1. Where we are Node level metrics Group level metrics Visualization
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
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
E =𝑚 𝑐 2
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:
Asymmetric
P1 models (Holland and Leinhardt, 1975) are based on probabilities of dyadic relations

7. P1 in UCINET Network->P1 Three equations: P1 on Class data
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 “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