degree” statistic; see Hunter and Handcock (2006).

  17. No mutuality term is included, since this is redundant with the edges term in an undirected network. Constraining the value of ρ when fitting the model with the GWESP term is often helpful; attaining adequate convergence is more difficult when it is estimated as a free parameter. We found that setting ρ = 1.2 served well here; the likelihood surface is relatively flat, so that using a value between 1.0 and 1.5 did not affect inferences about other parameters. Note, however, ρ was estimated at 0.93 when we left it as a free parameter in the third model in Table 9.

  18. Although the missing values are replaced with non-missing values during model fitting, the statistics measuring model fit are only evaluated using actors with non-missing values throughout the corresponding interval of time. Thus, standard imputation is not performed.

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Acknowledgements

We thank Nancy Keating for allowing us to re-analyze the data from the physician influence network. Research for the paper was supported by NIH grants R01 AG024448-02, P01 AG031093, and Robert Wood Johnson Foundation Award #58729.

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Authors and Affiliations

  1. Department of Health Care Policy, Harvard Medical School, 180 Longwood Avenue, Boston, MA, 02115-5899, USA

    A. James O’Malley

  2. Department of Sociology, Harvard University, 33 Kirkland Street, Cambridge, MA, 02138, USA

    Peter V. Marsden

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  1. A. James O’Malley
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Correspondence to A. James O’Malley.

Appendices

Appendices

Appendix A: R-code for fitting Individual Outcome Model

  • ## Functions used in the analysis ##

  • geodist <- function(adj) {

  • #Derive the geodesic distance between the actors

  • dist <- matrix(0,nr,nr)

  • matpow <- diag(1,nr)

  • for (k in 1:nr) {

  • matpow <- matpow %*% adj

  • ind <- ifelse(dist>0,1,0)

  • dist <- dist*ind+k*ifelse(matpow>0,1,0)*(1-ind)

  • }

  • ind <- ifelse(dist>0,1,0)

  • dist <- dist*ind-1*(1-ind) #-1 indicates of not connected

  • diag(dist) <- 0 #Always 0 distance on the diagonal!

  • return(dist)

  • }

  • like.auto <- function(al,y,x,w) {

  • #Evaluate log-likelihood function of outcome autocorrelation model

  • n <- nrow(x)

  • icov <- diag(1,n) - al*w

  • z <- icov %*% y

  • icov <- t(icov) %*% icov

  • be <- solve(t(x) %*% x) %*% (t(x) %*% z) #coefficients of exogeneous vbles in IV model

  • ri <- z - x %*% be #residuals

  • std2 <- as.vector(t(ri) %*% ri)/n #variance of outcome

  • icov <- icov/std2

  • return(log(det(icov)) - (t(ri) %*% ri)/std2)

  • }

  • parSE.auto<- function(al,y,x,w) {

  • #Standard errors of outcome autocorrelation model

  • n <- nrow(x)

  • p <- ncol(x)

  • icov1<- diag(1,n) - al*w

  • icov <- t(icov1) %*% icov1

  • z <- icov1%*% y

  • be.be <- (t(x) %*% x)

  • be <- solve(be.be) %*% (t(x) %*% z) #coefficients of x

  • ri <- z - x %*% be #residuals

  • std2 <- as.vector(t(ri) %*% ri)/n #variance of outcome

  • G <- w %*% solve(icov1)

  • Gxb <- G %*% x %*% be

  • #The following derivation is based on Harville ( 1997 , p. 309).

  • cov <- solve(icov)

  • wsym <- t(w)+w

  • wsqr <- t(w) %*% w

  • dicov.al <- 2*al*wsqr-wsym

  • dicov.alal <- 2*wsqr

  • t.alal <- cov %*% dicov.alal

  • t.al <- cov %*% dicov.al

  • detterm <- sum(diag(t.alal - (t.al %*% t.al)))/2

  • #Information matrix - extension of expression on page 370 of

  • #Doreian ( 1981 ) to asymmetric W

  • be.be <- be.be/std2

  • std2.std2 <- n/(2*(std2^2))

  • al.be <- (t(Gxb) %*% x)/std2

  • al.std2 <- sum(diag(G))/std2

  • al.al <- sum(diag(t(G) %*% G))+(t(Gxb) %*% Gxb)/std2 - detterm

  • Infm <- matrix(0,p+2,p+2)

  • Infm[1:p,1:p] <- be.be

  • Infm[p+2,1:p] <- al.be

  • Infm[1:p,p+2] <- t(al.be)

