Constructing and sampling partite, 3-uniform hypergraphs with given degree sequence.

Partite, 3-uniform hypergraphs are 3-uniform hypergraphs in which each hyperedge contains exactly one point from each of the 3 disjoint vertex classes. We consider the degree sequence problem of partite, 3-uniform hypergraphs, that is, to decide if such a hypergraph with prescribed degree sequences exists. We prove that this decision problem is NP-complete in general, and give a polynomial running time algorithm for third almost-regular degree sequences, that is, when each degree in one of the vertex classes is k or k - 1 for some fixed k, and there is no restriction for the other two vertex classes. We also consider the sampling problem, that is, to uniformly sample partite, 3-uniform hypergraphs with prescribed degree sequences. We propose a Parallel Tempering method, where the hypothetical energy of the hypergraphs measures the deviation from the prescribed degree sequence. The method has been implemented and tested on synthetic and real data. It can also be applied for χ2 testing of contingency tables. We have shown that this hypergraph-based χ2 test is more sensitive than the standard χ2 test. The extra sensitivity is especially advantageous on small data sets, where the proposed Parallel Tempering method shows promising performance.

Efficiently sampling the realizations of bounded, irregular degree sequences of bipartite and directed graphs.

Since 1997 a considerable effort has been spent on the study of the swap (switch) Markov chains on graphic degree sequences. All of these results assume some kind of regularity in the corresponding degree sequences. Recently, Greenhill and Sfragara published a breakthrough paper about irregular normal and directed degree sequences for which rapid mixing of the swap Markov chain is proved. In this paper we present two groups of results. An example from the first group is the following theorem: let [Formula: see text] be a directed degree sequence on n vertices. Denote by Δ the maximum value among all in- and out-degrees and denote by [Formula: see text] the number of edges in the realization. Assume furthermore that [Formula: see text]. Then the swap Markov chain on the realizations of [Formula: see text] is rapidly mixing. This result is a slight improvement on one of the results of Greenhill and Sfragara. An example from the second group is the following: let d be a bipartite degree sequence on the vertex set U ⊎ V, and let 0 < c1 ≤ c2 < |U| and 0 < d1 ≤ d2 < |V| be integers, where c1 ≤ d(v) ≤ c2: ∀v ∈ V and d1 ≤ d(u) ≤ d2: ∀u ∈ U. Furthermore assume that (c2 - c1 - 1)(d2 - d1 - 1) < max{c1(|V| - d2), d1(|U| - c2)}. Then the swap Markov chain on the realizations of d is rapidly mixing. A straightforward application of this latter result shows that when a random bipartite or directed graph is generated under the Erdos-Rényi G(n, p) model with mild assumptions on n and p then the degree sequence of the generated graph has, with high probability, a rapidly mixing swap Markov chain on its realizations.

Oral administration of pyrophosphate inhibits connective tissue calcification.

Various disorders including pseudoxanthoma elasticum (PXE) and generalized arterial calcification of infancy (GACI), which are caused by inactivating mutations in ABCC6 and ENPP1, respectively, present with extensive tissue calcification due to reduced plasma pyrophosphate (PPi). However, it has always been assumed that the bioavailability of orally administered PPi is negligible. Here, we demonstrate increased PPi concentration in the circulation of humans after oral PPi administration. Furthermore, in mouse models of PXE and GACI, oral PPi provided via drinking water attenuated their ectopic calcification phenotype. Noticeably, provision of drinking water with 0.3 mM PPi to mice heterozygous for inactivating mutations in Enpp1 during pregnancy robustly inhibited ectopic calcification in their Enpp1-/- offspring. Our work shows that orally administered PPi is readily absorbed in humans and mice and inhibits connective tissue calcification in mouse models of PXE and GACI PPi, which is recognized as safe by the FDA, therefore not only has great potential as an effective and extremely low-cost treatment for these currently intractable genetic disorders, but also in other conditions involving connective tissue calcification.