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.
Michael J. Bommarito II , Daniel Mar'n Katz
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
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
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
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
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
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
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
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
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
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
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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