en>Owais Khursheed: /* Equilibrium */

2014-12-31T14:09:22Z

Equilibrium

← Older revision		Revision as of 16:09, 31 December 2014
Line 1:		Line 1:
	~~{{Network Science}}~~		I like my hobby Nordic skating. Sounds boring? Not!<br>I also try to learn Swedish in my free time.<br><br>Have a look at my web site ... [http://qwe.net/ugg/ ugg boots on sale]
	~~'''Exponential random graph models''' (ERGMs) are a family of statistical models for analyzing data about [[social network\|social]] and [[network science\|other networks]].~~

	~~==Background==~~
	Many metrics exist to describe the structural features of an observed network such as the density, centrality, or assortativity.<ref>{{cite book\|last1=Wasserman \|first1=Stanley \|last2=Faust \|first2=Katherine \|title=Social Network Analysis: Methods and Applications \|year=1994 \|isbn=978-0-521-38707-1}}</ref><ref>{{cite journal\|last=Newman \|first=M.E.J. \|title=The Structure and Function of Complex Networks \|journal=SIAM Review \|volume=45 \|number=2 \|pages=167–256 \|doi=10.1137/S003614450342480}}</ref> However, these metrics describe the observed network which is only one instance of a large number of possible alternative networks. This set of alternative networks may have similar or dissimilar structural features. To support [[statistical inference]] on the processes influencing the formation of network structure, a [[statistical model]] should consider the set of all possible alternative networks weighted on their similarity to an observed network. However because network data is inherently relational, it violates the assumptions of independence and identical distribution of standard statistical models like ~~[[linear regression]]~~.<~~ref name="Contractor"~~>{{cite journal\|last1=Contractor \|first1=Noshir \|last2=Wasserman \|first2=Stanley \|last3=Faust \|first3=Katherine \|title=Testing Multitheoretical, Multilevel Hypotheses About Organizational Networks: An Analytic Framework and Empirical Example \|journal=Academy of Management Review \|volume=31 \|number=3 \|pages=681–703 \|doi=10.5465/AMR.2006.21318925}}</ref> Alternative statistical models should reflect the uncertainty associated with a given observation, permit inference about the relative frequency about network substructures of theoretical interest, disambiguating the influence of confounding processes, efficiently representing complex structures, and linking local-level processes to global-level properties.<ref name="Intro"> {{cite journal\|last1=Robins \|first1=G. \|last2=Pattison \|first2=P. \|last3=Kalish \|first3=Y. \|last4=Lusher \|first4=D.\|year=2007 \|title=An introduction to ~~exponential random graph models for social networks \|journal=Social Networks\|volume=29 \|pages=173–191 \|doi=10.1016/j.socnet.2006.08.002}} </ref>~~

	~~==Definition==~~
	~~The [[Exponential family]] is a broad family of models for covering many types of data, not just networks. An ERGM is a model from this family which describes networks.~~

	Formally a [[random graph]] <math>Y</math> consists of a set of <math>n</math> nodes and <math>m</math> dyads (edges) <math>\{ Y_{ij}: i=1,\dots,n; j=1,\dots,n\}</math> where <math>Y_{ij}=1</math> if the nodes <math>(i,j)</math> are connected and <math>Y_{ij}=0</math> otherwise.

	~~The basic assumption of these models is that the structure~~ in ~~an observed graph <math>y</math> can be explained by any statistics <math>s(y)</math> depending on the observed network and nodal attributes~~. ~~This way, it is possible to describe any kind of dependence between the dyadic variables:~~

	~~<math>~~
	~~P(Y = y \| \theta) = \frac{\exp(\theta^{T} s(y))}{c(\theta)}~~
	~~</math>~~

	~~where <math>\theta</math> is a vector of model parameters associated with <math>s(y)</math> and~~ <~~math~~>~~c(\theta)~~<~~/math~~> is a ~~normalising constant~~.

