1. Statistical Analysis of Social Networks 1)From description to Inference: Confidence intervals for measures 2)QAP Models – (review) 1)Networks as independent variables 2)Networks as dependent variables 3)ERGM / Markov Chain Monte Carlo (MCMC) 4)Simulations from process to network http://www.soc.duke.edu/~jmoody77/s884/notes/stochasticNets.ppt

2. Statistical Analysis of Social Networks Confidence Intervals: Bootstraps and Jackknifes (Snijders & Borgatti, 1999) Goal: “Useful to have an indication of how precise a given description is, particularly when making comparisons between groups.” Assumes that “a researcher is interested in some descriptive statistic … and wishes to have a standard error for this descriptive statistic without making implausibly strong assumptions about how the network came about.”

3. Confidence Intervals: Bootstraps and Jackknifes (Snijders & Borgatti, 1999) Jackknifes. Given a dataset w. N sample elements, N artificial datasets are created by deleting each sample element in turn from the observed dataset. In standard practice, the formula for the standard error is then:

4. Obs i x s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 1 1 0.85. 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 2 2 0.70 0.70. 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 3 3 1.00 1.00 1.00. 1.00 1.00 1.00 1.00 1.00 1.00 1.00 4 4 0.59 0.59 0.59 0.59. 0.59 0.59 0.59 0.59 0.59 0.59 5 5 0.22 0.22 0.22 0.22 0.22. 0.22 0.22 0.22 0.22 0.22 6 6 0.69 0.69 0.69 0.69 0.69 0.69. 0.69 0.69 0.69 0.69 7 7 0.43 0.43 0.43 0.43 0.43 0.43 0.43. 0.43 0.43 0.43 8 8 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32. 0.32 0.32 9 9 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50. 0.50 10 10 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67. MEAN: 0.60 0.57 0.58 0.55 0.60 0.64 0.59 0.61 0.63 0.61 0.59 Jackknifes: Example on regular data

5. SE j = 0.0753 SE = 0.0753

6. Jackknifes: For networks For networks,we need to adjust the scaling parameter: Where Z -i is the network statistic calculated without vertex i, and Z- is the average of Z -1 … Z -N. This procedure will work for any network statistic Z, and UCINET will use it to test differences in network density.

7. Jackknifes: For networks An example based on the Trade data. Density, Std. Errors and confidence intervals for each matrix. DIP_DEN DIP_SEJ DIP_UB DIP_LB 0.6684783 0.0636125 0.7931588 0.5437978 CRUDE_DEN CRUDE_SEJ CRUDE_UB CRUDE_LB 0.5561594 0.0676669 0.6887866 0.4235323 FOOD_DEN FOOD_SEJ FOOD_UB FOOD_LB 0.5561594 0.0633776 0.6803794 0.4319394 MAN_DEN MAN_SEJ MAN_UB MAN_LB 0.5615942 0.0724143 0.7035263 0.4196621 MIN_DEN MIN_SEJ MIN_UB MIN_LB 0.2445652 0.0530224 0.3484891 0.1406414

8. Bootstrap In general, bootstrap techniques effectively treat the given sample as the population, then draw samples, with replacement, from the observed distribution. For networks, we draw random samples of the vertices, creating a new network Y* If i(k) = i(h), then randomly fill in the dyads based from the set of all possible dyads (I.e. fill in this cell with a random draw from the population).

9. Bootstrap For each bootstrap sample: Draw N random numbers, with replacement, from 1 to N, denoted i(1)..i(N) Construct Y* based on i(1)..i(N) Calculate the statistic of interest, called Z *m, Repeat this process M (=thousands) of times.

10. Bootstraps: Comparing density

11. BOOTSTRAP PAIRED SAMPLE T-TEST -------------------------------------------------------------------------------- Density of trade_min is: 0.2446 Density of trade_dip is: 0.6685 Difference in density is: -0.4239 Number of bootstrap samples: 5000 Variance of ties for trade_min: 0.1851 Variance of ties for trade_dip: 0.2220 Classical standard error of difference: 0.0272 Classical t-test (indep samples): -15.6096 Estimated bootstrap standard error for density of trade_min: 0.0458 Estimated bootstrap standard error for density of trade_dip: 0.0553 Bootstrap standard error of the difference (indep samples): 0.0719 95% confidence interval for the difference (indep samples): [-0.5648, -0.2831] bootstrap t-statistic (indep samples): -5.8994 Bootstrap SE for the difference (paired samples): 0.0430 95% bootstrap CI for the difference (paired samples): [-0.5082, -0.3396] t-statistic: -9.8547 Average bootstrap difference: -0.3972 Proportion of absolute differences as large as observed: 0.0002 Proportion of differences as large as observed: 1.0000 Proportion of differences as large as observed: 0.0002 Bootstraps: Comparing density

12. Measurement Sensitivity A related question: How confident can you be in any measure on an observed network, given the likelihood that observed ties are, in fact, observed with error? Implies that some of the observed 0s are in fact 1s and some of the 1s are in fact 0s. Suggests that we view the network not as a binary array of 0s and 1s, but instead a set of probabilities, such that: P ij = f(A ij ) We can then calculate the statistic of interest M times under different realizations of the network given Pij and get a distribution of the statistic of interest.

