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Distributional limits for critical random graphs
 In preparation
, 2009
"... We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3, for some fixed λ ∈ R. Then, as a metric space with the graph distance rescaled by n −1/3, the sequence of connected components G(n, p) converges towards a sequence of continuous compact metri ..."
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We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3, for some fixed λ ∈ R. Then, as a metric space with the graph distance rescaled by n −1/3, the sequence of connected components G(n, p) converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n, p) rescaled by n −1/3 converges in distribution to an absolutely continuous random variable with finite mean. Keywords: Random graphs, GromovHausdorff distance, scaling limits, continuum random tree, diameter. 2000 Mathematics subject classification: 05C80, 60C05.
A sharp threshold for minimum boundeddepth and boundeddiameter spanning trees and Steiner trees in random networks
, 2008
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9. Bibliography............................................................................111. Team Research Scientist
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c t i v it y e p o r t 2009 Table of contents 1. Team.................................................................................... 1
9. Bibliography............................................................................101. Team
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c t i v it y e p o r t 2008 Table of contents 1. Team.................................................................................... 1
ProjectTeam Mascotte Méthodes Algorithmiques, Simulation et Combinatoire pour l’OpTimisation des
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Average Update Times for FullyDynamic AllPairs Shortest Paths ✩
"... We study the fullydynamicall pairsshortestpath problem forgraphswith arbitrary nonnegative edge weights. It is known for digraphs that an update of the distance matrix costs O(n 2.75 polylog(n)) worstcase time [Thorup, STOC ’05] and O(n 2 log 3 (n)) amortized time [Demetrescu and Italiano, J.ACM ..."
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We study the fullydynamicall pairsshortestpath problem forgraphswith arbitrary nonnegative edge weights. It is known for digraphs that an update of the distance matrix costs O(n 2.75 polylog(n)) worstcase time [Thorup, STOC ’05] and O(n 2 log 3 (n)) amortized time [Demetrescu and Italiano, J.ACM ’04] where n is the number of vertices. We present the first averagecase analysis of the undirected problem. For a random update we show that the expected time per update is bounded by O(n 4/3+ε) for all ε> 0. If the graph is outside the critical window, we prove even smaller bounds.