41 research outputs found
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 -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
. The other one shows that the switch Markov chain on the
degree sequence of an Erd\H{o}s-R\'enyi random graph is asymptotically
almost surely rapidly mixing if is bounded away from 0 and 1 by at least
.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.
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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 -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
-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