Algebraic Statistics for p1 Random Graph Models: Markov Bases and Their Uses

The term social network connotes a social structure composed of individuals (or organizations), typically labeled as nodes, linked by one or more relations, such as friendship, information sharing, financial transaction, and so on. Social network analysis uses a graphical representation where the individuals correspond to nodes in the graph and the presence of a relationship to edges. The term social network now also describes specific social structures on the World Wide Web and many authors have examined the Web itself in various forms as a social network.

[1]  A. Rinaldo,et al.  Algebraic Statistics and Contingency Table Problems: Log-Linear Models, Likelihood Estimation, and Disclosure Limitation , 2009 .

[2]  D. J. Strauss,et al.  Pseudolikelihood Estimation for Social Networks , 1990 .

[3]  P. Holland,et al.  Transitivity in Structural Models of Small Groups , 1971 .

[4]  Edoardo M. Airoldi,et al.  Mixed Membership Stochastic Blockmodels , 2007, NIPS.

[5]  Stephen E. Fienberg,et al.  An Exponential Family of Probability Distributions for Directed Graphs: Comment , 1981 .

[6]  A. Rinaldo,et al.  On the geometry of discrete exponential families with application to exponential random graph models , 2008, 0901.0026.

[7]  Gardner Murphy,et al.  Sociometry, experimental method and the science of society : an approach to a new political orientation , 1951 .

[8]  D. Geiger,et al.  On the toric algebra of graphical models , 2006, math/0608054.

[9]  Peter L. Bartlett,et al.  Classification with a Reject Option using a Hinge Loss , 2008, J. Mach. Learn. Res..

[10]  Eva Riccomagno,et al.  Algebraic and Geometric Methods in Statistics: Contingency tables , 2009 .

[11]  P. Holland,et al.  Local Structure in Social Networks , 1976 .

[12]  S. Fienberg,et al.  Categorical Data Analysis of Single Sociometric Relations , 1981 .

[13]  B. Bollobás.  The evolution of random graphs , 1984 .

[14]  David A. Cox,et al.  Using Algebraic Geometry , 1998 .

[15]  Shelby J. Haberman,et al.  Maximum Likelihood Estimates in Exponential Response Models , 1977 .

[16]  Nicholas Eriksson,et al.  Polyhedral conditions for the nonexistence of the MLE for hierarchical log-linear models , 2006, J. Symb. Comput..

[17]  Stephen E. Fienberg,et al.  Algebraic Statistics for a Directed Random Graph Model with Reciprocation , 2009, 0909.0073.

[18]  S. Haberman,et al.  The analysis of frequency data , 1974 .

[19]  Shelby J. Haberman,et al.  An Exponential Family of Probability Distributions for Directed Graphs: Comment , 1981 .

[20]  P. Holland,et al.  An Exponential Family of Probability Distributions for Directed Graphs , 1981 .

[21]  J. F. C. Kingman,et al.  Information and Exponential Families in Statistical Theory , 1980 .

[22]  Henry P. Wynn,et al.  Algebraic and geometric methods in statistics , 2009 .

[23]  Sharon L. Milgram,et al.  The Small World Problem , 1967 .

[24]  P. Diaconis,et al.  Algebraic algorithms for sampling from conditional distributions , 1998 .

[25]  Seth Sullivant,et al.  Lectures on Algebraic Statistics , 2008 .

[26]  D. Champion,et al.  Book Review:Assimilation in American Life: The Role of Race, Religion, and National Origins. Milton M. Gordon , 1965 .

[27]  B. Sturmfels.  Gröbner bases and convex polytopes , 1995 .

[28]  H. Wynn,et al.  Algebraic Statistics: Computational Commutative Algebra in Statistics , 2000 .

[29]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[30]  Stanley Milgram,et al.  An Experimental Study of the Small World Problem , 1969 .

[31]  Takayuki Hibi,et al.  Toric Ideals Generated by Quadratic Binomials , 1999 .

[32]  P. Holland,et al.  A Method for Detecting Structure in Sociometric Data , 1970, American Journal of Sociology.

[33]  L. Katz,et al.  PROBABILITY DISTRIBUTIONS OF RANDOM VARIABLES ASSOCIATED WITH A STRUCTURE OF THE SAMPLE SPACE OF SOCIOMETRIC INVESTIGATIONS , 1957 .

[34]  S. Wasserman,et al.  Logit models and logistic regressions for social networks: I. An introduction to Markov graphs andp , 1996 .

[35]  M. M. Meyer,et al.  Statistical Analysis of Multiple Sociometric Relations. , 1985 .

[36]  Stephen E. Fienberg,et al.  Maximum Likelihood Estimation in Network Models , 2011, ArXiv.

[37]  Lawrence D. Brown.  Fundamentals of Statistical Exponential Families , 1987 .

[38]  Gerhard H. Fischer,et al.  On the existence and uniqueness of maximum-likelihood estimates in the Rasch model , 1981 .

[39]  Alessandro Rinaldo,et al.  Computing Maximum Likelihood Estimates in Log-Linear Models , 2006 .

[40]  Stephen E. Fienberg,et al.  Discrete Multivariate Analysis: Theory and Practice , 1976 .

[41]  H. Jennings,et al.  Who Shall Survive , 2007 .

[42]  Leo Katz,et al.  The Distribution of the Number of Isolates in a Social Group , 1952 .

[43]  Samuel Leinhardt,et al.  An Omnibus Test for Social Structure Using Triads , 1978 .

[44]  D. Hunter,et al.  Goodness of Fit of Social Network Models , 2008 .