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Random multigraphs and aggregated triads with fixed degrees

Published online by Cambridge University Press:  28 December 2017

OVE FRANK  and
TERMEH SHAFIE
OVE FRANK
Affiliation:
Department of Statistics, Stockholm University, Sweden (e-mail: ove.frank@stat.su.se)
TERMEH SHAFIE
Affiliation:
Department of Computer & Information Science, University of Konstanz, Germany (e-mail: termeh.shafie@uni-konstanz.de)
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Abstract

Random multigraphs with fixed degrees are obtained by the configuration model or by so called random stub matching. New combinatorial results are given for the global probability distribution of edge multiplicities and its marginal local distributions of loops and edges. The number of multigraphs on triads is determined for arbitrary degrees, and aggregated triads are shown to be useful for analyzing regular and almost regular multigraphs. Relationships between entropy and complexity are given and numerically illustrated for multigraphs with different number of vertices and specified average and variance for the degrees.

Keywords

multigraphfixed degreesedge multiplicityconfiguration modelrandom stub matchingcomplexity and simplicityentropyaggregated triad

Information

Type
Research Article
Information
Network Science , Volume 6 , Issue 2 , June 2018 , pp. 232 - 250
Copyright
Copyright © Cambridge University Press 2017 

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References

Bender, E. A, & Canfield, R. E. (1978). The asymptotic number of labeled graphs with given degree sequences. Journal of Combinatorial Theory Series A, 24 (3), 296–307.CrossRefGoogle Scholar
Cover, T. M., & Thomas, J. A. (2006). Elements of information theory. Hoboken, NJ: John Wiley & Sons.Google Scholar
Frank, O. (1988). Triad count statistics. Annals of Discrete Mathematics, 38, 141–149.CrossRefGoogle Scholar
Frank, O. (2000). Structural plots of multivariate binary data. Journal of Social Structure, 1 (4), 1–19.Google Scholar
Frank, O. (2011). Statistical information tools for multivariate discrete data. In Pardo, L., Balakrishnan, N., & Gil, M. A. (Eds.), Modern mathematical tools and techniques in capturing complexity (pp. 177–190). Berlin Heidelberg: Springer.CrossRefGoogle Scholar
Frank, O., & Shafie, T. (2016). Multivariate entropy analysis of network data. Bulletin of Sociological Methodology/Bulletin de Méthodologie Sociologique, 129 (1), 45–63.CrossRefGoogle Scholar
Frank, O., & Strauss, D. (1986). Markov graphs. Journal of the American Statistical Association, 81 (395), 832–842.CrossRefGoogle Scholar
Holland, P. W., & Leinhardt, S. (1976). Local structure in social networks. Sociological Methodology, 7 (1), 1–46.CrossRefGoogle Scholar
Janson, S. (2009). The probability that a random multigraph is simple. Combinatorics, Probability and Computing, 18 (1–2), 205–225.CrossRefGoogle Scholar
Kolaczyk, E. D. (2009). Statistical analysis of network data: Methods and models. New York, NY: Springer Verlag.CrossRefGoogle Scholar
McKay, B. D., & Wormald, N. C. (1991). Asymptotic enumeration by degree sequence of graphs with degrees o(n 1/2). Combinatorica, 11 (4), 369–382.CrossRefGoogle Scholar
Shafie, T. (2012). Random multigraphs – complexity measures, probability models and statistical inference. Ph.D. thesis, Stockholm University.Google Scholar
Shafie, T. (2015). A multigraph approach to social network analysis. Journal of Social Structure, 16 (1), 21.CrossRefGoogle Scholar
Shafie, T. (2016). Analyzing local and global properties of multigraphs. Journal of Mathematical Sociology, 40 (4), 239–264.CrossRefGoogle Scholar
Wasserman, S., & Faust, K. (1994). Social network analysis: Methods and applications. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Wormald, N. C. (1999). Models of random regular graphs. London Mathematical Society Lecture Note Series, 239–298.CrossRefGoogle Scholar