Measurement Sensitivity

1. Measurement Sensitivity It seems a reasonable approach to assessing the effect of measurement error on the ties in a network is to ask how would the network measures change if the observed ties differed from those observed. This question can be answered simply with Monte Carlo simulations on the observed network. Thus, the procedure I propose is to: • Generate a probability matrix from the set of observed ties, • Generate many realizations of the network based on these underlying probabilities, and • Compare the distribution of generated statistics to those observed in the data. • How do we set pij? • Range based on observed features (Sensitivity analysis) • Outcome of a model based on observed patterns (ERGM)

2. Measurement Sensitivity As an example, consider the problem of defining “friendship” ties in highschools. Should we count nominations that are not reciprocated?

3. Measurement Sensitivity Reciprocated All ties

4. Measurement Sensitivity

5. Measurement Sensitivity

6. Measurement Sensitivity

7. Measurement Sensitivity

8. Measurement Sensitivity

9. Measurement Sensitivity

10. Statistical Analysis of Social Networks Comparing multiple networks: QAP The substantive question is how one set of relations (or dyadic attributes) relates to another. For example: • Do marriage ties correlate with business ties in the Medici family network? • Are friendship relations correlated with joint membership in a club? (review)

11. Modeling Social Networks parametrically: ERGM approaches The earliest approaches are based on simple random graph theory, but there’s been a flurry of activity in the last 10 years or so. Key historical references: - Holland and Leinhardt (1981) JASA - Frank and Strauss (1986) JASA - Wasserman and Faust (1994) – Chap 15 & 16 • Wasserman and Pattison (1996) Good practical overview: http://www.jstatsoft.org/v24 Great tutorial: http://statnet.csde.washington.edu/workshops/SUNBELT/EUSN/ergm/ergm_tutorial.html (last year’s sunbelt) Or • https://statnet.csde.washington.edu/trac/wiki/Sunbelt2014 (lots of how to slides)

12. Modeling Social Networks parametrically: ERGM approaches The “p1” model of Holland and Leinhardt is the classic foundation – the basic idea is that you can generate a statistical model of the network by predicting the counts of types of ties (asym, null, sym). They formulate a log-linear model for these counts; but the model is equivalent to a logit model on the dyads: Note the subscripts! This implies a distinct parameter for every node i and j in the model, plus one for reciprocity.

13. Modeling Social Networks parametrically: ERGM approaches

14. Modeling Social Networks parametrically: ERGM approaches Results from SAS version on PROSPER datasets

15. Modeling Social Networks parametrically: ERGM approaches Once you know the basic model format, you can imagine other specifications: Key is to ensure that the specification doesn’t imply a linear dependency of terms. Model fit is hard to judge – newer work shows that the se’s are “approximate” ;-)

16. Modeling Social Networks parametrically: ERGM approaches Where: q is a vector of parameters (like regression coefficients) z is a vector of network statistics, conditioning the graph k is a normalizing constant, to ensure the probabilities sum to 1.

17. Modeling Social Networks parametrically: ERGM approaches The simplest graph is a Bernoulli random graph,where each Xij is independent: Where: qij= logit[P(Xij= 1)] k(q) =P[1 + exp(ij )] Note this is one of the few cases where k(q) can be written.

18. Modeling Social Networks parametrically: ERGM approaches Typically, we add a homogeneity condition, so that all isomorphic graphs are equally likely. The homogeneous bernulli graph model: Where: k(q) =[1 + exp(q)]g

19. Modeling Social Networks parametrically: ERGM approaches If we want to condition on anything much more complicated than density, the normalizing constant ends up being a problem. We need a way to express the probability of the graph that doesn’t depend on that constant. First some terms:

20. Modeling Social Networks parametrically: ERGM approaches

21. Modeling Social Networks parametrically: ERGM approaches Note that we can now model the conditional probability of the graph, as a function of a set of difference statistics, without reference to the normalizing constant. The model, then, simply reduces to a logit model on the dyads.

22. Modeling Social Networks parametrically: ERGM approaches Consider the simplest possible model: the Bernoulli random graph model, which says the only feature of interest is the number of edges in the graph. What is the change statistic for that feature?

