1,759 research outputs found

    Topology of random simplicial complexes: a survey

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    This expository article is based on a lecture from the Stanford Symposium on Algebraic Topology: Application and New Directions, held in honor of Gunnar Carlsson, Ralph Cohen, and Ib Madsen.Comment: After revisions, now 21 pages, 5 figure

    Thresholds in Random Motif Graphs

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    We introduce a natural generalization of the Erd\H{o}s-R\'enyi random graph model in which random instances of a fixed motif are added independently. The binomial random motif graph G(H,n,p)G(H,n,p) is the random (multi)graph obtained by adding an instance of a fixed graph HH on each of the copies of HH in the complete graph on nn vertices, independently with probability pp. We establish that every monotone property has a threshold in this model, and determine the thresholds for connectivity, Hamiltonicity, the existence of a perfect matching, and subgraph appearance. Moreover, in the first three cases we give the analogous hitting time results; with high probability, the first graph in the random motif graph process that has minimum degree one (or two) is connected and contains a perfect matching (or Hamiltonian respectively).Comment: 19 page
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