  • Infm[p+1,p+1] <- std2.std2

  • Infm[p+1,p+2] <- al.std2

  • Infm[p+2,p+1] <- al.std2

  • Infm[p+2,p+2] <- al.al

  • #Estimates and covariance matrix

  • parms <- c(be=be, std2=std2, al=al)

  • covar <- solve(Infm)

  • SE <- sqrt(diag(covar))

  • names(SE) <- names(par)

  • tval=parms/SE

  • pval=2*(1-pt(abs(tval),n-p))

  • return(list(estimates=parms,SE=SE,tval=tval,pval=pval))

  • }

  • like.sar <- function(rho,y,x,w) {

  • #Evaluate log-likelihood function

  • n <- nrow(x)

  • icov <- diag(1,n) - rho*w

  • icov <- t(icov) %*% icov

  • be <- solve(t(x) %*% icov %*% x) %*% (t(x) %*% icov %*% y) #coefficients of exogeneous vbles in IV model

  • ri <- y - x %*% be #residuals

  • std2 <- as.vector(t(ri) %*% icov %*% ri)/n #variance of outcome

  • icov <- icov/std2

  • return(log(det(icov)) - t(ri) %*% icov %*% ri)

  • }

  • parSE.sar <- function(rho,y,x,w) {

  • n <- nrow(x)

  • p <- ncol(x)

  • icov1 <- diag(1,n) - rho*w

  • icov <- t(icov1) %*% icov1

  • be.be <- (t(x) %*% icov %*% x)

  • be <- solve(be.be) %*% (t(x) %*% icov %*% y) #coefficients of x

  • ri <- y - x %*% be #residuals

  • std2 <- as.vector(t(ri) %*% icov %*% ri)/n #variance of outcome

  • G <- w %*% solve(icov1)

  • #The following computation is based on Harville ( 1997 , p. 309).

  • cov <- solve(icov)

  • wsym <- t(w)+w

  • wsqr <- t(w) %*% w

  • dicov.rho <- 2*rho*wsqr-wsym

  • dicov.rhorho <- 2*wsqr

  • t.rhorho <- cov %*% dicov.rhorho

  • t.rho <- cov %*% dicov.rho

  • detterm <- sum(diag(t.rhorho - (t.rho %*% t.rho)))/2

  • #Information matrix - extension of expression on page 366 of

  • #Waller and Gotway ( 2004 ) to asymmetric W

  • be.be <- be.be/std2

  • std2.std2 <- n/(2*(std2^2))

  • rho.std2 <- sum(diag(G))/std2

  • rho.rho <- sum(diag(t(G) %*% G)) - detterm

  • Infm <- matrix(0,p+2,p+2)

  • Infm[1:p,1:p] <- be.be

  • Infm[p+1,p+1] <- std2.std2

  • Infm[p+1,p+2] <- rho.std2

  • Infm[p+2,p+1] <- rho.std2

  • Infm[p+2,p+2] <- rho.rho

  • #Estimates and covariance matrix

  • parms <- c(be=be, std2=std2, rho=rho)

  • covar <- solve(Infm)

  • SE <- sqrt(diag(covar))

  • names(SE) <- names(par)

  • tval=parms/SE

  • pval=2*(1-pt(abs(tval),n-p))

  • return(list(estimates=parms,SE=SE,tval=tval,pval=pval))

  • }

  • ## Code for loading data and fitting model ##

  • source(“NetAnalFns.r”)

  • #Load network

  • nr <- 33

  • adjdata <- scan(“your_network_file.txt”)

  • adjdata <- matrix(adjdata,nrow=nr,byrow=T)

  • adjdata[19,19] <- 0 #Tidy up data (self-connected nodes not possible)

  • iddata <- seq(1,33,1)

  • #Load node covariate data

  • covdata <- scan(“your_covariate_file.txt”)

  • covdata <- matrix(covdata,nrow=nr,byrow=T)

  • nodecov <- list(male=covdata[,2], whexpert=covdata[,3], pctwom=covdata[,4],

  • numsess=covdata[,5], practice=covdata[,6])

  • #Directed network

  • adjdir <- ifelse(adjdata>0,1,0)

  • #Standardize adjacency matrix so that row sums=1

  • numalters <- apply(adjdir,1,sum)

  • scale=as.vector(numalters^(-1))

  • noalters <- ifelse(is.infinite(scale)==1,1,0)