13. 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 p ij ? Range based on observed features (Sensitivity analysis) Outcome of a model based on observed patterns (ERGM)

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

15. Measurement Sensitivity All ties Reciprocated

16. Measurement Sensitivity

22. 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)

23. Assessing the correlation is straight forward, as we simply correlate each corresponding cell of the two matrices: Marriage 1 ACCIAIUOL 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 ALBIZZI 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 3 BARBADORI 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 4 BISCHERI 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 5 CASTELLAN 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 6 GINORI 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 GUADAGNI 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 8 LAMBERTES 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 9 MEDICI 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 10 PAZZI 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 11 PERUZZI 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 12 PUCCI 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 RIDOLFI 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 14 SALVIATI 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 15 STROZZI 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 16 TORNABUON 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 Business 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 4 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 5 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 6 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 7 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 8 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 9 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 10 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 11 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 Dyads: 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0 1 6 0 0 1 7 0 0 1 8 0 0 1 9 1 0 1 10 0 0 1 11 0 0 1 12 0 0 1 13 0 0 1 14 0 0 1 15 0 0 1 16 0 0 2 1 0 0 2 3 0 0 2 4 0 0 2 5 0 0 2 6 1 0 2 7 1 0 2 8 0 0 2 9 1 0 2 10 0 0 2 11 0 0 2 12 0 0 2 13 0 0 2 14 0 0 2 15 0 0 2 16 0 0 Correlation: 1 0.3718679 0.3718679 1 (review)

24. Comparing multiple networks: QAP But is the observed value statistically significant? Can’t use standard inference, since the assumptions are violated. Instead, we use a permutation approach. Essentially, we are asking whether the observed correlation is large (small) compared to that which we would get if the assignment of variables to nodes were random, but the interdependencies within variables were maintained. Do this by randomly sorting the rows and columns of the matrix, then re-estimating the correlation. (review)

25. Comparing multiple networks: QAP When you permute, you have to permute both the rows and the columns simultaneously to maintain the interdependencies in the data: ID ORIG A 0 1 2 3 4 B 0 0 1 2 3 C 0 0 0 1 2 D 0 0 0 0 1 E 0 0 0 0 0 Sorted A 0 3 1 2 4 D 0 0 0 0 1 B 0 2 0 1 3 C 0 1 0 0 2 E 0 0 0 0 0 (review)

26. Comparing multiple networks: QAP Procedure: 1.Calculate the observed correlation 2.for K iterations do: a) randomly sort one of the matrices b) recalculate the correlation c) store the outcome 3.compare the observed correlation to the distribution of correlations created by the random permutations. (review)

27. Comparing multiple networks: QAP (review)

28. Running QAP in UCINET: (review)

29. QAP MATRIX CORRELATION -------------------------------------------------------------------------------- Observed matrix: PadgBUS Structure matrix: PadgMAR # of Permutations: 2500 Random seed: 356 Univariate statistics 1 2 PadgBUS PadgMAR ------- ------- 1 Mean 0.125 0.167 2 Std Dev 0.331 0.373 3 Sum 30.000 40.000 4 Variance 0.109 0.139 5 SSQ 30.000 40.000 6 MCSSQ 26.250 33.333 7 Euc Norm 5.477 6.325 8 Minimum 0.000 0.000 9 Maximum 1.000 1.000 10 N of Obs 240.000 240.000 Hubert's gamma: 16.000 Bivariate Statistics 1 2 3 4 5 6 7 Value Signif Avg SD P(Large) P(Small) NPerm --------- --------- --------- --------- --------- --------- --------- 1 Pearson Correlation: 0.372 0.000 0.001 0.092 0.000 1.000 2500.000 2 Simple Matching: 0.842 0.000 0.750 0.027 0.000 1.000 2500.000 3 Jaccard Coefficient: 0.296 0.000 0.079 0.046 0.000 1.000 2500.000 4 Goodman-Kruskal Gamma: 0.797 0.000 -0.064 0.382 0.000 1.000 2500.000 5 Hamming Distance: 38.000 0.000 59.908 5.581 1.000 0.000 2500.000 (review)

