1,405 research outputs found
A notion of graph likelihood and an infinite monkey theorem
We play with a graph-theoretic analogue of the folklore infinite monkey
theorem. We define a notion of graph likelihood as the probability that a given
graph is constructed by a monkey in a number of time steps equal to the number
of vertices. We present an algorithm to compute this graph invariant and closed
formulas for some infinite classes. We have to leave the computational
complexity of the likelihood as an open problem.Comment: 6 pages, 1 EPS figur
Increased signaling entropy in cancer requires the scale-free property of protein interaction networks
One of the key characteristics of cancer cells is an increased phenotypic
plasticity, driven by underlying genetic and epigenetic perturbations. However,
at a systems-level it is unclear how these perturbations give rise to the
observed increased plasticity. Elucidating such systems-level principles is key
for an improved understanding of cancer. Recently, it has been shown that
signaling entropy, an overall measure of signaling pathway promiscuity, and
computable from integrating a sample's gene expression profile with a protein
interaction network, correlates with phenotypic plasticity and is increased in
cancer compared to normal tissue. Here we develop a computational framework for
studying the effects of network perturbations on signaling entropy. We
demonstrate that the increased signaling entropy of cancer is driven by two
factors: (i) the scale-free (or near scale-free) topology of the interaction
network, and (ii) a subtle positive correlation between differential gene
expression and node connectivity. Indeed, we show that if protein interaction
networks were random graphs, described by Poisson degree distributions, that
cancer would generally not exhibit an increased signaling entropy. In summary,
this work exposes a deep connection between cancer, signaling entropy and
interaction network topology.Comment: 20 pages, 5 figures. In Press in Sci Rep 201
Regular quantum graphs
We introduce the concept of regular quantum graphs and construct connected
quantum graphs with discrete symmetries. The method is based on a decomposition
of the quantum propagator in terms of permutation matrices which control the
way incoming and outgoing channels at vertex scattering processes are
connected. Symmetry properties of the quantum graph as well as its spectral
statistics depend on the particular choice of permutation matrices, also called
connectivity matrices, and can now be easily controlled. The method may find
applications in the study of quantum random walks networks and may also prove
to be useful in analysing universality in spectral statistics.Comment: 12 pages, 3 figure
Wick's theorem for q-deformed boson operators
In this paper combinatorial aspects of normal ordering arbitrary words in the
creation and annihilation operators of the q-deformed boson are discussed. In
particular, it is shown how by introducing appropriate q-weights for the
associated ``Feynman diagrams'' the normally ordered form of a general
expression in the creation and annihilation operators can be written as a sum
over all q-weighted Feynman diagrams, representing Wick's theorem in the
present context.Comment: 9 page
Estimation of pure qubits on circles
Gisin and Popescu [PRL, 83, 432 (1999)] have shown that more information
about their direction can be obtained from a pair of anti-parallel spins
compared to a pair of parallel spins, where the first member of the pair (which
we call the pointer member) can point equally along any direction in the Bloch
sphere. They argued that this was due to the difference in dimensionality
spanned by these two alphabets of states. Here we consider similar alphabets,
but with the first spin restricted to a fixed small circle of the Bloch sphere.
