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

COMPLEX SYSTEMS MODELS IN THE SOC IAL SC I ENCES

M ICHAEL J BOMMARITO I I DAN IEL MART IN KATZ

Exponen'al Random Graph Models (p*)

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

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.



�  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!



�  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!


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?


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?


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).


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!



�  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.



�  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/



�  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


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/


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)


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

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