30. Running QAP in UCINET: Regression Using the same logic,we can estimate alternative models, such as regression. Only complication is that you need to permute all of the independent matrices in the same way each iteration. (review)

31. NODE ADJMAT SAMERCE SAMESEX 1 0 1 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 2 1 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1 3 1 1 0 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 4 1 0 0 0 1 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 0 1 1 0 5 0 0 1 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 6 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 7 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 8 0 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 0 1 0 0 9 0 0 0 0 0 1 0 1 0 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 Simple Example: (review)

32. Distance (D ij =abs(Y i -Y j ).000.277.228.181.278.298.095.307.481.277.000.049.096.555.575.182.584.758.228.049.000.047.506.526.134.535.710.181.096.047.000.459.479.087.488.663.278.555.506.459.000.020.372.029.204.298.575.526.479.020.000.392.009.184.095.182.134.087.372.392.000.401.576.307.584.535.488.029.009.401.000.175.481.758.710.663.204.184.576.175.000 Y 0.32 0.59 0.54 0.50 0.04 0.02 0.41 0.01 -0.17 Simple Example: (review)

33. Simple Example: (review)

34. Simple Example, continuous dep. variable: # of permutations: 2000 Diagonal valid? NO Random seed: 995 Dependent variable: EX_SIM Expected values: C:\moody\Classes\soc884\examples\UCINET\mrqap-predicted Independent variables: EX_SSEX EX_SRCE EX_ADJ Number of valid observations among the X variables = 72 N = 72 Number of permutations performed: 1999 MODEL FIT R-square Adj R-Sqr Probability # of Obs -------- --------- ----------- ----------- 0.289 0.269 0.059 72 REGRESSION COEFFICIENTS Un-stdized Stdized Proportion Proportion Independent Coefficient Coefficient Significance As Large As Small ----------- ----------- ----------- ------------ ----------- ----------- Intercept 0.460139 0.000000 0.034 0.034 0.966 EX_SSEX -0.073787 -0.170620 0.140 0.860 0.140 EX_SRCE -0.020472 -0.047338 0.272 0.728 0.272 EX_ADJ -0.239896 -0.536211 0.012 0.988 0.012 (review)

35. We can also model the network as a dependent variable. In the next example, I model diplomacy as a function of trade, using the country data. Note that using UCINET, this gives a linear probability model, OLS on a 0-1 dependent variable. This is often not optimal… (review)

36. MULTIPLE REGRESSION QAP TRADE DIPLOMACY -------------------------------------------------------------------------------- # of permutations: 2000 Diagonal valid? NO Random seed: 1000 Dependent variable: trade_dip Expected values: C:\moody\Classes\soc884\examples\UCINET\mrqap-predicted Independent variables: TRADE_FOOD TRADE_CRUDE TRADE_MAN TRADE_MIN Number of valid observations among the X variables = 552 N = 552 Number of permutations performed: 1999 MODEL FIT R-square Adj R-Sqr Probability # of Obs -------- --------- ----------- ----------- 0.317 0.314 0.000 552 REGRESSION COEFFICIENTS Un-stdized Stdized Proportion Proportion Independent Coefficient Coefficient Significance As Large As Small -------------- ----------- ----------- ------------ ----------- ----------- Intercept 0.339308 0.000000 1.000 1.000 0.000 TRADE_FOOD 0.049975 0.052744 0.200 0.200 0.800 TRADE_CRUDE 0.109233 0.115284 0.033 0.033 0.967 TRADE_MAN 0.367435 0.387285 0.000 0.000 1.000 TRADE_MIN 0.140151 0.127965 0.058 0.058 0.942 Expected values saved as dataset mrqap-predicted Valid observations saved as dataset mrqap-valid (review)

37. One solution is to use a QAP Logit model. Note (yet) implemented in UCINET, but you can use DAMN or SAS. Let’s look at the country trade data again, using a logit model. DAMN *** REGRESSION PARAMETERS *** man coefficient= 1.80356 p-value=0.00000 food coefficient= 0.30928 p-value=0.14867 crude coefficient= 0.59624 p-value=0.00867 min coefficient= 1.61641 p-value=0.00000 constant coefficient= -0.78063 p-value=0.00000 Final Likelihood: -252.366402 QAP Results: Logit model coefficients Significance results are equal to the proportion of QAP parameters that are as *small* as the observed parameters. PARAMETERS COEF SIG INTERCEPT -.949.0010 TMAN 1.863.9980 TFOOD.3914.8420 TCRUDE.6667.9800 TMIN 1.658.9980 (review)

38. Modeling Social Networks parametrically: Exponential Random Graph models – ERGM/p* A long research tradition in statistics and random graph theory has lead to parametric models of networks. These are models of the entire graph, though as we will see they often work on the dyads in the graph to be estimated. Substantively, the approach is to ask whether the graph in question is an element of the class of all random graphs with the given known elements. For example, all graphs with 5 nodes and 3 edges, or, put probabilistically, the probability of observing the current graph given the conditions.