  • scale[noalters*seq(1,nr)] <- 0

  • on=matrix(1,ncol=nr,nrow=1)

  • wtadjdir <- adjdir * (scale %*% on)

  • #Compute mean value of dependent variables for directly connected

  • # actors - i.e., based on the adjacency matrix

  • hrtalt <- wtadjdir %*% as.vector(covdata$sumhrt)

  • regdata <- data.frame(covdata,hrtalt=hrtalt,noalters=noalters,numalters=numalters)

  • #Compute scaled geodesic distances having row sums equal to 1

  • gdist <- geodist(adjdir)

  • igdist <- gdist^(-1)

  • diag(igdist) <- 0 #Always 0 distance on the diagonal!

  • ind < ifelse(igdist <0,1,0) #Egos with no influential conversations

  • igdist <- igdist*(1-ind)

  • sumigeo <- apply(igdist,1,sum)

  • scale <- as.vector(sumigeo^(-1))

  • noalters <- ifelse(is.infinite(scale)==1,1,0)

  • scale[noalters*seq(1,nr)] <- 0 #Influence set equal to 0 if no influential conversations

  • wtigeo <- igdist * (scale %*% on)

  • #Augment analysis dataset with additional variables

  • hrtgeo <- wtigeo %*% as.vector(covdata$sumhrt)

  • regdata <- data.frame(regdata,hrtgeo=hrtgeo)

  • on <- as.vector(rep(1,nr))

  • x <- as.matrix(cbind(on,regdata[,c(“male”,”pctwom”,”numalters”)]))

  • #Use maximum likelihood to obtain MLEs for autoregressive outcomes and network autocorrelation (SAR) models

  • # - uses optim optimization function in R.

  • strtval <- 0

  • #Fit autoregressive outcomes model

  • #Use scaled adjacency matrix as weight matrix

  • mle <- optim(par=strtval, fn=like.auto, gr=NULL, method=“BFGS”,

  • control=list(fnscale=−1, trace=6, maxit=100, reltol=1e-16),

  • hessian=TRUE, y=regdata$sumhrt,x=x,w=wtadjdir)

  • #Compute estimates of all parameters and standard errors

  • Auto.adj <- parSE.auto(mle$par,y=regdata$sumhrt,x=x,w=wtadjdir)

  • #Use scaled geodesic matrix as weight matrix

  • mle <- optim(par=strtval, fn=like.auto, gr=NULL, method=“BFGS”,

  • control=list(fnscale=-1, trace=6, maxit=100, reltol=1e-16),

  • hessian=TRUE, y=regdata$sumhrt,x=x,w=wtigeo)

  • #Compute estimates of all parameters and standard errors

  • Auto.geo <- parSE.auto(mle$par,y=regdata$sumhrt,x=x,w=wtigeo)

  • #Fit corresponding SAR models

  • #Use scaled adjacency matrix as weight matrix

  • mle <- optim(par=strtval, fn=like.sar, gr=NULL, method=“BFGS”,

  • control=list(fnscale=-1, trace=6, maxit=100, reltol=1e-16),

  • hessian=TRUE, y=regdata$sumhrt,x=x,w=wtadjdir)

  • #Compute estimates of all parameters and standard errors

  • SAR.adj <- parSE.sar(mle$par,y=regdata$sumhrt,x=x,w=wtadjdir)

  • #Use scaled geodesic matrix as weight matrix

  • mle <- optim(par=strtval, fn=like.sar, gr=NULL, method=“BFGS”,

  • control=list(fnscale=-1, trace=6, maxit=100, reltol=1e-16),

  • hessian=TRUE, y=regdata$sumhrt,x=x,w=wtigeo)

  • #Compute estimates of all parameters and standard errors

  • SAR.geo <- parSE.sar(mle$par,y=regdata$sumhrt,x=x,w=wtigeo)

  • #Print out all results

  • print(data.frame(Auto.adj))

  • print(data.frame(Auto.geo))

  • print(data.frame(SAR.adj))

  • print(data.frame(SAR.geo))

  • ##Alternative code using lnam function in Carter Butt’s sna package. ##

  • library(sna)

  • library(numDeriv)

  • lnam1.adj <- lnam(regdata$sumhrt,x,wtadjdir)

  • lnam1.geo <- lnam(regdata$sumhrt,x,wtigeo)

  • lnam2.adj <- lnam(regdata$sumhrt,x,NULL,wtadjdir)

  • lnam2.geo <- lnam(regdata$sumhrt,x,NULL,wtigeo)

  • #Print out all results

  • summary(lnam1.adj)

  • summary(lnam1.geo)

  • summary(lnam2.adj)

  • summary(lnam2.geo)

Appendix B: R-code for Relational Data Model

  • #Install software on R: Can comment out after first use.

