Topology of random simplicial complexes: a survey

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

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 $G(H,n,p)$ is the random (multi)graph obtained by adding an instance of a fixed graph $H$ on each of the copies of $H$ in the complete graph on $n$ vertices, independently with probability $p$. 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