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ICPSR - Complex Systems Models in the Social Sciences - Lecture 5(b) - Professor Daniel Martin Katz
ICPSR - Complex Systems Models in the Social Sciences - Lecture 5(b) - Professor Daniel Martin Katz
ICPSR - Complex Systems Models in the Social Sciences - Lecture 5(b) - Professor Daniel Martin Katz
ICPSR - Complex Systems Models in the Social Sciences - Lecture 5(b) - Professor Daniel Martin Katz
ICPSR - Complex Systems Models in the Social Sciences - Lecture 5(b) - Professor Daniel Martin Katz
ICPSR - Complex Systems Models in the Social Sciences - Lecture 5(b) - Professor Daniel Martin Katz
ICPSR - Complex Systems Models in the Social Sciences - Lecture 5(b) - Professor Daniel Martin Katz
ICPSR - Complex Systems Models in the Social Sciences - Lecture 5(b) - Professor Daniel Martin Katz
ICPSR - Complex Systems Models in the Social Sciences - Lecture 5(b) - Professor Daniel Martin Katz
ICPSR - Complex Systems Models in the Social Sciences - Lecture 5(b) - Professor Daniel Martin Katz
ICPSR - Complex Systems Models in the Social Sciences - Lecture 5(b) - Professor Daniel Martin Katz
ICPSR - Complex Systems Models in the Social Sciences - Lecture 5(b) - Professor Daniel Martin Katz
ICPSR - Complex Systems Models in the Social Sciences - Lecture 5(b) - Professor Daniel Martin Katz
ICPSR - Complex Systems Models in the Social Sciences - Lecture 5(b) - Professor Daniel Martin Katz
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ICPSR - Complex Systems Models in the Social Sciences - Lecture 5(b) - Professor Daniel Martin Katz

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  • 1. COMPLEX  SYSTEMS  MODELS  IN  THE  SOCIAL  SCIENCES     MICHAEL  J  BOMMARITO  II   DANIEL  MARTIN  KATZ     Exponen'al  Random  Graph  Models  (p*)      
  • 2. And  now  a  very  quick  fly-­‐by…   Exponen'al  Random  Graph  Models  (p*)   —  Hunter,  Handcock,  BuFs,  Goodreau,  Morris.    ergm:  A  Package  to  Fit,  Simulate  and   Diagnose  Exponen<al-­‐Family  Models  for  Networks,  2008.   ¡  “ERGM  may  then  be  used  to  understand  a  par'cular   phenomenon  or  to  simulate  new  random  realiza'ons  of   networks  that  retain  the  essen'al  proper'es  of  the  original.”   ¡  “The  purpose  of  ERGM,  in  a  nutshell,  is  to  describe   parsimoniously  the  local  selec'on  forces  that  shape  the   global  structure  of  a  network.”       Michael  J.  Bommarito  II  ,  Daniel  Mar'n  Katz    
  • 3. Sta's'cal  Network  Models   —  Goal:  Explain  some  dependent  vector  Y  in  terms  of  a  set   of  independent  variables  in  X.   ¡  This  sounds  familiar  –  it’s  just  regression  analysis!   —  Dependent  Variable:  E,  the  set  of  edges   ¡  E  can  be  thought  of  as  a  matrix  Bernoulli  variables  ei,j    indica'ng  an   edge  exis'ng  between  ver'ces  i  and  j   ¡  Undirected  graphs  have  symmetric  E,  directed  graphs  do  not   necessarily.   Michael  J.  Bommarito  II  ,  Daniel  Mar'n  Katz    
  • 4. Sta's'cal  Network  Models   —  Dyad-­‐independent   ¡  ei,j    is  independent  of  ek,l   ¡  Easy  -­‐  this  model  is  just  standard  logis'c  regression!   —  Dyad-­‐dependent   ¡  ei,j    is  not  necessarily  independent  of  ek,l   ¡  Hard  –  this  model  requires  something  more  flexible  than   regression!   Michael  J.  Bommarito  II  ,  Daniel  Mar'n  Katz    
  • 5. Sta's'cal  Network  Models   —  How  do  we  deal  with   dyad-­‐dependence?   ¡  We  have  E  on  both  sides,  which   leads  to  complex  feedbacks.   ¡  Model  degeneracy  and  mis-­‐ specifica'on  abound!   Michael  J.  Bommarito  II  ,  Daniel  Mar'n  Katz    
  • 6. MCMC   —  MCMC:   ¡  MC1  =  Markov  Chain     ¡  MC2  =  Monte  Carlo   —  Basic  Idea:     ¡  Take  a  random  walk  through  distribu'on-­‐space  where  the  walk’s  equilibrium  is  our   target  likelihood  distribu'on   ¡  …but  how  do  we  decide  how  to  take  our  random  walk?   ¡  …and  how  many  random  steps  do  we  need  to  take?   Michael  J.  Bommarito  II  ,  Daniel  Mar'n  Katz    
  • 7. MCMC   —  How  to  walk?   ¡  Metropolis-­‐Has'ngs:   ÷  Move  an  epsilon  in  state-­‐space     ÷  Accept  or  reject  the  move  depending  on  the  “rejec'on  method”   ¡  Gibbs  Sampling   ÷  What  if  we  knew  the  condi'onal  distribu'ons?   ¢  …but  what  if  there  is  no  path  between  regions  of  the  state-­‐space  along   condi'onally  sampled  paths?   ¢  …or  what  if  the  right  path  occurs  with  such  a  low  probability  as  to  be  un-­‐ sampleable?   Michael  J.  Bommarito  II  ,  Daniel  Mar'n  Katz    
