152 research outputs found
Structure and enumeration of (3+1)-free posets
A poset is (3+1)-free if it does not contain the disjoint union of chains of
length 3 and 1 as an induced subposet. These posets play a central role in the
(3+1)-free conjecture of Stanley and Stembridge. Lewis and Zhang have
enumerated (3+1)-free posets in the graded case by decomposing them into
bipartite graphs, but until now the general enumeration problem has remained
open. We give a finer decomposition into bipartite graphs which applies to all
(3+1)-free posets and obtain generating functions which count (3+1)-free posets
with labelled or unlabelled vertices. Using this decomposition, we obtain a
decomposition of the automorphism group and asymptotics for the number of
(3+1)-free posets.Comment: 28 pages, 5 figures. New version includes substantial changes to
clarify the construction of skeleta and the enumeration. An extended abstract
of this paper appears as arXiv:1212.535
Uniform random generation of large acyclic digraphs
Directed acyclic graphs are the basic representation of the structure
underlying Bayesian networks, which represent multivariate probability
distributions. In many practical applications, such as the reverse engineering
of gene regulatory networks, not only the estimation of model parameters but
the reconstruction of the structure itself is of great interest. As well as for
the assessment of different structure learning algorithms in simulation
studies, a uniform sample from the space of directed acyclic graphs is required
to evaluate the prevalence of certain structural features. Here we analyse how
to sample acyclic digraphs uniformly at random through recursive enumeration,
an approach previously thought too computationally involved. Based on
complexity considerations, we discuss in particular how the enumeration
directly provides an exact method, which avoids the convergence issues of the
alternative Markov chain methods and is actually computationally much faster.
The limiting behaviour of the distribution of acyclic digraphs then allows us
to sample arbitrarily large graphs. Building on the ideas of recursive
enumeration based sampling we also introduce a novel hybrid Markov chain with
much faster convergence than current alternatives while still being easy to
adapt to various restrictions. Finally we discuss how to include such
restrictions in the combinatorial enumeration and the new hybrid Markov chain
method for efficient uniform sampling of the corresponding graphs.Comment: 15 pages, 2 figures. To appear in Statistics and Computin
Asymptotic analysis and efficient random sampling of directed ordered acyclic graphs
Directed acyclic graphs (DAGs) are directed graphs in which there is no path
from a vertex to itself. DAGs are an omnipresent data structure in computer
science and the problem of counting the DAGs of given number of vertices and to
sample them uniformly at random has been solved respectively in the 70's and
the 00's. In this paper, we propose to explore a new variation of this model
where DAGs are endowed with an independent ordering of the out-edges of each
vertex, thus allowing to model a wide range of existing data structures. We
provide efficient algorithms for sampling objects of this new class, both with
or without control on the number of edges, and obtain an asymptotic equivalent
of their number. We also show the applicability of our method by providing an
effective algorithm for the random generation of classical labelled DAGs with a
prescribed number of vertices and edges, based on a similar approach. This is
the first known algorithm for sampling labelled DAGs with full control on the
number of edges, and it meets a need in terms of applications, that had already
been acknowledged in the literature.Comment: 32 pages, 12 figures. For the implementation of the algorithms, see
https://github.com/Kerl13/randda
On the chromatic number of the preferential attachment graph
We prove that for every and every , the
chromatic number of the preferential attachment graph is
asymptotically almost surely equal to . The proof relies on a
combinatorial construction of a family of digraphs of chromatic number
followed by a proof that asymptotically almost surely there is a digraph in
this family, which is realised as a subgraph of the preferential attachment
graph.Comment: 14 pages, 3 figure
The birth of the strong components
Random directed graphs undergo a phase transition around the point
, and the width of the transition window has been known since the
works of Luczak and Seierstad. They have established that as
when , the asymptotic probability that the strongly
connected components of a random directed graph are only cycles and single
vertices decreases from 1 to 0 as goes from to .
By using techniques from analytic combinatorics, we establish the exact
limiting value of this probability as a function of and provide more
properties of the structure of a random digraph around, below and above its
transition point. We obtain the limiting probability that a random digraph is
acyclic and the probability that it has one strongly connected complex
component with a given difference between the number of edges and vertices
(called excess). Our result can be extended to the case of several complex
components with given excesses as well in the whole range of sparse digraphs.
Our study is based on a general symbolic method which can deal with a great
variety of possible digraph families, and a version of the saddle-point method
which can be systematically applied to the complex contour integrals appearing
from the symbolic method. While the technically easiest model is the model of
random multidigraphs, in which multiple edges are allowed, and where edge
multiplicities are sampled independently according to a Poisson distribution
with a fixed parameter , we also show how to systematically approach the
family of simple digraphs, where multiple edges are forbidden, and where
2-cycles are either allowed or not.
Our theoretical predictions are supported by numerical simulations, and we
provide tables of numerical values for the integrals of Airy functions that
appear in this study.Comment: 62 pages, 12 figures, 6 tables. Supplementary computer algebra
computations available at https://gitlab.com/vit.north/strong-components-au
Controllability in complex brain networks
Complex functional brain networks are large networks of brain regions and functional brain connections. Statistical characterizations of these networks aim to quantify global and local properties of brain activity with a small number of network measures. Recently it has been proposed to characterize brain networks in terms of their "controllability", drawing on concepts and methods of control theory. The thesis will review the control theory for networks and its application in neuroscience. In particular, the study will highlight important limitations and some warning and caveats in the brain controllability framework.ope
On the approximability of the maximum induced matching problem
In this paper we consider the approximability of the maximum induced matching problem (MIM). We give an approximation algorithm with asymptotic performance ratio <i>d</i>-1 for MIM in <i>d</i>-regular graphs, for each <i>d</i>≥3. We also prove that MIM is APX-complete in <i>d</i>-regular graphs, for each <i>d</i>≥3
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