	~~These models represent a probability distribution on each possible network on <math>n</math> nodes~~. ~~However, the size of the set of possible networks for an undirected network (simple graph) of size <math>n</math> is<math>2^{n(n-1)\over 2}</math>~~. ~~Because the number of possible networks in the set vastly outnumbers the number of parameters which can constrain the model, the ideal probability distribution is the one which maximizes the~~ [~~[Gibbs entropy]].<ref name="Newman">{{cite book\|last=Newman \|first=M.E.J.\|title=Networks\|chapter=Other Network Models\|pages=565–585 \|isbn=978-0-19-920665-0}}</ref>~~

	~~==References==~~
	~~{{reflist}}~~

	~~== Further reading ==~~
	*{{cite journal\|last1=Caimo \|first1=A. \|last2=Friel \|first2= N \|year=2011 \|title=Bayesian inference for exponential random graph models \|journal=Social Networks \|volume=33 \|pages=41–55 \|doi=10.1016/j.socnet.2010.09.004 }}
	*{{cite journal\|last1=Erdős \|first1=P. \|last2=Rényi \|first2= A \|year=1959 \|title=On random graphs \|journal=Publicationes Mathematicae \|volume=6 \|pages=290–297}}
	*{{cite journal\|last1=Fienberg \|first1=S. E. \|last2=Wasserman \|first2=S. \|year=1981 \|title=Discussion of An Exponential Family of Probability Distributions for Directed Graphs by Holland and Leinhardt \|journal=Journal of the American Statistical Association \|volume=76 \|pages=54–57}}
	*{{cite journal\|last1=Frank \|first1=O. \|last2=Strauss \|first2=D \|year=1986 \|title=Markov Graphs \|journal=Journal of the American Statistical Association \|volume=81 \|pages=832–842 \|doi=10.2307/2289017}}
	*{{cite journal\|last1=Handcock \|first1=M. S. \|last2=Hunter \|first2=D. R. \|last3=Butts \|first3=C. T. \|last4=Goodreau \|first4= S. M. \|last5=Morris \|first5=M. \|year=2008 \|title=statnet: Software Tools for the Representation, Visualization, Analysis and Simulation of Network Data \|journal=Journal of Statistical Software \|volume=24 \|pages=1–11 \|url=http://~~www.jstatsoft~~.~~org~~/~~v24~~/~~i01 }}~~
	*{{cite journal\|last1=Hunter \|first1=D. R. \|last2=Goodreau \|first2=S. M. \|last3=Handcock \|first3=M. S. \|year=2008 \|title=Goodness of Fit of Social Network Models \|journal=Journal of the American Statistical Association \|volume=103 \|pages=248–258 \|doi=10.1198/016214507000000446}}
	*{{cite journal\|last1=Hunter \|first1=D. R \|last2=Handcock \|first2=M. S. \|year=2006 \|title=Inference in curved exponential family models for networks \|journal=Journal of Computational and Graphical Statistics \|volume=15 \|pages=565–583 \|doi=10.1198/106186006X133069}}
	*{{cite journal\|last1=Hunter \|first1=D. R. \|last2=Handcock \|first2=M. S. \|last3=Butts \|first3=C. T. \|last4=Goodreau \|first4=S. M. \|last5=Morris \|first5=M. \|year=2008 \|title=ergm: A Package to Fit, Simulate and Diagnose Exponential-Family Models for Networks \|journal=Journal of Statistical Software \|volume=24 \|pages=1–29 \|url= http://www.jstatsoft.org/v24/i03}}
	*{{cite journal\|last1=Jin \|first1=I.H. \|last2=Liang \|first2=F. \|year=2012 \|title=Fitting social networks models using varying truncation stochastic approximation MCMC algorithm \|journal=Journal of Computational and Graphical Statistics \|doi=10.1080/10618600.2012.680851}}
	*{{cite journal\|last1=Koskinen \|first1=J. H. \|last2=Robins \|first2=G. L. \|last3=Pattison \|first3=P. E. \|year=2010 \|title=Analysing exponential random graph (p-star) models with missing data using Bayesian data augmentation \|journal=Statistical Methodology \|volume=7 \|pages=366–384 \|doi=10.1016/j.stamet.2009.09.007}}
	*{{cite journal\|last1=Morris \|first1=M. \|last2=Handcock \|first2=M. S. \|last3=Hunter \|first3=D. R. \|year=2008 \|title=Specification of Exponential-Family Random Graph Models: Terms and Computational Aspects \|journal=Journal of Statistical Software \|volume=24 \|url= http://www.jstatsoft.org/v24/i04}}
	*{{cite journal\|last1=Rinaldo \|first1=A. \|last2=Fienberg \|first2=S. E. \|last3=Zhou \|first3=Y. \|year=2009 \|title=On the geometry of descrete exponential random families with application to exponential random graph models \|journal=Electronic Journal of Statistics \|volume=3 \|pages=446–484 \|doi=10.1214/08-EJS350}}
	{{cite journal\|last1=Robins \|first1=G. \|last2=Snijders \|first2=T. \|last3=Wang \|first3=P. \|last4=Handcock \|first4=M. \|last5=Pattison \|first5=P \|year=2007 \|title=Recent developments in exponential random graph (p) models for social networks \|journal=Social Networks \|volume=29 \|pages=192–215 \|doi=10.1016/j.socnet.2006.08.003}}
	*{{cite journal\|last1=Snijders \|first1=T. A. B. \|year=2002 \|title=Markov chain Monte Carlo estimation of exponential random graph models \|journal=Journal of Social Structure \|volume=3 \|url=http://www.cmu.edu/joss/content/articles/volume3/Snijders.pdf}}
	*{{cite journal\|last1=Snijders \|first1=T. A. B. \|last2=Pattison \|first2=P. E. \|last3=Robins \|first3=G. L. \|year=2006 \|title=New specifications for exponential random graph models \|journal=Sociological Methodology \|volume=36 \|pages=99–153 \|doi=10.1111/j.1467-9531.2006.00176.x}}
	*{{cite journal\|last1=Strauss \|first1=D \|last2=Ikeda \|first2=M \|year=1990 \|title=Pseudolikelihood estimation for social networks \|journal=Journal of the American Statistical Association \|volume=5 \|pages=204–212 \|doi=10.2307/2289546}}
	*{{cite journal\|last1=van Duijn \|first1=M. A. \|last2=Snijders \|first2=T. A. B. \|last3=Zijlstra \|first3=B. H. \|year=2004\|title=p2: a random effects model with covariates for directed graphs \|journal=Statistica Neerlandica \|volume=58 \|pages=234–254 \|doi=10.1046/j.0039-0402.2003.00258.x}}
	*{{cite journal\|last1=van Duijn \|first1=M. A. J. \|last2=Gile \|first2=K. J. \|last3=Handcock \|first3=M. S. \|year=2009 \|title=A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models \|journal=Social Networks \|volume=31 \|pages=52–62 \|doi=10.1016/j.socnet.2008.10.003}}

