85 research outputs found
Transition to Reconstructibility in Weakly Coupled Networks
Across scientific disciplines, thresholded pairwise measures of statistical
dependence between time series are taken as proxies for the interactions
between the dynamical units of a network. Yet such correlation measures often
fail to reflect the underlying physical interactions accurately. Here we
systematically study the problem of reconstructing direct physical interaction
networks from thresholding correlations. We explicate how local common cause
and relay structures, heterogeneous in-degrees and non-local structural
properties of the network generally hinder reconstructibility. However, in the
limit of weak coupling strengths we prove that stationary systems with dynamics
close to a given operating point transition to universal reconstructiblity
across all network topologies.Comment: 15 pages, 4 figures, supplementary material include
Computational complexity of reconstruction and isomorphism testing for designs and line graphs
Graphs with high symmetry or regularity are the main source for
experimentally hard instances of the notoriously difficult graph isomorphism
problem. In this paper, we study the computational complexity of isomorphism
testing for line graphs of - designs. For this class of
highly regular graphs, we obtain a worst-case running time of for bounded parameters . In a first step, our approach
makes use of the Babai--Luks algorithm to compute canonical forms of
-designs. In a second step, we show that -designs can be reconstructed
from their line graphs in polynomial-time. The first is algebraic in nature,
the second purely combinatorial. For both, profound structural knowledge in
design theory is required. Our results extend earlier complexity results about
isomorphism testing of graphs generated from Steiner triple systems and block
designs.Comment: 12 pages; to appear in: "Journal of Combinatorial Theory, Series A
Reconstructing Extended Perfect Binary One-Error-Correcting Codes from Their Minimum Distance Graphs
The minimum distance graph of a code has the codewords as vertices and edges
exactly when the Hamming distance between two codewords equals the minimum
distance of the code. A constructive proof for reconstructibility of an
extended perfect binary one-error-correcting code from its minimum distance
graph is presented. Consequently, inequivalent such codes have nonisomorphic
minimum distance graphs. Moreover, it is shown that the automorphism group of a
minimum distance graph is isomorphic to that of the corresponding code.Comment: 4 pages. Accepted for publication in IEEE Transactions on Information
Theor
Assembly complexity of prokaryotic genomes using short reads
<p>Abstract</p> <p>Background</p> <p>De Bruijn graphs are a theoretical framework underlying several modern genome assembly programs, especially those that deal with very short reads. We describe an application of de Bruijn graphs to analyze the global repeat structure of prokaryotic genomes.</p> <p>Results</p> <p>We provide the first survey of the repeat structure of a large number of genomes. The analysis gives an upper-bound on the performance of genome assemblers for <it>de novo </it>reconstruction of genomes across a wide range of read lengths. Further, we demonstrate that the majority of genes in prokaryotic genomes can be reconstructed uniquely using very short reads even if the genomes themselves cannot. The non-reconstructible genes are overwhelmingly related to mobile elements (transposons, IS elements, and prophages).</p> <p>Conclusions</p> <p>Our results improve upon previous studies on the feasibility of assembly with short reads and provide a comprehensive benchmark against which to compare the performance of the short-read assemblers currently being developed.</p
Reconstruction of functions from minors
The central notion of this thesis is the minor relation on functions of several arguments. A function f: A^n→B is called a minor of another function g: A^m→B if f can be obtained from g by permutation of arguments, identification of arguments, and introduction of inessential arguments. We first provide some general background and context to this work by presenting a brief survey of basic facts and results concerning different aspects of the minor relation, placing some emphasis on the author’s contributions to the field.
The notions of functions of several arguments and minors give immediately rise to the following reconstruction problem: Is a function f: A^n→B uniquely determined, up to permutation of arguments, by its identification minors, i.e., the minors obtained by identifying a pair of arguments? We review known results – both positive and negative – about the reconstructibility of functions from identification minors, and we outline the main ideas of the proofs, which often amount to formulating and solving reconstruction problems for other kinds of mathematical objects.
We then turn our attention to functions determined by the order of first occurrence, and we are interested in the reconstructibility of such functions. One of the main results of this thesis states that the class of functions determined by the order of first occurrence is weakly reconstructible. Some reconstructible subclasses are identified; in particular, pseudo-Boolean functions determined by the order of first occurrence are reconstructible.
As our main tool, we introduce the notion of minor of permutation. This is a quotient-like construction for permutations that parallels minors of functions and has some similarities to permutation patterns. We develop the theory of minors of permutations, focusing on Galois connections induced by the minor relation and on the interplay between permutation groups and minors of permutations. Our results will then find applications in the analysis of the reconstruction problem of functions determined by the order of first occurrence
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