Random directed graph distributions and the triad census in social networks†

This paper is a revised version of a Ph.D. qualifying paper for the Department of Statistics at Harvard University. I am extremely grateful to Paul Holland who guided this research and made valuable comments. I would also like to thank Samuel Leinhardt, Frederick Mosteller and David Oakes for helpful suggestions. Support was partially provided by National Science Foundation Grants SOC73–05489 to Carnegie‐Mellon University and DCR70–03456‐A04 to the National Bureau of Economic Research, Inc. Current address: School of Urban and Public Affairs, Carnegie‐Mellon University, Pittsburgh, PA 15213.

This paper uses the concept of the triad census as, developed by Holland and Leinhardt, and describes several distributions on directed graphs. Methods are presented for calculating the mean and the covariance matrix of the triad census for the uniform distribution that conditions on the number of choices made by each individual in the social network. Several complex distributions on digraphs are approximated, and an application of these methods to a sociogram is given.

Notes

This paper is a revised version of a Ph.D. qualifying paper for the Department of Statistics at Harvard University. I am extremely grateful to Paul Holland who guided this research and made valuable comments. I would also like to thank Samuel Leinhardt, Frederick Mosteller and David Oakes for helpful suggestions. Support was partially provided by National Science Foundation Grants SOC73–05489 to Carnegie‐Mellon University and DCR70–03456‐A04 to the National Bureau of Economic Research, Inc. Current address: School of Urban and Public Affairs, Carnegie‐Mellon University, Pittsburgh, PA 15213.