Where we are

1. Where we are • Node level metrics • Degree centrality • Betweenness centrality • Group level metrics • Degree centralization • Betweenness centralization • Components • Subgroups • Visualization • None of these address the probability that a dyad or triad exists • They are broad summaries of structure

2. Mathematical versus Statistical Models E • Statistical models can tell you if the relationship observed between variables is due to chance • Mathematical models describe the relationship between variables and suggest what we should observe • This formula predicts: • Nuclear fission • Photoelectric cells • Black holes • The statistical analog would be to observe the characteristics of, say, a black hole and conclude they exist from those observations

3. Thinking about models • Models let us try to test why a structure exists rather than just describing it • QAP allows us to test whether a structure is explained by another structure or by an attribute or set of attributes • Equivalence begins to let us see how nodes have roles in network structure • Structural • Regular • Equivalence in Ucinet (Profile and CATREGE)

4. Network Models • Network models make it possible to test the probability that a dyad or triad exists due to chance or not • Dyads and triads are considered local structures • Network modeling is based on the concept that patterns of local structures may aggregate to a global structure • Ultimately, the global structure that is observed may in part emerge from local structures, from attributes or a combination of both

5. Five reasons to construct a network model(Garry Robins, Pip Pattison, Yuval Kalish, Dean Lusher (2007) An introduction to exponential random graph (p*) models for social networks Social Networks 29: 173–191) • Regularities in processes that give rise to ties. Models let you understand the uncertainty associated with observed data • Can determine if substructures are expected by chance • Can distinguish between structural effects versus node attribute effects • Simple measures (e.g. density, centrality) may not capture processes in complex networks • Can traverse the micro-macro gap – Does the distribution of local structures explain macro structures?

6. Local structures -- Dyads • Dyad – Two nodes • There are two types of dyads in an undirected graph: • Mutual • Null • There a re three types of dyads in a directed graph: • Mutual • Asymmetric • Null • P1 models (Holland and Leinhardt, 1975) are based on probabilities of dyadic relations

7. P1 in UCINET • Network->P1 • Three equations: • Probability of a reciprocated or asymmetric tie based on outdegree (expansiveness) • Probability of a reciprocated or asymmetric tie based on indegree (attractiveness) • Probability of a null tie (the residual of these two) • P1 on Class data • Analysis of residuals

8. Local structures -- Triads • Triads are sets of three nodes • Transitivity refers to the notion that if A knows B and B knows C then A should know C • This is not always the case • Some triads are transitive and some are intransitive

9. Transitivity and network models • If you take all possible sub-graphs of triads there is some distribution of transitive and intransitive triads • Holland, P.W., and Leinhardt, S. 1975. “Local structure in social networks." In D. Heise (ed.), Sociological Methodology. San Francisco: Jossey-Bass. • For undirected graphs there are four types • Empty • One edge • Two path • Triangle • For directed graphs there are 16 types • Snijders Transitivity slides 14-15

10. Triads in UCINET • Transitivity Index • Transitive ties/Potentially Transitive Ties • For random graphs the expected value is close to density of graph • For actual networks values between .3 and .6 are typical (from Tom Snijders) • Do Cohesion->Transitivity on class data • Do Triad Census on class data

11. Triads in Pajek • Info->Network->Triadic Census • Compare to UCINET Triad Census

12. ERGM (p*) models(Exponential Random Graph Models) • When observing a network there is the notion that the structure could have been different • The idea of modeling is to propose a process by which the observed data ended up as they did • For example, does the network demonstrate more reciprocity than you would expect due to chance – reciprocity can be a model parameter • Recall the triad census and the distribution of the different types • You can think of models as trying to explain that distribution, and in particular determining if the distribution is essentially random

13. p* models (cont.) • Networks are graphs of nodes and edges • The nodes are fixed – Meaning they are not a parameter to consider • With models you create a probability distribution of the possible graphs with the fixed nodes • The observed graph is located somewhere in this distribution • If the observed graph has many reciprocated ties, then a model that is a good fit will also have many reciprocated ties • Once you have a distribution of graphs it can be used to compare sampled graphs (from the distribution) to the observed one on other characteristics • The idea is to use the model to understand the processes underlying the observed structure • You can test whether node attributes (e.g. homophily) or local processes (e.g. transitivity) explain the global structure

14. Dependence assumptions • The possible set of configurations of the set of nodes is constrained by (dependent on) the statistics of the observed network • This limits the possibilities • Graphs in the distribution a consequence of potentially overlapping configurations • The evolution of ties is not random, it is in some way dependent on the environment around it • In considering a parameter like reciprocity, it could be further subdivided into other parameters that use node attributes, like girl-girl reciprocity, or girl-boy reciprocity

15. Different Models • Bernoulli graph – Assumes edges are independent • Dyadic model – Assumes dyads are independent • Markov random graphs – Assumes tie between two nodes is contingent on their ties to other nodes (conditional dependence)

16. ERGM on Class Data in R