Eric Klein and Ning Sa

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1 Week 12. Statistical Appraches t Netwrks: p1 and p* Wasserman and Faust Chapter 15: Statistical Analysis f Single Relatinal Netwrks There are fur tasks in psitinal analysis: 1) Define Equivalence 2) Measure hw clsely actrs adhere t this definitin 3) Represent the equivalences f the actrs 4) Measure the adequacy f this representatin Statistical mdels enable us t perfrm significance tests n data Mdels in this chapter are dyadic interactin mdels which use the natural lg f prbabilities as their basic mdeling unit This chapter fcuses n the Y-Array which cncentrates n dyads and is descriptive f individual actrs' ties t ther actrs. It cnsiders all pssible ties between actrs Fr a single, directinal relatin, we fcus n effects that represent the expansiveness f actrs and the ppularity f their partners and the reciprcatin f thse ties within dyads. Expansiveness reflects the likelihd that the actr selects ties t thers Ppularity reflects the tendency f an actr t be selected by thers Reciprcatin reflects if these ties are ne r tw directinal This data is then used t create a mdel f the netwrk. That mdel is then tested fr gdness f fit Mdels are deemed t be hierarchical if ne mdel can be btained frm the ther by setting sme parameters t 0. I think this means that, if yu eliminate sme ties in the larger netwrk, yu can islate the smaller ne, I will raise this issue in class and make sure we clarify it with Ynie This system is limited in that islated actrs (with n ties either way) and universally tied actrs (tied t everyne in every way) break the system by having infinite indegrees and utdegrees respectively. Anther way t break dwn the netwrk is with the W-Array. This is used fr attributes f actrs. In this methd, all f thse wh share the same attributes are cnsidered t have the same indegree/utdegree fr that attribute. This leads int the cncept f Stchastic Equivalence which is the assumptin that cmparable behavir f actrs within subsets has been viewed as a generalizatin f structural equivalence. Obviusly, these methds cannt wrk with nn-directinal ties as all actrs have the same indegree and utdegree 1

2 Week 12. Statistical Appraches t Netwrks: p1 and p* Wasserman and Faust Chapter 16: Stchastic Blckmdels and Gdness f Fit Indices These methds seek t answer the questin: Hw well des a blckmdel actually fit the data set? There are tw primary appraches t this questin: 1) Use standard data analytic techniques t cmpare the bserved data t the blckmdel 2) Use statistical measures and mdel based techniques t cnduct likelihd based tests The difference in these tw appraches cmes frm their different appraches t psitinal analyses The first states that psitinal analyses are nt statistical methds at all and any apprach that uses them cannt be subjected t statistical analysis The secnd states that if yu want t use statistical techniques t determine hw well a blckmdel fits data yu must d s frm the very beginning The authrs agree with Everett and Brgatti wh suggest that analytic methds need t be chsen based n hw perfrm in practice as ppsed t what I d nt knw The purpse f nnparametric tests is t check the data against certain null hyptheses, much like mre traditinal statistical tests Anther methd yu can use is stchastic blckmdeling which is based n the cncept f stchastic equivalence which was cvered earlier. These blckmdels can then be evaluated in tw ways: 1) Cmpare their densities 2) Cmpare their predictins Stchastic blckmdeling cnsists f a prbability distributin and a mapping f actrs int blcks. Garry Rbins, Pip Pattisn, Yuval Kalish, Dean Lusher (2007). An intrductin t expnential randm graph (p*) mdels fr scial netwrks Why mdel scial netwrk Understand the uncertainty assciated with bserved utcmes Whether certain netwrk substructures are bserved by chance Cmpare tw different scial prcesses which lead t similar predictins Analyze cmplex netwrk data structure Lcalized scial prcesses and structures glbal netwrk patterns The lgic behind p* mdels fr scial netwrks an utline Observed netwrk the netwrk data the researcher has cllected and is interested in mdeling 2

3 Week 12. Statistical Appraches t Netwrks: p1 and p* it is ne particular pattern f ties ut f a large set f pssible patterns a general framewrk fr mdel cnstructin each netwrk tie is regarded as a randm variable a dependencee hypthesis is prpsed, defining cntingencies amng ties the dependence hypthesis implies a particular frm t the mdel simplificatin f parameters thrugh hmgeneity r ther cnstraints estimate and interpret mdel parameters The general frm f the expnential randm graph mdel: dependence assumptins and parameter cnstraints expnential randm graph mdel A refers t a cnfiguratin; g A (y) is the netwrk statistic crrespnding t A; η A is the parameter f A; k is fr nrmalizatin Cnfiguratin: a small subset f pssible netwrk ties, like a single edge Cnstraints n parameters (t reduce the number f parameters) Hmgeneity assumptin: equating parameters when they refer t the same type f cnfiguratin, irrespective f whichh ndes are invlved Equating parameters fr ismrphic cnfiguratins invlving similar types f actrs Setting sme parameters t zer Dependence assumptins and mdels Bernulli graphs: the simplest dependence assumptin Dependence assumptin: all pssible distinct ties are independent f ne anther Single edges are the nly pssible cnfiguratins With hmgeneity assumptin number f edges Dyadic mdels: the dyadic independence assumptin where L(y) is the Markv randm graphs Markv dependence: tw pssible netwrk ties are cnditinally dependent when they have a cmmn actr Cnfiguratinn invlving mre than 2 ndes Tw star, three star, triangle, transitive. Dependence structures with nde-levehypthesis: scial ties tend t develp between actrs with the same attributes variables Hmphily 3