39. Random Graphs and Conditional Expectations The basis for the statistical modeling of graphs rests on random graph theory. Simply put, Random graph theory asks what properties do we expect when ties (X ij ) form at random. The simplest random graph is the Bernoulli random graph, where X ij is a constant and independent: says simply that each edge in the graph has an independent probability of being 1/0. Typically this is an uninteresting distribution of graphs, and we want to know what the graph looks like conditional on other features of the graph.

40. Random Graphs and Conditional Expectations A Bernoulli graph is only conditional on the expected number of edges. So effectively we ask “What is the probability of observing the graph we have, given the set of all possible graphs with the same number of edges.” We might, instead, want to condition on the degree distribution (sent or received) or all graphs with a particular dyad distribution (same number of Mutual, Asymmetric and Null dyads). Closed form solutions for some graph statistics (like the triad census) are known for out-degree, in-degree and MAN (but not all 3 simultaneously).

41. Random Graphs and Conditional Expectations PAJEK gives you the unconditional expected values: ------------------------------------------------------------------------------ Triadic Census 2. i:\people\jwm\s884\homework\prison.net (67) ------------------------------------------------------------------------------ Working... ---------------------------------------------------------------------------- Type Number of triads (ni) Expected (ei) (ni-ei)/ei ---------------------------------------------------------------------------- 1 - 003 39221 37227.47 0.05 2 - 012 5860 9587.83 -0.39 3 - 102 2336 205.78 10.35 4 - 021D 61 205.78 -0.70 5 - 021U 80 205.78 -0.61 6 - 021C 103 411.55 -0.75 7 - 111D 105 17.67 4.94 8 - 111U 69 17.67 2.91 9 - 030T 13 17.67 -0.26 10 - 030C 1 5.89 -0.83 11 - 201 12 0.38 30.65 12 - 120D 15 0.38 38.56 13 - 120U 7 0.38 17.46 14 - 120C 5 0.76 5.59 15 - 210 12 0.03 367.67 16 - 300 5 0.00 21471.04 ---------------------------------------------------------------------------- Chi-Square: 137414.3919*** 6 cells (37.50%) have expected frequencies less than 5. The minimum expected cell frequency is 0.00.

42. Random Graphs and Conditional Expectations SPAN gives you the (X|MAN) distributions: Triad Census T TPCNT PU EVT VARTU STDDIF 003 39221 0.8187 0.8194 39251 427.69 -1.472 012 5860 0.1223 0.1213 5810.8 1053.5 1.5156 102 2336 0.0488 0.0476 2278.7 321.01 3.1954 021D 61 0.0013 0.0015 70.949 67.37 -1.212 021U 80 0.0017 0.0015 70.949 67.37 1.1027 021C 103 0.0022 0.003 141.9 127.58 -3.444 111D 105 0.0022 0.0023 112.39 103.57 -0.727 111U 69 0.0014 0.0023 112.39 103.57 -4.264 030T 13 0.0003 0.0001 3.4292 3.3956 5.1939 030C 1 209E-7 239E-7 1.1431 1.1393 -0.134 201 12 0.0003 0.0009 42.974 38.123 -5.017 120D 15 0.0003 286E-7 1.3717 1.368 11.652 120U 7 0.0001 286E-7 1.3717 1.368 4.8122 120C 5 0.0001 573E-7 2.7433 2.7285 1.3662 210 12 0.0003 442E-7 2.1186 2.1023 6.8151 300 5 0.0001 549E-8 0.2631 0.2621 9.2522

43. 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) Best current update: http://www.jstatsoft.org/v24 http://www.jstatsoft.org/v24

44. 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.

45. Modeling Social Networks parametrically: ERGM approaches

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

47. 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” ;-)

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

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

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

51. 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: Modeling Social Networks parametrically: ERGM approaches

52. Modeling Social Networks parametrically: ERGM approaches

53. 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. Modeling Social Networks parametrically: ERGM approaches

54. 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?

