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- W1985770240 abstract "Abstract We consider the problem of the asymptotic size of the random maximum-weight matching of a sparse random graph, which we translate into dynamics of the operator in the space of distribution functions. A tight condition for the uniqueness of the globally attracting fixed point is provided, which extends the result of Karp and Sipser [Maximum matchings in sparse random graphs. 22nd Ann. Symp. on Foundations of Computer Science (Nashville, TN, 28 – 30 October, 1981) . IEEE, New York, 1981, pp. 364–375] from deterministic weight distributions (Dirac measures μ ) to general ones. Given a probability measure μ which corresponds to the weight distribution of a link of a random graph, we form a positive linear operator Φ μ (convolution) on distribution functions and then analyze a family of its exponents, with parameter λ , which corresponds to the connectivity of a sparse random graph. The operator 𝕋 relates the distribution F on the subtrees to the distribution 𝕋 F on the node of the tree by 𝕋 F =exp (− λΦ μ F ). We prove that for every probability measure μ and every λ < e , there exists a unique globally attracting fixed point of the operator; the probability measure corresponding to this fixed point can then be used to compute the expected maximum-weight matching on a sparse random graph. This result is called the e -cutoff phenomenon. For deterministic distributions and λ > e , there is no fixed point attractor. We further establish that the uniqueness of the invariant measure of the underlying operator is not a monotone property of the average connectivity; this parallels similar non-monotonicity results in the statistical physics context." @default.
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- W1985770240 date "2008-10-01" @default.
- W1985770240 modified "2023-09-25" @default.
- W1985770240 title "Invariant probability measures and dynamics of exponential linear type maps" @default.
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- W1985770240 doi "https://doi.org/10.1017/s014338570700106x" @default.
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