Where we are
Node level metricsDegree centralityBetweennesscentralityGroup level metricsDegree centralizationBetweennesscentralizationComponentsSubgroupsVisualization
None of these address the probability that a dyad or triad existsThey are broad summaries of structure
Mathematical versus Statistical Models
Statistical models can tell you if the relationship observed between variables is due to chanceMathematical models describe the relationship between variables and suggest what we should observe
This formula predicts:Nuclear fissionPhotoelectric cellsBlack holesThe 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 testwhya structure exists rather than justdescribingitQAP allows us to test whether a structure is explained by another structure or by an attribute or set of attributesEquivalence begins to let us see how nodes have roles in network structureStructuralRegularEquivalence inUcinet(Profile and CATREGE)
Network Models
Network models make it possible to test the probability that a dyad or triad exists due to chance or notDyads and triads are considered local structuresNetwork modeling is based on the concept that patterns of local structures may aggregate to a global structureUltimately, 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(GarryRobins,Pip Pattison, YuvalKalish, DeanLusher (2007) An introductionto exponential random graph (p*) modelsfor socialnetworks SocialNetworks29: 173–191)
Regularities in processes that give rise to ties. Models let you understand the uncertainty associated with observed dataCan determine if substructures are expected by chanceCan distinguish between structural effects versus node attribute effectsSimple measures (e.g. density, centrality) may not capture processes in complex networksCan traverse the micro-macro gap – Does the distribution of local structures explain macro structures?
Local structures -- Dyads
Dyad – Two nodesThere are two types of dyads in an undirected graph:MutualNullThere a re three types of dyads in a directed graph:MutualAsymmetricNullP1models(Holland andLeinhardt, 1975) arebased on probabilities of dyadic relations
P1 in UCINET
Network->P1Three equations:Probability of a reciprocated or asymmetric tie based onoutdegree(expansiveness)Probability of a reciprocated or asymmetric tie based onindegree(attractiveness)Probability of a null tie (the residual of these two)P1 on Class dataAnalysis of residuals
Local structures -- Triads
Triads are sets of three nodesTransitivity refers to the notion that if A knows B and B knows C then A should know CThis is not always the caseSome 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 triadsHolland, P.W., and Leinhardt, S. 1975.“Local structure in social networks." In D.Heise(ed.), Sociological Methodology. San Francisco:Jossey-Bass.Forundirected graphs there are four typesEmptyOne edgeTwo pathTriangleFor directed graphs there are 16 typesSnijdersTransitivity slides 14-15
Triads in UCINET
Transitivity IndexTransitive ties/Potentially Transitive TiesFor random graphs the expected value is close to density of graphFor actual networks values between .3 and .6 are typical (from TomSnijders)Do Cohesion->Transitivity on class dataDo Triad Censusonclass data
Triads inPajek
Info->Network->Triadic CensusCompare to UCINET Triad Census
ERGM (p*) models(Exponential Random Graph Models)
When observing a network there is the notion that thestructurecouldhave beendifferentThe idea of modeling is to propose a process by which the observed data ended up as they didFor example, does the network demonstrate more reciprocity than you would expect due to chance – reciprocity can be a model parameterRecall the triad census and the distribution of the different typesYou 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 edgesThe nodes are fixed – Meaning they are not a parameter to considerWith models you create a probability distribution of the possible graphs with the fixed nodesThe observed graph is located somewhere in this distributionIf the observed graph has many reciprocated ties, then a model that is a good fit will also have many reciprocated tiesOnce you have a distribution of graphs it can be used to compare sampled graphs (from the distribution) to the observed one on other characteristicsThe idea is to use the model to understand the processes underlying the observed structureYou 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 networkThis limits the possibilitiesGraphs in the distribution a consequence of potentially overlapping configurationsThe evolution of ties is not random, it is in some way dependent on the environment around itIn 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 independentDyadic model – Assumes dyads are independentMarkov random graphs – Assumes tie between two nodes is contingent on their ties to other nodes (conditional dependence)
ERGM on Class Data in R
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