  • install.package(“statnet”)

  • install.packages(“coda”)

  • #Attached libraries each time use StatNet

  • library(statnet)

  • library(coda)

  • #Load network

  • nr <- 33

  • adjdata <- scan(“your_network_file.txt”)

  • adjdata <- matrix(adjdata,nrow=nr,byrow=T)

  • adjdata[19,19] <- 0 #Tidy up data (self-connected nodes not possible)

  • iddata <- seq(1,33,1)

  • #Load node covariate data

  • covdata <- scan(“your_covariate_file.txt”)

  • covdata <- matrix(covdata,nrow=nr,byrow=T)

  • nodecov <- list(male=covdata[,2], whexpert=covdata[,3], pctwom=covdata[,4],

  • numsess=covdata[,5], practice=covdata[,6])

  • #Make directed network

  • pnet <- network(adjdata,directed=TRUE,matrixtype=“adjacency”,

  • vertex.attr=nodecov,

  • vertex.attrnames=c(“male”,”whexpert”,”pctwom”,”numsess”,”practice”,

  • “bcma”,”bima”,”bpp”,”wnhlth”,”numcat”,”pctcat”))

  • #Plot directed network

  • plot(pnet,mode=“fruchtermanreingold”,displaylabels=T)

  • #Fit models to directed network

  • model1a <- ergm(pnet~edges)

  • model1b <- ergm(pnet~edges+mutual)

  • model1c <- ergm(pnet~edges+mutual+ttriad, theta0=“MPLE”, MPLEonly=TRUE)

  • model1d <- ergm(pnet~edges+mutual +receivercov(“whexpert”) +

  • receivercov(“pctwom”)+receivercov(“numsess”))

  • #Evaluate Goodness of fit with respect to distance and degrees

  • dist.gof <- gof(model1d~distance, nsim=10, verbose=T)

  • plot(dist.gof)

  • ideg.gof <- gof(model1d~idegree, nsim=10, verbose=T)

  • plot(ideg.gof)

  • odeg.gof <- gof(model1d~odegree, nsim=10, verbose=T)

  • plot(odeg.gof)

  • #Define network using mutual ties.

  • adjmut <- ifelse(adjdata>0,1,0)+ifelse(t(adjdata)>0,1,0)

  • adjmut <- ifelse(adjmut>0,1,0); #One possible definition of mutual tie

  • #Make undirected network

  • pnetmut <- network(adjmut,directed=FALSE,matrixtype=“adjacency”,

  • vertex.attr=nodecov,

  • vertex.attrnames=c(“male”,”whexpert”,”pctwom”,”numsess”,”practice”,

  • “bcma”,”bima”,”bpp”,”wnhlth”,”numcat”,”pctcat”))

  • #Plot undirected network

  • plot(pnetmut,mode=“fruchtermanreingold”,displaylabels=T)

  • #Fit models to undirected network

  • model2a <- ergm(pnetmut~edges)

  • model2b <- ergm(pnetmut~edges+gwesp(1.2,fixed=T),

  • burnin=10000, MCMCsamplesize=10000, interval=100, maxit=3)

  • model2c <- ergm(pnetmut~edges+gwesp(1.2,fixed=F)+kstar(2:3),

  • burnin=10000, MCMCsamplesize=10000, interval=100, maxit=3)

  • model2d <- ergm(pnetmut~edges+gwesp(1.2,fixed=T)+

  • nodematch(“male”,diff=F)+nodematch(“practice”,diff=F),

  • burnin=10000, MCMCsamplesize=10000, interval=100, maxit=3)

  • model2e <- ergm(pnetmut~edges+gwesp(1.2,fixed=T) +

  • nodematch(“male”,diff=F)+nodematch(“practice”,diff=T),

  • burnin=10000, MCMCsamplesize=10000, interval=100, maxit=3)

  • #Test goodness of fit with respect to espartners statistic

  • esppart.gof <- gof(model2e~espartners,nsim=10,verbose=T)

  • plot(esppart.gof)

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O’Malley, A.J., Marsden, P.V. The analysis of social networks. Health Serv Outcomes Res Method 8, 222–269 (2008). https://doi.org/10.1007/s10742-008-0041-z

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