	~~[[Category:Network theory]~~]

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2012-04-20T15:52:01Z

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← Older revision		Revision as of 16:09, 31 December 2014
Line 1:		Line 1:
	~~{{Network Science}}~~		I like my hobby Nordic skating. Sounds boring? Not!<br>I also try to learn Swedish in my free time.<br><br>Have a look at my web site ... [http://qwe.net/ugg/ ugg boots on sale]
	~~'''Exponential random graph models''' (ERGMs) are a family of statistical models for analyzing data about [[social network\|social]] and [[network science\|other networks]].~~

	~~==Background==~~
	Many metrics exist to describe the structural features of an observed network such as the density, centrality, or assortativity.<ref>{{cite book\|last1=Wasserman \|first1=Stanley \|last2=Faust \|first2=Katherine \|title=Social Network Analysis: Methods and Applications \|year=1994 \|isbn=978-0-521-38707-1}}</ref><ref>{{cite journal\|last=Newman \|first=M.E.J. \|title=The Structure and Function of Complex Networks \|journal=SIAM Review \|volume=45 \|number=2 \|pages=167–256 \|doi=10.1137/S003614450342480}}</ref> However, these metrics describe the observed network which is only one instance of a large number of possible alternative networks. This set of alternative networks may have similar or dissimilar structural features. To support [[statistical inference]] on the processes influencing the formation of network structure, a [[statistical model]] should consider the set of all possible alternative networks weighted on their similarity to an observed network. However because network data is inherently relational, it violates the assumptions of independence and identical distribution of standard statistical models like ~~[[linear regression]]~~.<~~ref name="Contractor"~~>{{cite journal\|last1=Contractor \|first1=Noshir \|last2=Wasserman \|first2=Stanley \|last3=Faust \|first3=Katherine \|title=Testing Multitheoretical, Multilevel Hypotheses About Organizational Networks: An Analytic Framework and Empirical Example \|journal=Academy of Management Review \|volume=31 \|number=3 \|pages=681–703 \|doi=10.5465/AMR.2006.21318925}}</ref> Alternative statistical models should reflect the uncertainty associated with a given observation, permit inference about the relative frequency about network substructures of theoretical interest, disambiguating the influence of confounding processes, efficiently representing complex structures, and linking local-level processes to global-level properties.<ref name="Intro"> {{cite journal\|last1=Robins \|first1=G. \|last2=Pattison \|first2=P. \|last3=Kalish \|first3=Y. \|last4=Lusher \|first4=D.\|year=2007 \|title=An introduction to ~~exponential random graph models for social networks \|journal=Social Networks\|volume=29 \|pages=173–191 \|doi=10.1016/j.socnet.2006.08.002}} </ref>~~

