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# The random triangle model

Abstract

The random triangle model is a Markov random graph model which, for parameters p ∊ (0,1) and q ≥ 1 and a graph G = (V,E), assigns to a subset, η, of E, a probability which is proportional to p |η|(1-p)|E|-|η| q t(η), where t(η) is the number of triangles in η. It is shown that this model has maximum entropy in the class of distributions with given edge and triangle probabilities.

Using an analogue of the correspondence between the Fortuin-Kesteleyn random cluster model and the Potts model, the asymptotic behavior of the random triangle model on the complete graph is examined for p of order n −α, α > 0, and different values of q, where q is written in the form q = 1 + h(n) / n. It is shown that the model exhibits an explosive behavior in the sense that if h(n) ≤ c log n for c < 3α, then the edge probability and the triangle probability are asymptotically the same as for the ordinary G(n,p) model, whereas if h(n) ≥ c' log n for c' > 3α, then these quantities both tend to 1. For critical values, h(n) = 3α log n + o(log n), the probability mass divides between these two extremes.

Moreover, if h(n) is of higher order than log n, then the probability that η = E tends to 1, whereas if h(n) = o(log n) and α > 2/3, then, with a probability tending to 1, the resulting graph can be coupled with a graph resulting from the G(n,p) model. In particular these facts mean that for values of p in the range critical for the appearance of the giant component and the connectivity of the graph, the way in which triangles are rewarded can only have a degenerate influence.

Corresponding author
Postal address: Department of Mathematics, Chalmers University of Technology, S-412 96, Göteborg, Sweden. Email address: expect@math.chalmers.se.
References
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[1] Bollobás, B. (1985). Random Graphs. Academic Press, London.
[2] Bollobás, B., Grimmett, G., and Janson, S. (1996). The random cluster model on the complete graph. Prob. Theory Rel. Fields 104, 283317.
[3] Edwards, R. G., and Sokal, A. D. (1988). Generalization of the Fortuin–Kasteleyn–Swendsen–Wang representation and Monte Carlo algorithm. Phys. Rev. 38, 20092012.
[4] Erdős, P. and Rényi, A. (1959). On random graphs I. Publ. Math. Debrecen 6, 290297.
[5] Faust, K., and Wasserman, S. (1994). Social Network Analysis. Cambridge University Press, Cambridge.
[6] Fortuin, C. M., and Kasteleyn, P. W. (1972). On the random cluster model I, introduction and relation to other models. Physica 58, 393418.
[7] Frank, O. (1988). Random sampling and social networks: a survey of various approaches, Math. Sci. Humaines 104, 1933.
[8] Häggström, O. (1996). Random cluster representations in the study of phase transitions. Preprint, Chalmers University of Technology and Göteborg University. Markov Proc. Random Fields, 4, 275321.
[9] Häggström, O., and Jonasson, J. (1997). Phase transition in the random triangle model. Preliminary version. To appear in J. Appl. Prob. 36.
[10] Janson, S., Knuth, D., Luczak, T., and Pittel, B. (1993). The birth of the giant component. Rand. Struct. Alg. 4, 233358.
[11] Karonski, M., and Luczak, T. (1996). Random hypergraphs. Combinatorics, Paul Erdős is Eighty, Vol 2, pp. 283293. Janos Bolyai Math Soc., Budapest.
[12] Karonski, M., Scheinerman, E., and Singer, K. (1997). On random intersection graphs: the subgraph problem. Preprint. Preliminary version.
[13] Strauss, D. (1986). On a general class of models for interaction. SIAM Review 28, 513527.
[14] Swendsen, R. H., and Wang, J. S. (1987). Nonuniversal critical dynamics in Monte Carlo simulation. Phys. Rev. Lett. 59, 8688.
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Journal of Applied Probability
• ISSN: 0021-9002
• EISSN: 1475-6072
• URL: /core/journals/journal-of-applied-probability
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