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 p ij ? 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 All ties Reciprocated

4. 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 Key historical references: - Holland and Leinhardt (1981) JASA - Frank and Strauss (1986) JASA - Wasserman and Faust (1994) – Chap 15 & 16 -Wasserman and Pattison (1996) Good practical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 http://www.jstatsoft.org/v24 Great tutorial: http://statnet.csde.washington.edu/workshops/SUNBELT/EUSN/ergm/er gm_tutorial.html (last year’s sunbelt) http://statnet.csde.washington.edu/workshops/SUNBELT/EUSN/ergm/er gm_tutorial.html Or -https://statnet.csde.washington.edu/trac/wiki/Sunbelt2014 (lots of how to slides)https://statnet.csde.washington.edu/trac/wiki/Sunbelt2014

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. 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

17. 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

18. 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

19. 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

20. Modeling Social Networks parametrically: ERGM approaches

21. 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

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. 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;

25. Modeling Social Networks parametrically: ERGM approaches

26. 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:

27. 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.

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

51. Modeling Social Networks parametrically: Exponential Random Graph Models: Quick example (demo)

52. Modeling Social Networks parametrically: Latent Space Models

53. Modeling Social Networks parametrically: Latent Space Models Z = a dimension in some unknown space that, once accounted for makes ties independent. Z is effectively chosen with respect to some latent cluster-space, G. These “groups” define different social sources for association.

54. Modeling Social Networks parametrically: Latent Space Models Z = a dimension in some unknown space that, once accounted for makes ties independent. Z is effectively chosen with respect to some latent cluster-space, G. These “groups” define different social sources for association.

55. Modeling Social Networks parametrically: Latent Space Models

56. Modeling Social Networks parametrically: Latent Space Models Prosper data, with three groups

57. Modeling Social Networks parametrically: Latent Space Models Prosper data, with three groups (posterior density plots)

58. Modeling Social Networks parametrically: Latent Space Models …note there is a non-R option.,..

59. 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.

60. 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.

61. Romantic Networks Generating Random Graph Samples

62. Romantic Networks Generating Random Graph Samples

63. Romantic Networks Generating Random Graph Samples A draw from the simulation, this is what appeared in “Glamour”

64. Edge-matching random permutation Can easily generate networks with appropriate degree distributions by generating “edge stems” and sorting: a Degree: 1: 2 2: 2 3: 1 b d i =1 cc d i =2 ddff d i =3 f (need to ensure you have a valid edge list!) Generating Random Graph Samples

65. Edge-matching random permutation Generating Random Graph Samples

66. Partner Distribution Component Size/Shape Emergent Connectivity in low-degree networks Generating Random Graph Samples

67. Development of STD cores in low-degree networks: rapid transition without stars. Complete Network Analysis Network Connections: Connectivity

68. Extend this view across the space of low-degree distributions defined by shape and volume... Complete Network Analysis Network Connections: Connectivity

69. Extend this view across the space of low-degree distributions defined by shape and volume... Complete Network Analysis Network Connections: Connectivity

70. Complete Network Analysis Network Connections: Connectivity ERGMs make it (fairly) easy to simulate networks from models. Simple: simulation from an estimated ERGM (this is how the GOF function works) Simple II: simulate from a pre-defined ERGM formula (i.e. set the parameters by hand) A little harder: Simulate from EGO networks. Here you can use ERGM to match the observed distribution for mixing by node characteristics reported in an ego-network survey. Can use degree, attribute mixing, A bit harder: fit global structure features using ego-nets by modeling distribution of sub-structures (see Jeff Smith’s work)

71. Generating Random Graph Samples Model based estimates ERGM to simulate networks from Add Health

72. Modeling Network Dynamics Rule-based simulation models Rule-Based simulation models: The network-science approach to dynamic networks has been to identify toy behavioral models and play out the implications of these models for network dynamics. Focus is typically on how the network evolves (or reaches a steady stat). dynamics OF networks Balance, preferential attachment, voter models dynamics ON networks diffusion simulations These are usually agent-based models, difficult to specify – tradeoff in simplicity & realism.

73. Modeling Network Dynamics Descriptive dynamic techniques Goal here is to make sense of how networks change or how things flow through them using a clear measurement / metrics approach. Challenge is defining the network.

