Surface Effects in Dense Random Graphs with Sharp Edge Constraint
نویسندگان
چکیده
We show that the random number Tn of triangles in a random graph on n vertices, with a strict constraint on the total number of edges, admits an expansion Tn = an 3 + bn + Fn, where a and b are numbers, with the mean 〈Fn〉 = O(n) and the standard deviation σ(Tn) = σ(Fn) = O(n ). The presence of a ‘surface term’ bn has a significance analogous to the macroscopic surface effects of materials, and is missing in the model where the edge constraint is removed. We also find the surface effect in other graph models using similar edge constraints. 1. A random graph model with dependent edges Consider the spaces Gn, n = 1, . . . , of simple graphs on n labeled vertices, on which we will define probability distributions, giving us random graph models of increasing ‘size’ n. Let H be an arbitrary but fixed graph, and for g ∈ Gn let TH(g) denote the number of copies of H found in g. We will compute the growth rates of the expectation and variance of TH , and will show that the expectation has both ‘volume’ and ‘surface’ rates of growth, which are not overshadowed by the lower rate of growth of its standard deviation. This is analogous to the volume and surface components of macroscopic materials, and indeed our models were chosen to mimick the statistical mechanics model of macroscopic materials. We define our probability distributions as follow. For each 0 < p < 1 fix some sequence En = p (
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تاریخ انتشار 2017