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The Random Triangle Model

Johan Jonasson
Journal of Applied Probability
Vol. 36, No. 3 (Sep., 1999), pp. 852-867
Stable URL: http://www.jstor.org/stable/3215446
Page Count: 16
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
The Random Triangle Model
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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|qt(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 $n^{-\alpha}, \alpha > 0$, 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 $c < 3\alpha$, 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 $c^\prime > 3\alpha$, 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 $\alpha > 2/3$, 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.

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