	~~==Definition==~~
	~~The [[Exponential family]] is a broad family of models for covering many types of data, not just networks. An ERGM is a model from this family which describes networks.~~

	Formally a [[random graph]] <math>Y</math> consists of a set of <math>n</math> nodes and <math>m</math> dyads (edges) <math>\{ Y_{ij}: i=1,\dots,n; j=1,\dots,n\}</math> where <math>Y_{ij}=1</math> if the nodes <math>(i,j)</math> are connected and <math>Y_{ij}=0</math> otherwise.

	~~The basic assumption of these models is that the structure~~ in ~~an observed graph <math>y</math> can be explained by any statistics <math>s(y)</math> depending on the observed network and nodal attributes~~. ~~This way, it is possible to describe any kind of dependence between the dyadic variables:~~

	~~<math>~~
	~~P(Y = y \| \theta) = \frac{\exp(\theta^{T} s(y))}{c(\theta)}~~
	~~</math>~~

	~~where <math>\theta</math> is a vector of model parameters associated with <math>s(y)</math> and~~ <~~math~~>~~c(\theta)~~<~~/math~~> is a ~~normalising constant~~.

	~~These models represent a probability distribution on each possible network on <math>n</math> nodes~~. ~~However, the size of the set of possible networks for an undirected network (simple graph) of size <math>n</math> is<math>2^{n(n-1)\over 2}</math>~~. ~~Because the number of possible networks in the set vastly outnumbers the number of parameters which can constrain the model, the ideal probability distribution is the one which maximizes the~~ [~~[Gibbs entropy]].<ref name="Newman">{{cite book\|last=Newman \|first=M.E.J.\|title=Networks\|chapter=Other Network Models\|pages=565–585 \|isbn=978-0-19-920665-0}}</ref>~~