4 Week 12. Statistical Appraches t Netwrks: p1 and p* Mre cmplex dependence assumptins Cntext Estimatin Pseud-likelihd estimatin Prperties nt well understd; nt accurate Mnte Carl maximum likelihd Simulatin Start with a set f parameter values; cmpare the simulatin results with the bserved netwrk; adjust the parameter values; repeat until cnvergence. Larger psitive parameterr Garry Rbins, Tm Snijders, Peng Wang, Mark Handcck, Philippa Pattisn (2007). Recent develpments in expnential randm graph (p*) mdels fr scial netwrks Expnential randm graph mdels Markv randm graphs Edges, 2-stars, 3-stars, and triangles Near degeneracy Refers t a graph distributin implying nly a very few distinct graphs with substantial nn-zer prbabilities New specificatins Alternating k-stars, with λ>1 The higher rder star parameter estimate was ppsite sign t, and smaller than, a lwer rder star parameter. In the article, λ=2 A lse cre-periphery k-triangles structure Alternating, with λ>1 4

5 Week 12. Statistical Appraches t Netwrks: p1 and p* example f 2-triangle Alternating tw-paths K-triangles withut base New specificatins fr directed graphs Based n the directin f the ties Estimatin Mnte Carl Markv Chain (MCMC) maximum likelihd estimatin One can als btain reliable standard errrs fr the estimates Prgrams fr MCMCC maximum likelihd estimatin SIENA Estimates and standard errr T-rati: hw welll the estimates has cnverges (the clser t 0 the better) pnet statnet very few simulatin runs Mdels with cmbinatins f parameters cmbine r separate Markv randm mdel parameters and new cnfiguratins in this article, separate fr cmparisn mdel interpretatin and gdness f fit psitive larger parameter suggests the presence f the cnfiguratin significant: estimates >= 2xStd.Err (1.65 fr triangles and k-triangles) Fitting the new specificatins t UCINET data sets 20 well-knwn datasets 12 nn-directed netwrks and 8 directed netwrks # nde: Markv mdels nt fit fr 5/12 nn-directed netwrks and 3/8 directed netwrks Mdels incrprating the new specificatins fit fr all Cmpare pseud-likelihd and MC maximum likelihd PL estimates d give sme infrmatin but the values cannt be relied n 5

6 Week 12. Statistical Appraches t Netwrks: p1 and p* Nshir Cntractr, Stanley Wasserman, Katherine Faust (1999). Testing multi-level, multitheretical hyptheses abut netwrks in 21 st century rganizatinal frms: An analytic framewrk and empirical example Organizatins as netwrk recnfiguratin f new technlgy n rganizatins It is nt netwrk in rganizatin, but rganizatin as netwrk The fcus f rganizatin study shift frm emergent netwrks t the emergence f netwrks Multiple theries develped at multiple levels f analysis A framewrk needed T extend the fcus frm descriptive netwrk metrics t statistical testing T incrprate theretical explanatins at multiple levels f analysis T incrprate theretical explanatins based n the actr s attribute Analytic framewrk fr specifying and testing rganizatinal netwrk hyptheses Endgenus variables and exgenus variables Endgenus: prperties f the fcal netwrk itself that influence the prbability f ties in the same netwrk Exgenus: prperties utside the fcal netwrk Attributes Netwrks built n ther relatins The same netwrk but at previus pint in time Different levels f analysis Actr level/dyad level/triad level/glbal level Structural autnmy/reciprcity/transitivity and cyclicity/centralizatin Empirical example: the CRADA netwrk 17 members representing the varius private and gvernment rganizatins Tw sets f hyptheses Endgenus prperties nly at dyadic, cyclical triadic, transitive triadic, and glbal levels One exgenus attribute: gv r nt n actr, dyad, transitive triadic, and glbal levels Analysis p* mdel + lgistic regressin Results Fcus n imprvement f each mdel based n the baseline Cnfirmed the hyptheses 1,2,4, and 5 6