An official website of the United States government
Given a fixed graph H, what is the (exponentially small) probability that the number XHof copies of Hin the binomial random graph Gn,pis at least twice its mean? Studied intensively since the mid 1990s, this so‐called infamous upper tail problem remains a challenging testbed for concentration inequalities. In 2011 DeMarco and Kahn formulated an intriguing conjecture about the exponential rate of decay of for fixed ε > 0. We show that this upper tail conjecture is false, by exhibiting an infinite family of graphs violating the conjectured bound.
more »
« less
Expansion of random 0/1 polytopes
Abstract A conjecture of Milena Mihail and Umesh Vazirani (Proc. 24th Annu. ACM Symp. Theory Comput., ACM, Victoria, BC, 1992, pp. 26–38.) states that the edge expansion of the graph of every polytope is at least one. Any lower bound on the edge expansion gives an upper bound for the mixing time of a random walk on the graph of the polytope. Such random walks are important because they can be used to generate an element from a set of combinatorial objects uniformly at random. A weaker form of the conjecture of Mihail and Vazirani says that the edge expansion of the graph of a polytope in is greater than one over some polynomial function of . This weaker version of the conjecture would suffice for all applications. Our main result is that the edge expansion of the graph of arandompolytope in is at least with high probability.
more »
« less
- PAR ID:
- 10521272
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Random Structures & Algorithms
- Volume:
- 64
- Issue:
- 2
- ISSN:
- 1042-9832
- Page Range / eLocation ID:
- 309 to 319
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract A conjecture of Kalai asserts that for $$d\geq 4$$, the affine type of a prime simplicial $$d$$-polytope $$P$$ can be reconstructed from the space of affine $$2$$-stresses of $$P$$. We prove this conjecture for all $$d\geq 5$$. We also prove the following generalization: for all pairs $(i,d)$ with $$2\leq i\leq \lceil \frac d 2\rceil -1$$, the affine type of a simplicial $$d$$-polytope $$P$$ that has no missing faces of dimension $$\geq d-i+1$$ can be reconstructed from the space of affine $$i$$-stresses of $$P$$. A consequence of our proofs is a strengthening of the Generalized Lower Bound Theorem: it was proved by Nagel that for any simplicial $(d-1)$-sphere $$\Delta $$ and $$1\leq k\leq \lceil \frac {d}{2}\rceil -1$$, $$g_{k}(\Delta )$$ is at least as large as the number of missing $(d-k)$-faces of $$\Delta $$; here we show that, for $$1\leq k\leq \lfloor \frac {d}{2}\rfloor -1$$, equality holds if and only if $$\Delta $$ is $$k$$-stacked. Finally, we show that for $$d\geq 4$$, any simplicial $$d$$-polytope $$P$$ that has no missing faces of dimension $$\geq d-1$$ is redundantly rigid, that is, for each edge $$e$$ of $$P$$, there exists an affine $$2$$-stress on $$P$$ with a non-zero value on $$e$$.more » « less
-
Abstract We show that for every integer and large , every properly edge‐colored graph on vertices with at least edges contains a rainbow subdivision of . This is sharp up to a polylogarithmic factor. Our proof method exploits the connection between the mixing time of random walks and expansion in graphs.more » « less
-
Symmetric edge polytopes are lattice polytopes associated with finite simplegraphs that are of interest in both theory and applications. We investigate thefacet structure of symmetric edge polytopes for various models of randomgraphs. For an Erd\H{o}s-Renyi random graph, we identify a thresholdprobability at which with high probability the symmetric edge polytope sharesmany facet-supporting hyperplanes with that of a complete graph. We alsoinvestigate the relationship between the average local clustering, also knownas the Watts-Strogatz clustering coefficient, and the number of facets forgraphs with either a fixed number of edges or a fixed degree sequence. We usewell-known Markov Chain Monte Carlo sampling methods to generate empiricalevidence that for a fixed degree sequence, higher average local clustering in aconnected graph corresponds to higher facet numbers in the associated symmetricedge polytope.more » « less
-
The upper tail problem in a random graph asks to estimate the probability that the number of copies of some fixed subgraph in an Erdős‐Rényi random graph exceeds its expectation by some constant factor. There has been much exciting recent progress on this problem. We study the corresponding problem for hypergraphs, for which less is known about the large deviation rate. We present new phenomena in upper tail large deviations for sparse random hypergraphs that are not seen in random graphs. We conjecture a formula for the large deviation rate, that is, the first order asymptotics of the log‐probability that the number of copies of fixed subgraphHin a sparse Erdős‐Rényi randomk‐uniform hypergraph exceeds its expectation by a constant factor. This conjecture turns out to be significantly more intricate compared to the case for graphs. We verify our conjecture when the fixed subgraphHbeing counted is a clique, as well as whenHis the 3‐uniform 6‐vertex 4‐edge hypergraph consisting of alternating faces of an octahedron, where new techniques are required.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1002/rsa.21184
