Skip to main content Accessibility help

Random multigraphs and aggregated triads with fixed degrees

  • OVE FRANK (a1) and TERMEH SHAFIE (a2)


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.



Hide All
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), 296307.
Cover, T. M., & Thomas, J. A. (2006). Elements of information theory. Hoboken, NJ: John Wiley & Sons.
Frank, O. (1988). Triad count statistics. Annals of Discrete Mathematics, 38, 141149.
Frank, O. (2000). Structural plots of multivariate binary data. Journal of Social Structure, 1 (4), 119.
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. 177190). Berlin Heidelberg: Springer.
Frank, O., & Shafie, T. (2016). Multivariate entropy analysis of network data. Bulletin of Sociological Methodology/Bulletin de Méthodologie Sociologique, 129 (1), 4563.
Frank, O., & Strauss, D. (1986). Markov graphs. Journal of the American Statistical Association, 81 (395), 832842.
Holland, P. W., & Leinhardt, S. (1976). Local structure in social networks. Sociological Methodology, 7 (1), 146.
Janson, S. (2009). The probability that a random multigraph is simple. Combinatorics, Probability and Computing, 18 (1–2), 205225.
Kolaczyk, E. D. (2009). Statistical analysis of network data: Methods and models. New York, NY: Springer Verlag.
McKay, B. D., & Wormald, N. C. (1991). Asymptotic enumeration by degree sequence of graphs with degrees o(n 1/2). Combinatorica, 11 (4), 369382.
Shafie, T. (2012). Random multigraphs – complexity measures, probability models and statistical inference. Ph.D. thesis, Stockholm University.
Shafie, T. (2015). A multigraph approach to social network analysis. Journal of Social Structure, 16 (1), 21.
Shafie, T. (2016). Analyzing local and global properties of multigraphs. Journal of Mathematical Sociology, 40 (4), 239264.
Wasserman, S., & Faust, K. (1994). Social network analysis: Methods and applications. Cambridge, UK: Cambridge University Press.
Wormald, N. C. (1999). Models of random regular graphs. London Mathematical Society Lecture Note Series, 239–298.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed