2,590 research outputs found
On the swap-distances of different realizations of a graphical degree sequence
One of the first graph theoretical problems which got serious attention
(already in the fifties of the last century) was to decide whether a given
integer sequence is equal to the degree sequence of a simple graph (or it is
{\em graphical} for short). One method to solve this problem is the greedy
algorithm of Havel and Hakimi, which is based on the {\em swap} operation.
Another, closely related question is to find a sequence of swap operations to
transform one graphical realization into another one of the same degree
sequence. This latter problem got particular emphases in connection of fast
mixing Markov chain approaches to sample uniformly all possible realizations of
a given degree sequence. (This becomes a matter of interest in connection of --
among others -- the study of large social networks.) Earlier there were only
crude upper bounds on the shortest possible length of such swap sequences
between two realizations. In this paper we develop formulae (Gallai-type
identities) for these {\em swap-distance}s of any two realizations of simple
undirected or directed degree sequences. These identities improves considerably
the known upper bounds on the swap-distances.Comment: to be publishe
On realization graphs of degree sequences
Given the degree sequence of a graph, the realization graph of is the
graph having as its vertices the labeled realizations of , with two vertices
adjacent if one realization may be obtained from the other via an
edge-switching operation. We describe a connection between Cartesian products
in realization graphs and the canonical decomposition of degree sequences
described by R.I. Tyshkevich and others. As applications, we characterize the
degree sequences whose realization graphs are triangle-free graphs or
hypercubes.Comment: 10 pages, 5 figure
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
Algebraic matroids with graph symmetry
This paper studies the properties of two kinds of matroids: (a) algebraic
matroids and (b) finite and infinite matroids whose ground set have some
canonical symmetry, for example row and column symmetry and transposition
symmetry.
For (a) algebraic matroids, we expose cryptomorphisms making them accessible
to techniques from commutative algebra. This allows us to introduce for each
circuit in an algebraic matroid an invariant called circuit polynomial,
generalizing the minimal poly- nomial in classical Galois theory, and studying
the matroid structure with multivariate methods.
For (b) matroids with symmetries we introduce combinatorial invariants
capturing structural properties of the rank function and its limit behavior,
and obtain proofs which are purely combinatorial and do not assume algebraicity
of the matroid; these imply and generalize known results in some specific cases
where the matroid is also algebraic. These results are motivated by, and
readily applicable to framework rigidity, low-rank matrix completion and
determinantal varieties, which lie in the intersection of (a) and (b) where
additional results can be derived. We study the corresponding matroids and
their associated invariants, and for selected cases, we characterize the
matroidal structure and the circuit polynomials completely
Graph realizations constrained by skeleton graphs
In 2008 Amanatidis, Green and Mihail introduced the Joint Degree Matrix (JDM)
model to capture the fundamental difference in assortativity of networks in
nature studied by the physical and life sciences and social networks studied in
the social sciences. In 2014 Czabarka proposed a direct generalization of the
JDM model, the Partition Adjacency Matrix (PAM) model. In the PAM model the
vertices have specified degrees, and the vertex set itself is partitioned into
classes. For each pair of vertex classes the number of edges between the
classes in a graph realization is prescribed. In this paper we apply the new
{\em skeleton graph} model to describe the same information as the PAM model.
Our model is more convenient for handling problems with low number of partition
classes or with special topological restrictions among the classes. We
investigate two particular cases in detail: (i) when there are only two vertex
classes and (ii) when the skeleton graph contains at most one cycle.Comment: 19 page
Marathon: An open source software library for the analysis of Markov-Chain Monte Carlo algorithms
In this paper, we consider the Markov-Chain Monte Carlo (MCMC) approach for
random sampling of combinatorial objects. The running time of such an algorithm
depends on the total mixing time of the underlying Markov chain and is unknown
in general. For some Markov chains, upper bounds on this total mixing time
exist but are too large to be applicable in practice. We try to answer the
question, whether the total mixing time is close to its upper bounds, or if
there is a significant gap between them. In doing so, we present the software
library marathon which is designed to support the analysis of MCMC based
sampling algorithms. The main application of this library is to compute
properties of so-called state graphs which represent the structure of Markov
chains. We use marathon to investigate the quality of several bounding methods
on four well-known Markov chains for sampling perfect matchings and bipartite
graph realizations. In a set of experiments, we compute the total mixing time
and several of its bounds for a large number of input instances. We find that
the upper bound gained by the famous canonical path method is several
magnitudes larger than the total mixing time and deteriorates with growing
input size. In contrast, the spectral bound is found to be a precise
approximation of the total mixing time
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