976 research outputs found
The number of Hamiltonian decompositions of regular graphs
A Hamilton cycle in a graph is a cycle passing through every vertex
of . A Hamiltonian decomposition of is a partition of its edge
set into disjoint Hamilton cycles. One of the oldest results in graph theory is
Walecki's theorem from the 19th century, showing that a complete graph on
an odd number of vertices has a Hamiltonian decomposition. This result was
recently greatly extended by K\"{u}hn and Osthus. They proved that every
-regular -vertex graph with even degree for some fixed
has a Hamiltonian decomposition, provided is sufficiently
large. In this paper we address the natural question of estimating ,
the number of such decompositions of . Our main result is that
. In particular, the number of Hamiltonian
decompositions of is
Community detection for correlation matrices
A challenging problem in the study of complex systems is that of resolving,
without prior information, the emergent, mesoscopic organization determined by
groups of units whose dynamical activity is more strongly correlated internally
than with the rest of the system. The existing techniques to filter
correlations are not explicitly oriented towards identifying such modules and
can suffer from an unavoidable information loss. A promising alternative is
that of employing community detection techniques developed in network theory.
Unfortunately, this approach has focused predominantly on replacing network
data with correlation matrices, a procedure that tends to be intrinsically
biased due to its inconsistency with the null hypotheses underlying the
existing algorithms. Here we introduce, via a consistent redefinition of null
models based on random matrix theory, the appropriate correlation-based
counterparts of the most popular community detection techniques. Our methods
can filter out both unit-specific noise and system-wide dependencies, and the
resulting communities are internally correlated and mutually anti-correlated.
We also implement multiresolution and multifrequency approaches revealing
hierarchically nested sub-communities with `hard' cores and `soft' peripheries.
We apply our techniques to several financial time series and identify
mesoscopic groups of stocks which are irreducible to a standard, sectorial
taxonomy, detect `soft stocks' that alternate between communities, and discuss
implications for portfolio optimization and risk management.Comment: Final version, accepted for publication on PR
Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators
The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing
approximation algorithms for local Hamiltonian problems. In this paper we
attack this problem using the algebraic structure of QMC, in particular the
relationship between the quantum max cut Hamiltonian and the representation
theory of the symmetric group.
The first major contribution of this paper is an extension of non-commutative
Sum of Squares (ncSoS) optimization techniques to give a new hierarchy of
relaxations to Quantum Max Cut. The hierarchy we present is based on
optimizations over polynomials in the qubit swap operators. This is contrast to
the ``standard'' quantum Lasserre Hierarchy, which is based on polynomials
expressed in terms of the Pauli matrices. To prove correctness of this
hierarchy, we give a finite presentation of the algebra generated by the qubit
swap operators. This presentation allows for the use of computer algebraic
techniques to manipulate simplify polynomials written in terms of the swap
operators, and may be of independent interest. Surprisingly, we find that
level-2 of this new hierarchy is exact (up to tolerance ) on all QMC
instances with uniform edge weights on graphs with at most 8 vertices.
The second major contribution of this paper is a polynomial-time algorithm
that exactly computes the maximum eigenvalue of the QMC Hamiltonian for certain
graphs, including graphs that can be ``decomposed'' as a signed combination of
cliques. A special case of the latter are complete bipartite graphs with
uniform edge-weights, for which exact solutions are known from the work of Lieb
and Mattis. Our methods, which use representation theory of the symmetric
group, can be seen as a generalization of the Lieb-Mattis result.Comment: 75 pages, 6 figure
Information Theory in Molecular Evolution: From Models to Structures and Dynamics
This Special Issue collects novel contributions from scientists in the interdisciplinary field of biomolecular evolution. Works listed here use information theoretical concepts as a core but are tightly integrated with the study of molecular processes. Applications include the analysis of phylogenetic signals to elucidate biomolecular structure and function, the study and quantification of structural dynamics and allostery, as well as models of molecular interaction specificity inspired by evolutionary cues
Random multi-index matching problems
The multi-index matching problem (MIMP) generalizes the well known matching
problem by going from pairs to d-uplets. We use the cavity method from
statistical physics to analyze its properties when the costs of the d-uplets
are random. At low temperatures we find for d>2 a frozen glassy phase with
vanishing entropy. We also investigate some properties of small samples by
enumerating the lowest cost matchings to compare with our theoretical
predictions.Comment: 22 pages, 16 figure
Reconstruction on trees and spin glass transition
Consider an information source generating a symbol at the root of a tree
network whose links correspond to noisy communication channels, and
broadcasting it through the network. We study the problem of reconstructing the
transmitted symbol from the information received at the leaves. In the large
system limit, reconstruction is possible when the channel noise is smaller than
a threshold.
We show that this threshold coincides with the dynamical (replica symmetry
breaking) glass transition for an associated statistical physics problem.
Motivated by this correspondence, we derive a variational principle which
implies new rigorous bounds on the reconstruction threshold. Finally, we apply
a standard numerical procedure used in statistical physics, to predict the
reconstruction thresholds in various channels. In particular, we prove a bound
on the reconstruction problem for the antiferromagnetic ``Potts'' channels,
which implies, in the noiseless limit, new results on random proper colorings
of infinite regular trees.
This relation to the reconstruction problem also offers interesting
perspective for putting on a clean mathematical basis the theory of glasses on
random graphs.Comment: 34 pages, 16 eps figure
Statistical physics methods in computational biology
The interest of statistical physics for combinatorial optimization is not new, it suffices to think of a famous tool as
simulated annealing. Recently, it has also resorted to statistical inference to address some "hard" optimization problems, developing a new class of message passing algorithms. Three applications to computational biology are presented in this thesis, namely:
1) Boolean networks, a model for gene regulatory networks;
2) haplotype inference, to study the genetic information present in a population;
3) clustering, a general machine learning tool
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