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

Dependent Random Graphs And Multi-Party Pointer Jumping

We initiate a study of a relaxed version of the standard Erdos-Renyi random graph model, where each edge may depend on a few other edges. We call such graphs dependent random graphs . Our main result in this direction is a thorough understanding of the clique number of dependent random graphs. We also obtain bounds for the chromatic number. Surprisingly, many of the standard properties of random graphs also hold in this relaxed setting. We show that with high probability, a dependent random graph will contain a clique of size ((1-o(1))log(n))/log(1/p), and the chromatic number will be at most (nlog(1/(1-p)))/log(n). We expect these results to be of independent interest. As an application and second main result, we give a new communication protocol for the k-player Multi-Party Pointer Jumping problem (MPJk) in the number-on-the-forehead (NOF) model. Multi-Party Pointer Jumping is one of the canonical NOF communication problems, yet even for three players, its communication complexity is not well understood. Our protocol for MPJ3 costs O((n * log(log(n)))/log(n)) communication, improving on a bound from [BrodyChakrabarti08]. We extend our protocol to the non-Boolean pointer jumping problem, achieving an upper bound which is o(n) for any k \u3e= 4 players. This is the first o(n) protocol and improves on a bound of Damm, Jukna, and Sgall, which has stood for almost twenty years

Random Interval Graphs

In this thesis, which is supervised by Dr. David Penman, we examine random interval graphs. Recall that such a graph is defined by letting $X_{1},\ldots X_{n},Y_{1},\ldots Y_{n}$ be $2n$ independent random variables, with uniform distribution on $[0,1]$. We then say that the $i$th of the $n$ vertices is the interval $[X_{i},Y_{i}]$ if $X_{i}<Y_{i}$ and the interval $[Y_{i},X_{i}]$ if $Y_{i}<X_{i}$. We then say that two vertices are adjacent if and only if the corresponding intervals intersect. We recall from our MA902 essay that fact that in such a graph, each edge arises with probability $2/3$, and use this fact to obtain estimates of the number of edges. Next, we turn to how these edges are spread out, seeing that (for example) the range of degrees for the vertices is much larger than classically, by use of an interesting geometrical lemma. We further investigate the maximum degree, showing it is always very close to the maximum possible value $(n-1)$, and the striking result that it is equal to $(n-1)$ with probability exactly $2/3$. We also recall a result on the minimum degree, and contrast all these results with the much narrower range of values obtained in the alternative \lq comparable\rq\, model $G(n,2/3)$ (defined later). We then study clique numbers, chromatic numbers and independence numbers in the Random Interval Graphs, presenting (for example) a result on independence numbers which is proved by considering the largest chain in the associated interval order. Last, we make some brief remarks about other ways to define random interval graphs, and extensions of random interval graphs, including random dot product graphs and other ways to define random interval graphs. We also discuss some areas these ideas should be usable in. We close with a summary and some comments

On the Giant Component of Geometric Inhomogeneous Random Graphs

In this paper we study the threshold model of geometric inhomogeneous random graphs (GIRGs); a generative random graph model that is closely related to hyperbolic random graphs (HRGs). These models have been observed to capture complex real-world networks well with respect to the structural and algorithmic properties. Following comprehensive studies regarding their connectivity, i.e., which parts of the graphs are connected, we have a good understanding under which circumstances a giant component (containing a constant fraction of the graph) emerges. While previous results are rather technical and challenging to work with, the goal of this paper is to provide more accessible proofs. At the same time we significantly improve the previously known probabilistic guarantees, showing that GIRGs contain a giant component with probability 1 - exp(-?(n^{(3-?)/2})) for graph size n and a degree distribution with power-law exponent ? ? (2, 3). Based on that we additionally derive insights about the connectivity of certain induced subgraphs of GIRGs

Bounds on Mixing Time for Time-Inhomogeneous Markov Chains

Mixing of finite time-homogeneous Markov chains is well understood nowadays, with a rich set of techniques to estimate their mixing time. In this paper, we study the mixing time of random walks in dynamic random environments. To that end, we propose a concept of mixing time for time-inhomogeneous Markov chains. We then develop techniques to estimate this mixing time by extending the evolving set method of Morris and Peres (2003). We apply these techniques to study a random walk on a dynamic Erd\H{o}s-R\'enyi graph, proving that the mixing time is $O(\log(n))$ when the graph is well above the connectivity threshold. We also give an almost matching lower bound