In this case, the dimensionality spanned by the anti-parallel versus parallel
alphabet is now equal. However, the anti-parallel alphabet is found to still
contain more information in general. We generalize this to having N parallel
spins and M anti-parallel spins. When the pointer member is restricted to a
small circle these alphabets again span spaces of equal dimension, yet in
general, more directional information can be found for sets with smaller |N-M|
for any fixed total number of spins. We find that the optimal POVMs for
extracting directional information in these cases can always be expressed in
terms of the Fourier basis. Our results show that dimensionality alone cannot
explain the greater information content in anti-parallel combinations of spins
compared to parallel combinations. In addition, we describe an LOCC protocol
which extract optimal directional information when the pointer member is
restricted to a small circle and a pair of parallel spins are supplied.Comment: 23 pages, 8 figure
A quantum Bose-Hubbard model with evolving graph as toy model for emergent spacetime
We present a toy model for interacting matter and geometry that explores
quantum dynamics in a spin system as a precursor to a quantum theory of
gravity. The model has no a priori geometric properties, instead, locality is
inferred from the more fundamental notion of interaction between the matter
degrees of freedom. The interaction terms are themselves quantum degrees of
freedom so that the structure of interactions and hence the resulting local and
causal structures are dynamical. The system is a Hubbard model where the graph
of the interactions is a set of quantum evolving variables. We show
entanglement between spatial and matter degrees of freedom. We study
numerically the quantum system and analyze its entanglement dynamics. We
analyze the asymptotic behavior of the classical model. Finally, we discuss
analogues of trapped surfaces and gravitational attraction in this simple
model.Comment: 23 pages, 6 figures; updated to published versio
A mathematical model of kinetoplastid mitochondrial gene scrambling advantage
We model and discuss advantages of pan-editing, the complex way of expressing
mitochondrial genes in kinetoplastids. The rapid spread and preservation of pan-editing
seems to be due to its concomitant fragment dispersal. Such dispersal prevents losing
temporarily non expressed mitochondrial genes upon intense intraspecific competition,
by linking non expressed fragments to parts which are still needed. We mathematically
modelled protection against gene loss, due to the absence of selection, by this kind of
fragment association. This gives ranges of values for parameters like scrambling extent,
population size, and number of generations still retaining full genomes despite limited
selection. Values obtained seem consistent with those observed. We find a quasi-linear
correlation between dispersal and number of generations after which populations lose
genes, showing that pan-editing can be selected to effectively limit gene loss under
relaxed selective pressure
Locality for quantum systems on graphs depends on the number field
Adapting a definition of Aaronson and Ambainis [Theory Comput. 1 (2005),
47--79], we call a quantum dynamics on a digraph "saturated Z-local" if the
nonzero transition amplitudes specifying the unitary evolution are in exact
correspondence with the directed edges (including loops) of the digraph. This
idea appears recurrently in a variety of contexts including angular momentum,
quantum chaos, and combinatorial matrix theory. Complete characterization of
the digraph properties that allow such a process to exist is a long-standing
open question that can also be formulated in terms of minimum rank problems. We
prove that saturated Z-local dynamics involving complex amplitudes occur on a
proper superset of the digraphs that allow restriction to the real numbers or,
even further, the rationals. Consequently, among these fields, complex numbers
guarantee the largest possible choice of topologies supporting a discrete
quantum evolution. A similar construction separates complex numbers from the
skew field of quaternions. The result proposes a concrete ground for
distinguishing between complex and quaternionic quantum mechanics.Comment: 9 page
Increased entropy of signal transduction in the cancer metastasis phenotype
Studies into the statistical properties of biological networks have led to
important biological insights, such as the presence of hubs and hierarchical
modularity. There is also a growing interest in studying the statistical
properties of networks in the context of cancer genomics. However, relatively
little is known as to what network features differ between the cancer and
normal cell physiologies, or between different cancer cell phenotypes. Based on
the observation that frequent genomic alterations underlie a more aggressive
cancer phenotype, we asked if such an effect could be detectable as an increase
in the randomness of local gene expression patterns. Using a breast cancer gene
expression data set and a model network of protein interactions we derive
constrained weighted networks defined by a stochastic information flux matrix
reflecting expression correlations between interacting proteins. Based on this
stochastic matrix we propose and compute an entropy measure that quantifies the
degree of randomness in the local pattern of information flux around single
genes. By comparing the local entropies in the non-metastatic versus metastatic
breast cancer networks, we here show that breast cancers that metastasize are
characterised by a small yet significant increase in the degree of randomness
of local expression patterns. We validate this result in three additional
breast cancer expression data sets and demonstrate that local entropy better
characterises the metastatic phenotype than other non-entropy based measures.
We show that increases in entropy can be used to identify genes and signalling
pathways implicated in breast cancer metastasis. Further exploration of such
integrated cancer expression and protein interaction networks will therefore be
a fruitful endeavour.Comment: 5 figures, 2 Supplementary Figures and Table
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