Statistical Methods for Network Analysis: Exponential Random Graph Models

Transcription

1 Day 2: Network Modeling Statistical Methods for Network Analysis: Exponential Random Graph Models NMID workshop September 17 21, 2012 Prof. Martina Morris Prof. Steven Goodreau Supported by the US National Institutes of Health

2 Example: clustering of ties One of the defining features of social networks they are more clustered than a simple random graph Color = grade Color = race Add Health school friendship network Descriptive statistic in the literature: clustering coefficient (the % of all 2-stars that are closed 3-cycles)

3 But why? Friend of a friend, or birds of a feather? Two theories about the process that generates 3cycles: 1. Homophily: People tend to chose friends who are like them, in grade, race, etc. ( birds of a feather ), triad closure is a by-product 2. Transitivity: People who have friends in common tend to become friends ( friend of a friend ), closure is the key process So, for three actors in the same grade Cycle-closing tie forms because of transitivity but also homophily

4 Transitivity and homophily are confounded But not completely Any tie may be : Triangle forming: Grade Pair: Yes No Within Both Homophily Across Transitivity NA Suggests we should be able to disentangle these statistically

5 Generative Model: Basic idea We want to model the probability of a tie as a function of things like: Nodal attributes (that influence degree and mixing) The propensity for certain configurations (like 3-cycles) The ties may be dependent Note that nodal attribute effects do not induce dyad dependence So we model the joint distribution directly

6 Generative Model: Testing Suppose kids have a tendency to become friends with their friends friends, and this is the only generative process occurring. Presumably, this would mean that you would observe more triangles than expected by chance in the graph. How would you test this in an empirical dataset? You begin by observing x triangles. You then determine the probability of observing x or more triangles in a network of your size for which ties are distributed randomly (a random graph ), and see if this is less than 5%. To do this, you might try enumerating all possible networks for a fixed number of nodes, and count their triangles.

7 Example: for 4 nodes: # of dyads is 4*3/2 = 6 # of possible networks = 2 6 = 64 for 10 nodes: # of dyads is 10*9/2 = 45 # of possible networks = trillion for 20 nodes: # of dyads is 20*19/2 = 190 # of possible networks = If one is only interested in calculating probabilities of triangle counts in a random graph, one can use theorems from graph theory. However, we have already seen that other processes (for example, homophily) can affect this statistic as well. Testing arbitrary combinations of generative processes requires a general statistical framework

8 Exponential Random Graph Model (ERGM) Probability of observing a graph (set of relationships) y on a set of nodes X : exp where: z = vector of network statistics = vector of model parameters c = numerator summed over all possible networks on node set X Exponential family model, so has well understood statistical properties Bahadur (1961), Besag (1974), Frank (1986) Very general and flexible

9 ERG Model specification z ( x) The z y terms in the model represent network statistics Counts of network configurations e.g., edges, 2-stars, 3-cycles, ties where nodal attributes match The number of terms can range from minimal (# of edges) to saturated (one term for every dyad) Model specification involves: 1. choosing the set of network statistics 2. choosing the homogeneity constraints on the parameter

10 ERG Model specification: Network Statistics Example models: p1 (Holland and Leinhardt 1981) z(x) I i (x) = number of in-ties for each actor O i (x) = number of out-ties for each actor M(x) = number of mutual dyads (i j, j i) Markov graphs (Frank and Strauss 1986) E(x) = number of ties (undirected) S k (x) = number of k-stars T(x) = number of 3cycles p* / ERGM (Wasserman and Pattison 1996) the most general form

11 Equivalent representation: The conditional probability of a tie This expression for the probability of a network under a given model is equivalent to an expression for the probability of an individual tie conditional on the rest of the network. That is, PY ( y) exp z( y) c can be re expressed as logit(p( Y ij PY ( ij 1 rest of the graph) 1 rest of the graph)) = log PY ( ij 0 rest of the graph) z( y) where zy ( ) represents the change in z y) when Y ij is toggled between 0 and 1

12 ERG Models: Homogeneity Constraints on p1 models have two types of network statistics: degrees, and mutuality no homogeneity constraints on degree parameters every node has a unique parameter, fits nodal degrees exactly homogeneity constraint on the mutuality term so all dyads are homogeneous with respect to reciprocity ERGMs in general have many more network configuration terms but researchers have typically imposed homogeneity constraints on all of these terms (not always a good idea, it turns out) There is also a middle ground: heterogeneity by group # edges # of edges by race # of edges between nodes of the same race # of 3-cycles # of racially homogenous 3-cycles # of racially heterogeneous 3-cycles

13 ERG Model Estimation exp zx ( ) P( X x) exp This is the likelihood function to maximize c - Normalizing constant c makes direct ML estimation of the θ vector impossible - e.g., with 50 nodes, there are graphs - Old approach: pseudolikelihood based on logistic regression approximation Besag 1974, Strauss & Ikeda 1990) - not good when dependence among ties is strong (Geyer and Thompson 1992, Handcock 2003) - MCMC is really the right approach (Geyer and Thompson 1992, Crouch et al. 1998) - theoretically guarantees estimation, but implementation for networks has been a challenge

14 The stumbling block: Model Degeneracy Definition: The graph used to estimate a model is in fact extremely unlikely under this model. Instead, the model places almost all of its probability mass on graphs that bear no resemblance to the original graph (Handcock 2003).

15 What degeneracy looks like The average is exactly what we wanted. All possible 3cycles But none of the graphs will have this average density and clustering. So this model would not have produced a graph with the values we want. And if you try to estimate this model on a graph with those features, it will bounce back and forth between the two regions and not converge.

