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The random triangle model

Part of: Multivariate analysis Graph theory

Published online by Cambridge University Press:  14 July 2016

Johan Jonasson
Johan Jonasson*
Affiliation:
Chalmers University of Technology
*
∗Postal address: Department of Mathematics, Chalmers University of Technology, S-412 96, Göteborg, Sweden. Email address: expect@math.chalmers.se.
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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−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.

Keywords

Random graphrandom cluster modelSwendsen–Wang correspondenceentropy

MSC classification

Primary: 05C80: Random graphs
Secondary: 62H30: Classification and discrimination; cluster analysis
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 852 - 867
Copyright
Copyright © Applied Probability Trust 1999 

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