1. Statistical Analysis of Social Networks
From description to Inference: Confidence intervals for measures
QAP models:
Networks as independent variables
Networks as dependent variables
P* and Markov Chain Monte Carlo (MCMC)

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. Jackknifes: Example on regular data
Obs i x s1 s2 s3 s4 s5 s6 s7 s8 s9 s10
MEAN:

5. Jackknifes: Example on regular data
SEj =
SE =

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
CRUDE_DEN CRUDE_SEJ CRUDE_UB CRUDE_LB
FOOD_DEN FOOD_SEJ FOOD_UB FOOD_LB
MAN_DEN MAN_SEJ MAN_UB MAN_LB
MIN_DEN MIN_SEJ MIN_UB MIN_LB

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. Bootstraps: Comparing density
BOOTSTRAP PAIRED SAMPLE T-TEST
Density of trade_min is:
Density of trade_dip is:
Difference in density is:
Number of bootstrap samples: 5000
Variance of ties for trade_min:
Variance of ties for trade_dip:
Classical standard error of difference:
Classical t-test (indep samples):
Estimated bootstrap standard error for density of trade_min:
Estimated bootstrap standard error for density of trade_dip:
Bootstrap standard error of the difference (indep samples):
95% confidence interval for the difference (indep samples): [ , ]
bootstrap t-statistic (indep samples):
Bootstrap SE for the difference (paired samples):
95% bootstrap CI for the difference (paired samples): [ , ]
t-statistic:
Average bootstrap difference:
Proportion of absolute differences as large as observed:
Proportion of differences as large as observed:
Proportion of differences as large as observed:

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 probabiliites, such that:
Pij = f(Aij)
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 pij?
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

17. Measurement Sensitivity

18. Measurement Sensitivity

19. Measurement Sensitivity

20. Measurement Sensitivity

21. Measurement Sensitivity

22. Modeling Social Networks parametrically:
p* approaches
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.

23. 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 (Xij) form at random.
The simplest random graph is the Bernoulli random graph, where Xij is a constant and independent: says simply that each edge in the graph has an independent probability of being “on.”
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.

24. 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).

25. 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 D U C D U T C D U C
Chi-Square: ***
6 cells (37.50%) have expected frequencies less than 5.
The minimum expected cell frequency is 0.00.

26. Random Graphs and Conditional Expectations
SPAN gives you the (X|MAN) distributions:
Triad Census
T TPCNT PU EVT VARTU STDDIF
021D
021U
021C
111D
111U
030T
030C E-7 239E
120D E
120U E
120C E E E

27. Modeling Social Networks parametrically: p* 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 references:
- Holland and Leinhardt (1981) JASA
- Frank and Strauss (1986) JASA
- Wasserman and Faust (1994) – Chap 15 & 16
- Wasserman and Pattison (1996)
Thanks to Mark Handcock for sharing some figures/slides about these models.

28. Modeling Social Networks parametrically:
p* 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.

29. Modeling Social Networks parametrically:
p* 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.

30. Modeling Social Networks parametrically:
p* 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

31. Modeling Social Networks parametrically:
p* 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:

32. Modeling Social Networks parametrically:
p* approaches

33. Modeling Social Networks parametrically:
p* 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.

34. Modeling Social Networks parametrically: p* approaches
Fitting p* models.
I highly recommend working through the p* primer examples, which can be found at:
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.

35. Modeling Social Networks parametrically: p* approaches: Fitting models
We can model this network based on parameters for overall degree of Choice (), Differential Choice Within Positions (W), Mutuality(), Differential Mutuality Within Positions (W), and Transitivity (T).
The vector of model parameters to be estimated is:  = {  W  W T }.

36. Modeling Social Networks parametrically:
p* approaches: Fitting models
The first step is to calculate the vector of change statistics. This is done by first calculating the value of the statistic if the ij tie is present, then if it is absent, then take the difference. The program PREPSTAR does this for you (see also pspar – for large networks:
For example, the simple choice parameter is Xij, so if forced present Xij=1, if absent, Xij=0, the difference is going to be 1. Since this is true for every dyad, it is a constant, equivalent to the model intercept.

37. The model described above would be written in W&P notation as:
z1(x) = L = i,j Xij is the statistic for the Choice parameter, ,
z2(x) = LW = i,j Xij ij is the statistic for the Choice Within Positions parameter, W,
z3(x) = M = i<j Xij Xji is the statistic for the Mutuality parameter, ,
z4(x) = MW = i<j Xij Xji ij is the statistic for the Mutuality Within Positions parameter, W,
z5(x) = TT = i,j,k Xij Xjk Xik is the statistic for the Transitivity parameter, T.
Note that the indicator variable ij=1 if actors i and j are in the same position, and 0 otherwise.

38. Looking over the first few cases:
Obs i j tie L L_W M M_W T_T

39. Modeling Social Networks parametrically: p* approaches: Fitting models
proc logistic descending ;
tie = l lw m mw tt / noint;
run;

40. Modeling Social Networks parametrically: p* approaches: Fitting models
One practical problem is that the resulting values are often quite correlated, making estimation difficult. This is particularly difficult with “star” parameters.
lw m mw tt lw
< m
< mw
< < tt

41. Modeling Social Networks parametrically:
p* approaches: Fitting models
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

42. Modeling Social Networks parametrically:
Exponential Random Graph Models
In practice, p* models are difficult to estimate, and we have no good sense of how approximate the PMLE is.
A recent generalization is to use MCMC methods to better estimate the parameters. The following slides are courtesy of David Hunter at Penn State.

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

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

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

57. 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.

58. 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.

59. Generating Random Graph Samples
Tom Snijders has a program called ZO (Zero-One) for doing this.
The program only works well for smallish networks (less than ~100)

60. 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 2 6 7 7 1 3 6 5 3 4 2 5

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

62. 3) Check to see if selected cell meets the structural condition.
Generating Random Networks with Structural Constraints.
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 5 3 3 4 4 5 5 1 1 3 3 2 2 6 6 7 7 7 1 3 6 5 3 4 2 5 7 1 3 6 5 3 4 2 5

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

64. 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.

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

66. Number of Mutual Dyads 250 200 150 Z.O. Number of Networks 100 RANFIX
2000 Networks, with fixed In and Out Degree
250
200
150
Z.O.
Number of Networks
100
RANFIX 50 5 10 15 20 25 30 35 40
Number of Mutual Dyads

67. Distribution of Selected Triad Types
Simulations compared to Observed
2000 random networks, with fixed in and out degree.
350
300
250
200
Count
Z.O.
150
100
RANFIX 50
Observed.
030T
201
120D
120U
120C
210
300

68. Romantic Networks

69. Romantic Networks