16 Morals: Network models do not work like linear models. the dyadic dependent terms have highly nonlinear effects Start simple and slowly add complexity does not work here. simple models are most likely degenerate It is going to take some time to develop intuition.

17 ERG Models: new model statistics To capture clustering use a different function of the triad counts 1. Old: t x = # of completed 3-cycles in the graph or C x = 3-cycles as a percent of all 2-stars Here, every 3-cycle here has the same impact, 2. New : Weight the marginal impact of the number of shared partners geometrically weighted edgewise shared partner (GWESP) statistic 1 1 p i = # of ties btwn nodes with i partners in common 3. The more partners in common, the more likely two nodes are linked, but with a declining marginal return (Snijders et al. 2004, Hunter and Handcock 2004)

18 Back to the friendship networks Grade 7 Grade 8 Grade 9 Grade 10 Grade 11 Grade 12 Grade NA White (non-hispanic) Black (non-hispanic) Hispanic (of any race) Asian / Native Am / Other (non-hispanic) Race NA How much of the clustering is due to homophily, and how much to transitivity?

19 Test this by comparing four models Model Edges Network Statistics z(x) # of edges Edges + Attributes (homophily) Edges + GWESP (transitivity) Edges + Attributes + GWESP (both) # of edges # of edges for each race, sex,grade # of edges that are within-race, within-grade, within-sex # of edges weighted shared partners # of edges # of edges for each race, sex,grade # of edges that are within-race, within-grade, within-sex weighted shared partners

20 Testing goodness of fit Traditional GOF stats can be used (AIC, BIC) We also take another approach We are interested in how well we fit aggregate properties of the network structure that we did not include as model terms This helps to identify what the model gets wrong We use 3 higher order statistics: Degree distribution Shared partner distribution (non-parametric) (local clustering) Geodesic distance distribution (global clustering)

21 DATA MODEL MODEL COEFFICIENTS SIMULATED DATA (draws from the prob. dist.) HIGHER ORDER GRAPH STATISTICS OF DATA HIGHER ORDER GRAPH STATISTICS OF SIMULATED DATA GOODNESS OF FIT OF MODEL TO DATA

22 Goodness of fit measure 1: degree distribution Data: node Model: Bernoulli (i.e. edges only)

23 Goodness of fit measure 1: degree distribution Data: node Model: Bernoulli (i.e. edges only)

24 Goodness of fit measure 2: ESP distribution (local clustering) Data: Model: Bernoulli (i.e. edges only) This edge has an ESP value of 3

25 Goodness of fit measure 2: ESP distribution (local clustering) Data: Model: Bernoulli (i.e. edges only) This edge has an ESP value of 3

26 Goodness of fit measure 3: geodesic distribution (global clustering) Data: Model: Bernoulli (i.e. edges only) C A B A/B have geodesic 2 A/C have geodesic

27 Goodness of fit measure 3: geodesic distribution (global clustering) Data: Model: Bernoulli (i.e. edges only) C A B A/B have geodesic 2 A/C have geodesic

28 Goodness of fit measures assembled School 10, Model: Bernoulli degree edgewise shared partner geodesic

29 School 10 degree edgewise shared partner geodesic Model: Edges AIC: Model: Edges + Attributes AIC: Model: Edges + GWESP AIC: Model: Edges + Attributes + GWESP AIC:

30 Attributes + GWESP Model as network size increases n: degree edgewise shared partner geodesic

31 Parameter values: Across all 59 schools Grade 8 Grade 9 Grade 10 Grade 11 Grade 12 Black Hispanic Asian Native Am Other Female Grade 7 Grade 8 Grade 9 Grade 10 Grade 11 Grade 12 White Black Hispanic Asian Native Am Other Sex GWESP Nodal Sociality Attribute Effects parameters Homophily Homophily parameters Transitivity While there is variation across the schools, there are also clear and systematic patterns that are shared. And the homophily effects are generally stronger than the transitivity effects.

32 Transitivity: before and after homophily controls Shared partner parameters homophily coefficients with shared partner Shared partner coefficients with attributes Transitivity estimates fall by nearly 25% once you control for the homophily effect. So some of the observed triad closure is driven by homophily Shared homophily partner coefficients coefficients w/o shared w/o attributes partner

33 Grade mixing: before and after transitivity controls Homophily parameters for grade homophily coefficients with shared partner Grade 7 Grade 8 Grade 9 Grade 10 Grade 11 Grade 12 Grade based homophily estimates fall by about 14% after controlling for transitivity. So part of the observed assortative mixing by grade is driven by friend of a friend homophily coefficients w/o shared partner

34 Race Mixing: before and after transitivity controls Homophily parameters for race homophily coefficients with shared partner White (non-hispanic) Black (non-hispanic) Hispanic (of any race) Asian (non-hispanic) Native American (non-hispanic) Other (non-hispanic) Race based homophily estimates do not consistently fall, and often rise once you control for the transitivity effect. So the assortative mixing here is driven by birds of a feather. Triad closure actually helps to reduce segregation homophily coefficients w/o shared partner

35 Summary Both transitivity and homophily play a role in clustering friendships Homophily alone would generate the distribution of path lengths A simple parametric form of transitivity captures local clustering 25% of the transitivity effect is a by-product of homophily Grade mixing is typically stronger than race mixing but also less robust to the transitivity confound All 4 of these models begin to fail on the largest schools There is more clustering than these models predict Suggests additional sources of heterogeneity Or perhaps an endogeneous fissioning of groups