	~~==References==~~
	~~{{reflist}}~~

	~~== Further reading ==~~
	*{{cite journal\|last1=Caimo \|first1=A. \|last2=Friel \|first2= N \|year=2011 \|title=Bayesian inference for exponential random graph models \|journal=Social Networks \|volume=33 \|pages=41–55 \|doi=10.1016/j.socnet.2010.09.004 }}
	*{{cite journal\|last1=Erdős \|first1=P. \|last2=Rényi \|first2= A \|year=1959 \|title=On random graphs \|journal=Publicationes Mathematicae \|volume=6 \|pages=290–297}}
	*{{cite journal\|last1=Fienberg \|first1=S. E. \|last2=Wasserman \|first2=S. \|year=1981 \|title=Discussion of An Exponential Family of Probability Distributions for Directed Graphs by Holland and Leinhardt \|journal=Journal of the American Statistical Association \|volume=76 \|pages=54–57}}
	*{{cite journal\|last1=Frank \|first1=O. \|last2=Strauss \|first2=D \|year=1986 \|title=Markov Graphs \|journal=Journal of the American Statistical Association \|volume=81 \|pages=832–842 \|doi=10.2307/2289017}}
	*{{cite journal\|last1=Handcock \|first1=M. S. \|last2=Hunter \|first2=D. R. \|last3=Butts \|first3=C. T. \|last4=Goodreau \|first4= S. M. \|last5=Morris \|first5=M. \|year=2008 \|title=statnet: Software Tools for the Representation, Visualization, Analysis and Simulation of Network Data \|journal=Journal of Statistical Software \|volume=24 \|pages=1–11 \|url=http://~~www.jstatsoft~~.~~org~~/~~v24~~/~~i01 }}~~
	*{{cite journal\|last1=Hunter \|first1=D. R. \|last2=Goodreau \|first2=S. M. \|last3=Handcock \|first3=M. S. \|year=2008 \|title=Goodness of Fit of Social Network Models \|journal=Journal of the American Statistical Association \|volume=103 \|pages=248–258 \|doi=10.1198/016214507000000446}}
	*{{cite journal\|last1=Hunter \|first1=D. R \|last2=Handcock \|first2=M. S. \|year=2006 \|title=Inference in curved exponential family models for networks \|journal=Journal of Computational and Graphical Statistics \|volume=15 \|pages=565–583 \|doi=10.1198/106186006X133069}}
	*{{cite journal\|last1=Hunter \|first1=D. R. \|last2=Handcock \|first2=M. S. \|last3=Butts \|first3=C. T. \|last4=Goodreau \|first4=S. M. \|last5=Morris \|first5=M. \|year=2008 \|title=ergm: A Package to Fit, Simulate and Diagnose Exponential-Family Models for Networks \|journal=Journal of Statistical Software \|volume=24 \|pages=1–29 \|url= http://www.jstatsoft.org/v24/i03}}
	*{{cite journal\|last1=Jin \|first1=I.H. \|last2=Liang \|first2=F. \|year=2012 \|title=Fitting social networks models using varying truncation stochastic approximation MCMC algorithm \|journal=Journal of Computational and Graphical Statistics \|doi=10.1080/10618600.2012.680851}}
	*{{cite journal\|last1=Koskinen \|first1=J. H. \|last2=Robins \|first2=G. L. \|last3=Pattison \|first3=P. E. \|year=2010 \|title=Analysing exponential random graph (p-star) models with missing data using Bayesian data augmentation \|journal=Statistical Methodology \|volume=7 \|pages=366–384 \|doi=10.1016/j.stamet.2009.09.007}}
	*{{cite journal\|last1=Morris \|first1=M. \|last2=Handcock \|first2=M. S. \|last3=Hunter \|first3=D. R. \|year=2008 \|title=Specification of Exponential-Family Random Graph Models: Terms and Computational Aspects \|journal=Journal of Statistical Software \|volume=24 \|url= http://www.jstatsoft.org/v24/i04}}
	*{{cite journal\|last1=Rinaldo \|first1=A. \|last2=Fienberg \|first2=S. E. \|last3=Zhou \|first3=Y. \|year=2009 \|title=On the geometry of descrete exponential random families with application to exponential random graph models \|journal=Electronic Journal of Statistics \|volume=3 \|pages=446–484 \|doi=10.1214/08-EJS350}}
	{{cite journal\|last1=Robins \|first1=G. \|last2=Snijders \|first2=T. \|last3=Wang \|first3=P. \|last4=Handcock \|first4=M. \|last5=Pattison \|first5=P \|year=2007 \|title=Recent developments in exponential random graph (p) models for social networks \|journal=Social Networks \|volume=29 \|pages=192–215 \|doi=10.1016/j.socnet.2006.08.003}}
	*{{cite journal\|last1=Snijders \|first1=T. A. B. \|year=2002 \|title=Markov chain Monte Carlo estimation of exponential random graph models \|journal=Journal of Social Structure \|volume=3 \|url=http://www.cmu.edu/joss/content/articles/volume3/Snijders.pdf}}
	*{{cite journal\|last1=Snijders \|first1=T. A. B. \|last2=Pattison \|first2=P. E. \|last3=Robins \|first3=G. L. \|year=2006 \|title=New specifications for exponential random graph models \|journal=Sociological Methodology \|volume=36 \|pages=99–153 \|doi=10.1111/j.1467-9531.2006.00176.x}}
	*{{cite journal\|last1=Strauss \|first1=D \|last2=Ikeda \|first2=M \|year=1990 \|title=Pseudolikelihood estimation for social networks \|journal=Journal of the American Statistical Association \|volume=5 \|pages=204–212 \|doi=10.2307/2289546}}
	*{{cite journal\|last1=van Duijn \|first1=M. A. \|last2=Snijders \|first2=T. A. B. \|last3=Zijlstra \|first3=B. H. \|year=2004\|title=p2: a random effects model with covariates for directed graphs \|journal=Statistica Neerlandica \|volume=58 \|pages=234–254 \|doi=10.1046/j.0039-0402.2003.00258.x}}
	*{{cite journal\|last1=van Duijn \|first1=M. A. J. \|last2=Gile \|first2=K. J. \|last3=Handcock \|first3=M. S. \|year=2009 \|title=A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models \|journal=Social Networks \|volume=31 \|pages=52–62 \|doi=10.1016/j.socnet.2008.10.003}}

	~~[[Category:Network theory]~~]

Conjugate beam method - Revision history

en>Owais Khursheed: /* Equilibrium */

en>Fraggle81: Reverted 2 edits by 14.139.209.20 identified as test/vandalism using STiki

68.185.86.139: typo