The mixing time of the switch Markov chains: a unified approach

Since 1997 a considerable effort has been spent to study the mixing time of switch Markov chains on the realizations of graphic degree sequences of simple graphs. Several results were proved on rapidly mixing Markov chains on unconstrained, bipartite, and directed sequences, using different mechanisms. The aim of this paper is to unify these approaches. We will illustrate the strength of the unified method by showing that on any $P$-stable family of unconstrained/bipartite/directed degree sequences the switch Markov chain is rapidly mixing. This is a common generalization of every known result that shows the rapid mixing nature of the switch Markov chain on a region of degree sequences. Two applications of this general result will be presented. One is an almost uniform sampler for power-law degree sequences with exponent $\gamma>1+\sqrt{3}$. The other one shows that the switch Markov chain on the degree sequence of an Erd\H{o}s-R\'enyi random graph $G(n,p)$ is asymptotically almost surely rapidly mixing if $p$ is bounded away from 0 and 1 by at least $\frac{5\log n}{n-1}$.Comment: Clarification

Városi közlekedés monitorozása közösségi média segítségével

A városok közlekedésének optimalizációja a mai napig megoldásra váró problémát jelent a népesen lakott területek számára. Továbbá a folyamatosan fejlődő világunkban a pontosság és a határidők betartás elengedhetetlen. Így az utazási idő, fontos információ a lakosság számára. A közlekedés finomhangolásához nagy segítség, ha tudjuk az utazások célját, a tömegek mozgását, adott időszakra levetítve. A tanulmányban a közösségi média által szolgáltatott adatokat használjuk fel, ezek megfigyelésére

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 Erdős-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

Thalassaemiás betegen észlelt anaemia perniciosa – a diagnózis nehézségei = Pernicious anemia in a thalassemic patient – difficulties of the diagnosis

Absztrakt: A thalassaemia minoros betegünk progresszív, hypochrom, microcytás, hiporegeneratív anaemiája, leukopeniája miatt történt kivizsgálása során elvégzett csontvelő-aspiráció megaloblastos vérképzést igazolt. Az alacsony szérum-B12-vitamin-szint ezt támogatta, és a B12-vitamin-pótlást követően reticulocytakrízist és a vérszegénység jelentős javulását észleltük. A helyes diagnózis felállítását nehezítette, hogy a B12-vitamin-hiányra jellegzetes macrocytás, hyperchrom vörösvérsejt-morfológia helyett a thalassaemia minor miatt vészes vérszegénységben is hypochrom, microcytás eltérést észleltünk. Orv Hetil. 2018; 159(33): 1368–1371. | Abstract: The bone marrow aspiration, which was done in a leukopenic, hypochromic, microcytic, progressive anemic, thalassemic patient, revealed megaloblastic morphology. The low level of vitamin B12 and the reticulocytosis following the B12 supportation strenghtened the diagnosis of pernicious anemia. The set of the right diagnosis has been delayed by the fact that even in severe anemia one could not obtain the typical signs of B12 deficiency, having a hypochromic, microcytic erythrocyte morphology, due to the thalassemia minor disorder. Orv Hetil. 2018; 159(33): 1368–1371

Half-graphs, other non-stable degree sequences, and the switch Markov chain

One of the simplest methods of generating a random graph with a given degree sequence is provided by the Monte Carlo Markov Chain method using switches. The switch Markov chain converges to the uniform distribution, but generally the rate of convergence is not known. After a number of results concerning various degree sequences, rapid mixing was established for so-called $P$-stable degree sequences (including that of directed graphs), which covers every previously known rapidly mixing region of degree sequences. In this paper we give a non-trivial family of degree sequences that are not $P$-stable and the switch Markov chain is still rapidly mixing on them. This family has an intimate connection to Tyshkevich-decompositions and strong stability as well.Comment: Generalized the main theorem, paper extended with a number of corroborating result