  • 8. ERGM  &  MCMC   —  What  does  MCMC  mean  for  ERGM?   ¡  Imagine  if  each  state  were  a  possible  graph…   ¡  We  could  generate  a  likelihood  distribu'on  over  possible  graph!   ¡  We  also  obtain  MCMC  standard  errors,  lecng  us  think  about  our   coefficient  es'mates  as  more  than  just  points.   —  This  allows  us  to  use  likelihood  in  all  the  regular  ways  (with   a  properly  specified  model).   Michael  J.  Bommarito  II  ,  Daniel  Mar'n  Katz    
  • 9. What  about  the  RHS?   —  So  what  interes'ng  things  can  we  throw  on  the  RHS?   ¡  Assorta've  mixing  with  shared  vertex  aFributes   ¡  Density     ¡  Clustering  coefficient  /  number  of  triangles   ¡  Path  length  distribu'on   ¡  Edgewise  shared  partners   ¡  Geometrically-­‐weighted  edgewise  shared  partners  (safer!)   ¡  …   ¡  Any  variable  you  can  code  yourself!   Michael  J.  Bommarito  II  ,  Daniel  Mar'n  Katz    
  • 10. Exponen'al  Random  Graph  Models  (p*)   —  DocumentaLon!   ¡  Statnet  webpage:  hQp://csde.washington.edu/statnet/resources.shtml   ¡  Usergroup:  hQp://csde.washington.edu/statnet/statnet_users_group.shtml   —  Papers  You  Should  Consult:   ¡  Frank,  O.,  &  Strauss,  D.  (1986).  Markov  graphs.  Journal  of  the  American  Sta<s<cal  Associa<on,  81,   832-­‐842.         ¡  Wasserman,  S.,  &  Pacson,  P.  E.  (1996).  Logit  models  and  logis'c  regressions  for  social  networks:  I.   An  introduc'on  to  Markov  graphs  and  p*.  Psychometrika,  61,  401-­‐425.     ¡  Anderson,  C.J.,  Wasserman,  S.,  &  Crouch,  B.  (1999).  A  p*  primer:  Logit  models  for  social  networks.   Social  Networks,  21,  37-­‐66.     ¡  Snijders,  T.A.B.  (2002).  Markov  chain  Monte  Carlo  es'ma'on  of  exponen'al  random  graph  models.   Journal  of  Social  Structure,  3,  2.     ¡  Garry  Robins,  Tom  Snijders,  Peng  Wang,  Mark  Handcock  &    Philippa  Pacson  (2007).  Recent   developments  in  exponen<al  random  graph  (p*)  models  for  social  networks,  Social  Networks,  29   192–215.       Michael  J.  Bommarito  II  ,  Daniel  Mar'n  Katz    
  • 11. Exponen'al  Random  Graph  Models  (p*)   —  SoVware  You  Might  Consider:       ÷  R  Siena  (Now  Available  for  R)       ¢  Runs  ERGM  models   ÷  Has  some  computa'onal  limita'ons  (~  1000  nodes)     ÷  Also,  allows  for  Longitudinal  Network  Analysis     •  Including  analysis  of  longitudinal  data  of  networks  and  behavior     hFp://www.stats.ox.ac.uk/~snijders/siena/       Michael  J.  Bommarito  II  ,  Daniel  Mar'n  Katz    
  • 12. Exponen'al  Random  Graph  Models  (p*)   —  SoVware  You  Might  Consider:       ÷  R  package  developed  by  some  of  the  leading  scholars  (hFp:// statnet.org/)   ÷  Statnet  is  a  suite  of  sorware  packages  for  sta's'cal  network  analysis   ÷  Func'onality  is  powered  by  a  Markov  chain  Monte  Carlo  (MCMC)   ÷  hFp://cran.r-­‐project.org/web/packages/statnet/index.html       Michael  J.  Bommarito  II  ,  Daniel  Mar'n  Katz    
  • 13. Statnet  Tutorial   —  Statnet  Tutorial     ÷  A  Statnet  Tutorial    Steven  M.  Goodreau,      Mark  S.  Handcock,      David  R.  Hunter,      Carter  T.  BuQs,  and      MarLna  Morris,              24  Journal  of  StaLsLcal                     SoVware  1  (2008).     hFp://www.ncbi.nlm.nih.gov/pmc/ar'cles/PMC2443947/   Michael  J.  Bommarito  II  ,  Daniel  Mar'n  Katz    
  • 14. Video  Based  Tutorial   —  Video  You  Might  Consider:     ÷  Carter  BuFs  Tutorial  @  Poli'cal  Networks  Conference  (DUKE  2010)     ÷  Descrip'on:    hFp://www.poli.duke.edu/poli'calnetworks/day01.html       ÷  Morning  Session:    hFp://'nyurl.com/23r3v9t   •  Available  in  both  Flash  &  Quick'me     •  (hFp://lectopia.oit.duke.edu/ilectures/ilectures.lasso?ut=1065&id=27646)   ÷  Arernoon  Session:    hFp://'nyurl.com/2bpxnud   •  Available  in  both  Flash  &  Quick'me     •  (hFp://lectopia.oit.duke.edu/ilectures/ilectures.lasso?ut=1065&id=27647)         Michael  J.  Bommarito  II  ,  Daniel  Mar'n  Katz    

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