74. Time and Social Networks Examples of looking at change in networks: Roy and interlocking directorates (ASR 1983, 248-257) Non-financial interlocks: 1886 - 1890

75. Time and Social Networks Examples of looking at change in networks: Roy and interlocking directorates (ASR 1983, 248-257) Non-financial interlocks: 1891 - 1895

76. Time and Social Networks Examples of looking at change in networks: Roy and interlocking directorates (ASR 1983, 248-257) Non-financial interlocks: 1896 - 1900

77. Time and Social Networks Examples of looking at change in networks: Roy and interlocking directorates (ASR 1983, 248-257) Non-financial interlocks: 1901 - 1905

78. Bearman and Everett: The Structure of Social Protest 1 3 2 4 5 6 1 3 2 4 5 7 6 1 3 2 4 5 (‘61-63) (‘66-68) (‘71-73) 7 6 1 3 2 4 5 (‘76-78) (‘81-83) 7 5 1 6 3 4 2 See paper for group compositions

79. Data on drug users in Colorado Springs, over 5 years

84. http://csde.washington.edu/statnet/movies/ConcurrencyAndReachability.mov Animation captures much of the dynamism we care about: STD Diffusion Representing dynamic networks?

85. Animation captures much of the dynamism we care about: Representing dynamic networks?

86. Animation captures much of the dynamism we care about: Representing dynamic networks?

87. Modeling Network Dynamics Random Graph models Panel ERGM: Simply want to account for effect of past structures, you can add temporal covariates to the standard ERGM. Really only good for two waves. STERGM: Separable Temporal ERGM. This is a two-equation model, with one equation for the formation of ties, a 2 nd for the dissolution of ties. Goal is like ERGM, to explain the dynamics of the network. http://statnet.csde.washington.edu/workshops/SUNBELT/current/tergm/tergm_tutori al.pdf RELEVENT: Relational Events Model. This is really a model of action on a network  think of conversation events or similar. Dynamic networks of very short duration events. http://statnet.csde.washington.edu/workshops/SUNBELT/current/relevent/statnet_su nbelt2014_relevent.pdf SIENA: Stochastic Actor Oriented Model (SAOM). Used to disentangle selection from influence, by jointly modeling both as functions of each other. Multi-equation model, simplest is one for behavior & one for network formation. Intro: https://www.stats.ox.ac.uk/~snijders/siena/SnijdersSteglichVdBunt2009.pdfhttps://www.stats.ox.ac.uk/~snijders/siena/SnijdersSteglichVdBunt2009.pdf Manual: https://www.stats.ox.ac.uk/~snijders/siena/RSiena_Manual.pdfhttps://www.stats.ox.ac.uk/~snijders/siena/RSiena_Manual.pdf

88. Modeling Network Dynamics Random Graph models: STERGM http://statnet.csde.washington.edu/workshops/SUNBELT/current/tergm/tergm_tutorial.html slides adapted from the workshop materials: http://statnet.csde.washington.edu/EpiModel/nme/index.htmlhttp://statnet.csde.washington.edu/EpiModel/nme/index.html

89. Modeling Network Dynamics Random Graph models: STERGM http://statnet.csde.washington.edu/workshops/SUNBELT/current/tergm/tergm_tutorial.html slides adapted from the workshop materials: http://statnet.csde.washington.edu/EpiModel/nme/index.htmlhttp://statnet.csde.washington.edu/EpiModel/nme/index.html Under certain assumptions, you can model a single network w. average duration information (assumes an equilibrium process)

90. Modeling Network Dynamics Random Graph models: STERGM samp.fit <- stergm(samp, formation= ~edges+mutual+cyclicalties+transitiveties, dissolution = ~edges+mutual+cyclicalties+transitiveties, estimate = "CMLE", times=1:3 )

91. SIENA

92. SIENA: Key Assumptions of the model

93. SIENA

95. Key element is how actors make changes. This is based on an evaluation of “utility” functions, similar to discrete choice models. The model is then implemented as an actor-simulation, where actors are striving to maximize their utility. note Tom is adamant that this is an “as if” model – no clear ontological commitment to a “choice” model!

96. Modeling Network Dynamics Random Graph models: Siena

97. Modeling Network Dynamics Random Graph models: Siena Osgood, D. W., Ragan, D. T., Wallace, L., Gest, S. D., Feinberg, M. E., & Moody, J. 2013. “Peers and the emergence of alcohol use: Influence and selection processes in adolescent friendship networks.” Journal of Research on Adolescence 23:500–512.

99. Modeling Network Dynamics Random Graph models: RelEvent For repeated interactions amongst nodes