23. Modeling Social Networks parametrically: ERGM approaches Consider the simplest possible model: the Bernoulli random graph model, which says the only feature of interest is the number of edges in the graph. What is the change statistic for that feature? The “Edges” parameter is simply an intercept-only model. NODE ADJMAT 1 0 1 1 1 0 0 0 0 0 2 1 0 1 0 0 0 1 0 0 3 1 1 0 0 1 0 1 0 0 4 1 0 0 0 1 0 0 0 0 5 0 0 1 1 0 1 0 1 0 6 0 0 0 0 1 0 0 1 1 7 0 1 1 0 0 0 0 0 0 8 0 0 0 0 1 1 0 0 1 9 0 0 0 0 0 1 0 1 0 Density: 0.311

24. Modeling Social Networks parametrically: ERGM approaches Consider the simplest possible model: the Bernoulli random graph model, which says the only feature of interest is the number of edges in the graph. What is the change statistic for that feature? The “Edges” parameter is simply an intercept-only model. proclogisticdescendingdata=dydat; model nom =; run; quit; ---see results copy coef --- data chk; x=exp(-0.5705)/(1+exp(-0.5705)); run; procprintdata=chk; run;

25. Modeling Social Networks parametrically: ERGM approaches

26. Modeling Social Networks parametrically: ERGM approaches The logit model estimation procedure was popularized by Wasserman & colleagues, and a good guide to this approach is: Including: A Practical Guide To Fitting p* Social Network Models Via Logistic Regression The site includes the PREPSTAR program for creating the variables of interest. The following example draws from this work. – this bit nicely walks you through the logic of constructing change variables, model fit and so forth. But the estimates are not very good for any parameters other than “dyad independent” parameters!

27. Modeling Social Networks parametrically: ERGM approaches Parameters that are often fit include: • Expansiveness and attractiveness parameters. = dummies for each sender/receiver in the network • Degree distribution • Mutuality • Group membership (and all other parameters by group) • Transitivity / Intransitivity • K-in-stars, k-out-stars • Cyclicity • Node-level covariates (Matching, difference) • Edge-level covariates (dyad-level features such as exposure) • Temporal data – such as relations in prior waves.

28. Modeling Social Networks parametrically: Exponential Random Graph Models

29. Modeling Social Networks parametrically: Exponential Random Graph Models …and there are LOTS of terms…

30. Modeling Social Networks parametrically: Exponential Random Graph Models

31. Modeling Social Networks parametrically: Exponential Random Graph Models

32. Modeling Social Networks parametrically: Exponential Random Graph Models

33. Modeling Social Networks parametrically: Exponential Random Graph Models

34. Modeling Social Networks parametrically: Exponential Random Graph Models

35. Modeling Social Networks parametrically: Exponential Random Graph Models

36. Modeling Social Networks parametrically: Exponential Random Graph Models In practice, logit estimated models are difficult to estimate, and we have no good sense of how approximate the PMLE is. The STATNET generalization is to use MCMC methods to better estimate the parameters. This is essentially a simulation procedure working “under the hood” to explore the space of graphs described by the model parameters; searching for the best fit to the observed data.

37. Modeling Social Networks parametrically: Exponential Random Graph Models:

38. Modeling Social Networks parametrically: Exponential Random Graph Models:

39. Modeling Social Networks parametrically: Exponential Random Graph Models You can specify a model as a simple statement on terms:

40. Modeling Social Networks parametrically: Exponential Random Graph Models A simple example: One of the schools in PROSPER library(statnet); library(foreign); g <- read.paj("C:/jwmdata/prosper/Network_data_files/PAJEK/MATCHED/SC1C1W1Sch101.net"); g %v% "indegree" <- degree(g,cmode="indegree"); g %v% "outdegree" <- degree(g,cmode="outdegree"); atr<-read.table("C:/jwmdata/prosper/Network_data_files/Rfiles/ergmfiles/n111101.txt"); g %v% "sex" <- atr[,2 ]; g %v% "white" <- atr[,3 ]; g %v% "slun" <- atr[,4 ]; g %v% "irtuse" <- atr[,5 ]; g %v% "irtdev" <- atr[,6 ]; g %v% "tgrad" <- atr[,7 ]; g %v% "discip" <- atr[,8 ]; g %v% "church" <- atr[,9 ]; g %v% "sens" <- atr[,10 ]; plot(g,vertex.col="sex"); plot(g,vertex.col="slun"); plot(g,vertex.col="white");

41. Dynamics 1: Simple time-lag model: Prosper Peers

42. Modeling Social Networks parametrically: Exponential Random Graph Models

43. Complete Network Analysis Stochastic Network Analysis An example: Panel model in PROSPER

44. Complete Network Analysis Stochastic Network Analysis

50. Modeling Social Networks parametrically: Exponential Random Graph Models: Degeneracy "Assessing Degeneracy in Statistical Models of Social Networks" Mark S. Handcock, CSSS Working Paper #39