Summary
In certain types of populations, one is interested in the relationships among the individuals, e.g., animals and plants in a food web, persons in a social group, cities linked by various transportation routes, etc. These can be modeled by graphs or digraphs (directed graphs). Trying to infer something about these structures from a subgraph (sample) constitutes statistical inference in graphs (what we have called “stagra- phics”). There are many distributions which occur in graph theory which could be useful in inference. This paper is a survey of some of these distributions, specifically, degree distributions, path length distributions, distance distributions, cycle distributions, triad distributions, and distributions connected with pairs of stars. Various properties of these distributions will be discussed, as will their connections with inference. Several open problems will be presented.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Battle, J., Harary, F., Kodama, Y. (1962). Every planar graph with nine points has a non-planar complement. Bulletin of the American Mathematical Society, 68, 569–571.
Berge, C. (1962). The Theory of Graphs and Its Applications. Wiley, New York.
Capobianco, M. (1970). Statistical inference in finite populations having structure. Transactions of the New York Academy of Sciences, 32, 401–413.
Capobianco, M. (1972). Estimating the connectivity of a graph. In Graph Theory and Applications, Y. Alavi, D. Lick, A. White, eds. Springer-Verlag, New York.
Capobianco, M. (1978). Some problems in stagraphics. Presented at the Joint Statistical Meetings, San Diego.
Capobianco, M. (1979). Exploring graphs using pairs of stars in a sample. Presented at the Twelfth Annual Meetinjg of European Statisticians, Varna.
Capobianco, M. (1980a). Collinearity, distance, and connectedness of graphs. To appear.
Capobianco, M. (1980b). Introduction to statistical inference in graphs (stagraphics) and its applications. Under revision for SIAM Review.
Capobianco, M., Frank, O. (1979). Comparison of statistical graph size estimators. Submitted.
Capobianco, M., Molluzzo, J. (1978). Examples and Counter-examples in Graph Theory. Elsevier, North Holland, New York.
Cook, R. (1979). Vertex degrees of planar graphs. Journal of Combinatorial Theory B, 26, 337–345.
Frank, O. (1978). Sampling and estimation in large social networks, 1, 91–101.
Frank, O. (1979). Estimating a graph from triad counts. Journal of Statistical Computing and Simulation, 9, 31–46.
Frank, O. (1980). Estimating the number of vertices of different degrees in a graph. Journal of Statistical Planning and Inference. To appear.
Frank, O. (1980b). Sampling and inference in a population graph. International Statistical Review, 48, 33–41.
Frank, O. (1980c). A survey of statistical methods for graph and analysis. Sociological Methodology 1981, S. Leinhardt, ed. Jossey-Bass, San Francosco.
Hakimi, S. (1962). On the realizability of a set of integers as the degrees of the vertices of a graph. SIAM Journal of Applied Mathematics, 10, 496–506.
Holland, P., Leinhardt, S. (1970). A method for detecting structure in sociometric data. American Journal of Sociology, 70, 492–513.
Holland, P., Leinhardt, S. (1975). Local structure in social networks. In Sociological Methodology 1976, D. R. Heise, ed. Jossey-Bass, San Francisco.
Karónski, M. (1978). A Review of Random Graphs. Uniwersytet Im. Adama Mickiewicza W Poznaniu, Instytut Matematyki Poznan.
Mier, A., Moon, J. (1975). Climing certain types of rooted trees, I. In Proceedings of the Fifth British Combinatorial Conference, Aberdeen, 461–469
Meir, A., Moon, J. (1978a). Climbing certain types of rooted trees, II. Acta Mathematica Hungarian Academy of Sciences, 31, 43–54.
Meir, A., Moon, J. (1978b). On the altitude of nodes in random trees. Canadian Journal of Mathematics, 30, 997–1015.
Moon, J. (1971). The lengths of limbs of random trees. Mathematika, 18, 78–81.
Robinson, R., Schwenk, A. (1975). The distribution of degrees in a large random tree. Discrete Mathematics, 12, 359–372.
Wasserman, S. (1977). Random directed graph distribution and the triad census in social networks. Journal of Mathematical Sociology, 5, 61–86
Wright, E. (1974). Large cycles in large labelled graphs. Mathematical Proceedings of the Cambridge Philosophical Society, 78, 7–17.
Wright, E. (1977). Large cycles in large labeled graphs II. Mathematical Proceedings of the Cambridge Philosophical Society, 81, 1–2.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1981 D. Reidel Publishing Company
About this chapter
Cite this chapter
Capobianco, M. (1981). Some Distributions in the Theory of Graphs. In: Taillie, C., Patil, G.P., Baldessari, B.A. (eds) Statistical Distributions in Scientific Work. NATO Advanced study Institutes Series, vol 79. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-8552-0_31
Download citation
DOI: https://doi.org/10.1007/978-94-009-8552-0_31
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-009-8554-4
Online ISBN: 978-94-009-8552-0
eBook Packages: Springer Book Archive