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

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

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)

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

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?

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

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

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

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

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

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

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

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

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

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)

ERGM on Class Data in R