55. 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

56. 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. proc logistic descending data=dydat; model nom =; run; quit; ---see results copy coef --- data chk; x=exp(-0.5705)/(1+exp(-0.5705)); run; proc print data=chk; run;

57. Modeling Social Networks parametrically: ERGM approaches

58. 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! Modeling Social Networks parametrically: ERGM approaches The logit model estimation procedure was popularized by Wasserman & colleagues, and a good guide to this approach is:

59. Modeling Social Networks parametrically: ERGM approaches Parameters that are often fit include: 1)Expansiveness and attractiveness parameters. = dummies for each sender/receiver in the network 2)Degree distribution 3)Mutuality 4)Group membership (and all other parameters by group) 5)Transitivity / Intransitivity 6)K-in-stars, k-out-stars 7)Cyclicity 8)Node-level covariates (Matching, difference) 9)Edge-level covariates (dyad-level features such as exposure) 10)Temporal data – such as relations in prior waves.

60. Modeling Social Networks parametrically: Exponential Random Graph Models

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

62. -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Same Race SES GPA Both Smoke College Drinking Fight Reciprocity Same Sex Same Clubs Transitivity Intransitivity Same Grade Network Model Coefficients, In school Networks Complete Network Analysis Stochastic Network Analysis An example:

63. 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.

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

65. Modeling Social Networks parametrically: Exponential Random Graph Models

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

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

68. Modeling Social Networks parametrically: Exponential Random Graph Models: Example 2: Hierarchy & Context in AH

69. Modeling Social Networks parametrically: Exponential Random Graph Models: Example 2: Hierarchy & Context in AH

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

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

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

73. Generating Random Graph Samples A conceptual merge between exponential random graph models and QAP/sensitivity models is to attempt to identify a sample of graphs from the universe you are trying to model. That is, generate X empirically, then compare z(x) to see how likely a measure on x would be given X. The difficulty, however, is generating X.

74. Generating Random Graph Samples The first option would be to generate all isomorphic graphs within a given constraint. This is possible for small graphs, but the number gets large fast. For a network with 3 nodes, there are 16 possible directed graphs. For a network with 4 nodes, there are 218, for 5 nodes 9608, for 6 nodes1,540,944, and so on… So, the best approach is to sample from the universe, but, of course, if you had the universe you wouldn’t need to sample from it. How do you sample from a population you haven’t observed? (a) use a construction algorithm that generates a random graph with known constraints (b) use a ERGM model like above.

75. Generating Random Graph Samples Tom Snijders has a program called ZO (Zero-One) for doing this. http://stat.gamma.rug.nl/snijders/ The program only works well for smallish networks (less than ~100)

76. Generating Random Networks with Structural Constraints. General strategy: Assign arcs at random within the cells of an adjacency matrix until the desired graph is achieved. Process. 1) Define the pool of open arcs. Any cells of the g by g matrix which are structurally zero are not allowed. 5 3 4 5 1 3 0 2 6 7 5345130267

77. Generating Random Networks with Structural Constraints. 2) Randomly draw an element from the available set. 5 3 4 5 1 3 0 2 6 7 5345130267

78. Generating Random Networks with Structural Constraints. 5 3 4 5 1 3 0 2 6 7 5345130267 3) Check to see if selected cell meets the structural condition. 4) If a condition is met,then remove any implicated cells from the pool. 5 3 4 5 1 3 0 2 6 7 5345130267

79. 5) Check for Identification: Does the last arc imply the set of arcs for another? 5 3 4 5 1 3 0 2 6 7 5345130267 Generating Random Networks with Structural Constraints. In this example, there are only 7 available spots left in the last row, equal to the number needed to fill that row condition.

80. Generating Random Networks with Structural Constraints. Process: 1) Identify the pool of open cells. 2) Randomly draw an arc from this pool. 3) Check the structural conditions against this arc. 4) If structural conditions are met, then remove implied cells from the pool. 5) Check for identification of other arcs. Types of constraints: Structural Patterns, such as the in and out degree, prohibition against cycles, etc. Category Mixing Constraints. Nodes in category i restricted to nodes from category j. Event Counts. Number of mutual arcs, number of ties between group i and j, etc.

81. Social Relations at “Holy Trinity” School. 7th Grade 8th Grade 9th Grade 10th Grade 11th Grade 12th Grade g = 74 l = 466  =.086 M=108 Transitivity:.357 Mean Degree: 6.3

82. 0 50 100 150 200 250 Number of Networks 510152025303540 Number of Mutual Dyads Z.O. RANFIX Number of Mutual Dyads 2000 Networks, with fixed In and Out Degree

83. Distribution of Selected Triad Types Simulations compared to Observed 0 50 100 150 200 250 300 350 Count 300210 120C 120U120D 201 030T Z.O. RANFIX Observed. 2000 random networks, with fixed in and out degree.

84. Romantic Networks